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CONFORMALLY INVARIANT SYSTEMS OF DIFFERENTIAL OPERATORS ASSOCIATED TO TWOSTEP NILPOTENT MAXIMAL PARABOLICS OF NONHEISENBERG TYPE By TOSHIHISA KUBO Bachelor of Science in Mathematics University of Central Oklahoma Edmond, Oklahoma, USA 2003 Master of Science in Mathematics Oklahoma State University Stillwater, Oklahoma, USA 2005 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 2012 COPYRIGHT c⃝ By TOSHIHISA KUBO May, 2012 CONFORMALLY INVARIANT SYSTEMS OF DIFFERENTIAL OPERATORS ASSOCIATED TO TWOSTEP NILPOTENT MAXIMAL PARABOLICS OF NONHEISENBERG TYPE Dissertation Approved: Dr. Leticia Barchini Dissertation Advisor Dr. Anthony C. Kable Dr. Roger Zierau Dr. K.S. Babu Dr. Sheryl A. Tucker Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Conformally Invariant Systems and the Ωk Systems 8 2.1 Conformally Invariant Systems . . . . . . . . . . . . . . . . . . . . . . 8 2.2 A Specialization on a gmanifold and gbundle . . . . . . . . . . . . . 10 2.3 A gmanifold N 0 and gbundle Ls . . . . . . . . . . . . . . . . . . . 13 2.4 Useful Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 The Ωk Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 The Ωk Systems and Generalized Verma Modules . . . . . . . . . . . 28 3 Parabolic Subalgebras and Zgradings 32 3.1 kstep Nilpotent Parabolic Subalgebras . . . . . . . . . . . . . . . . . 32 3.2 Maximal TwoStep Nilpotent Parabolic q of NonHeisenberg type . . 36 3.3 The Simple Subalgebras l and ln . . . . . . . . . . . . . . . . . . . . 41 3.4 Technical Facts on the Highest Weights for l , ln , g(1), and z(n) . . . 43 4 The Ω1 System 48 4.1 Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 The Ω1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 Irreducible Decomposition of l z(n) 54 5.1 Irreducible Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 54 iv 5.2 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Special Constituents of l z(n) 68 6.1 Special Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.2 Types of Special Constituents . . . . . . . . . . . . . . . . . . . . . . 70 6.3 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 The Ω2 Systems 89 7.1 Covariant Map 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2 The Ω2jV ( +ϵ) Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3 Special Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8 The Homomorphisms between Generalized Verma Modules induced by the Ω1 System and Ω2 Systems 106 8.1 The Standard Map between Generalized Verma Modules . . . . . . . 107 8.2 The Homomorphism φΩ1 induced by the Ω1 System . . . . . . . . . . 111 8.3 The Homomorphisms φΩ2 induced by the Ω2 Systems . . . . . . . . . 112 8.3.1 The Type 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.3.2 The Positive Integer Special Value Case . . . . . . . . . . . . 115 8.3.3 The V ( + ϵ ) Case for Bn(i) for 3 i n 1 . . . . . . . . 122 BIBLIOGRAPHY 138 A Reducibility Points 141 A.1 Verma modules and Generalized Verma Modules . . . . . . . . . . . . 141 A.2 Jantzen's Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.3 Necessary Conditions of the Reducibility of Mq( t) . . . . . . . . . . 149 A.4 Reducibility Criteria for SimplyLaced Case . . . . . . . . . . . . . . 151 A.5 Reducibility Points of Mq( t) for Exceptional Algebras . . . . . . . . 156 v A.5.1 E6(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.5.2 E7(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 A.5.3 E7(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 A.5.4 E8(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.5.5 F4(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.6 The Special Values and The Reducibility Points . . . . . . . . . . . . 173 B Dynkin Diagrams and Extended Dynkin Diagrams 176 C Basic Data 183 D Lists of Positive Roots for Exceptional Algebras 198 vi LIST OF TABLES Table Page 6.1 Highest Weights for Special Constituents (Classical Cases) . . . . . . 74 6.2 Highest Weights for Special Constituents (Exceptional Cases) . . . . 75 6.3 The Roots , ϵ , and ϵn (Classical Cases) . . . . . . . . . . . . . . . 75 6.4 The Roots , ϵ , and ϵn (Exceptional Cases) . . . . . . . . . . . . . 76 6.5 Types of Special Constituents . . . . . . . . . . . . . . . . . . . . . . 78 7.1 Line Bundles with Special Values . . . . . . . . . . . . . . . . . . . . 104 7.2 The Generalized Verma Modules corresponding to L(s0 q) in Table 7.1 105 8.1 The Homomorphism φΩ2 for the NonHeisenberg Case . . . . . . . . . 137 A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 vii LIST OF FIGURES Figure Page B.1 The Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.2 The Dynkin diagrams with the multiplicities of the simple roots in the highest root of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.3 The extended Dynkin diagrams with the highest root of g . . . . . 181 viii CHAPTER 1 Introduction The main work of this thesis concerns systems of differential operators that are equiv ariant under an action of a Lie algebra. We call such systems conformally invariant. To explain the meaning of the equivariance condition, suppose that V ! M is a vec tor bundle over a smooth manifold M and g is a Lie algebra of rstorder differential operators that act on sections of V. A linearly independent list D1; : : : ;Dn of linear differential operators on sections of V is called a conformally invariant system if, for each X 2 g, there are smooth functions CX ij (m) on M so that, for all 1 i n, and sections f of V, we have ( [X;Di] f ) (m) = Σn j=1 CX ji (m)(Dj f)(m); (1.0.1) where [X;Dj ] = XDjDjX, and the dot denotes the action of differential operators on smooth functions. (See De nition 2.1.4 for the precise de nition.) A typical example for a conformally invariant system of one differential operator is the wave operator □ = @2 @x21 + @2 @x22 + @2 @x23 @2 @x24 on the Minkowski space R3;1. If X is an element of g = so(4; 2) acting as a rstorder differential operators on sections of an appropriate line bundle over R3;1 then there is a smooth function CX on R3;1 so that [X;□] = CX□: An important consequence of the de nition (1.0.1) is that the common kernel of the operators in the conformally invariant system D1; : : : ;Dn is invariant under a Lie algebra action. The representation theoretic question of understanding the common 1 kernel as a gmodule is an open question (except for a small number of very special examples). The notion of conformally invariant systems generalizes that of quasiinvariant differential operators introduced by Kostant in [19] and is related to a work of Huang ([8]). It is also compatible with the de nition given by Ehrenpreis in [6]. Confor mally invariant systems are explicitly or implicitly presented in the work of Davidson EnrightStanke ([5]), Kable ([12], [13]), Kobayashi rsted ([16], [17], [18]), Wallach ([25]), among others. Much of the published work is for the case that M = G=Q with Q = LN, N abelian. The systematic study of conformally invariant systems started with the work of BarchiniKableZierau in [1] and [2] Although the theory of conformally invariant systems can be viewed as a geometric analytic theory, it is closely related to algebraic objects such as generalized Verma modules. It has been shown in [2] that a conformally invariant system yields a ho momorphism between certain generalized Verma modules. The classi cation of non standard homomorphisms between generalized Verma modules is an open problem. The main goal of this thesis is to build systems of differential operators that satisfy the condition (1.0.1), when M is a homogeneous manifold G=Q with Q a maximal twostep nilpotent parabolic subgroup. This is to construct systems D1; : : : ;Dn acting on sections of bundles Vs ! G=Q over G=Q in a systematic manner and to determine the bundles Vs on which the systems are conformally invariant. The method that we use is different from one used by BarchiniKableZierau in [1]. The systems that we build yield explicit homomorphisms between appropriate generalized Verma modules. We show that the most of those homomorphisms are nonstandard. To describe our work more precisely, let G be a complex, simple, connected, simplyconnected Lie group with Lie algebra g. It is known that g has a Zgrading g = ⊕r j=r g(j) so that q = g(0) ⊕ j>0 g(j) = l n is a parabolic subalgebra of g. Let Q = NG(q) = LN. For a real form g0 of g, de ne G0 to be an analytic subgroup 2 of G with Lie algebra g. Set Q0 = NG0(q). Our manifold is M = G0=Q0 and we consider a line bundle Ls ! G0=Q0 for each s 2 C. It is known, by the Bruhat theory, that G0=Q0 admits an open dense submanifold N 0Q0=Q0. We restrict our bundle to this submanifold. The systems that we study act on sections of the restricted bundle. To build systems of differential operators we observe that L acts by the adjoint representation on g(1) with a unique open orbit. This makes g(1) a prehomogeneous vector space. Our construction is based on the invariant theory of a prehomogeneous vector space. It is natural to associate Lequivariant polynomial maps called covariant maps to the prehomogeneous vector space (L; Ad; g(1)). To de ne our systems of differential operators, we use covariant maps that are associated to g(1). We denote the covariant maps by k. Each k can be thought of as giving the symbols of the differential operators that we study. For 0 k 2r, the maps k are de ned by k : g(1) ! g(r + k) g(r) (1.0.2) X 7! 1 k! ad(X)k!0; where !0 is a certain element in g(r + k) g(r). (See De nition 2.5.1.) Let g(r + k) g(r) = V1 Vm (1.0.3) be the irreducible decomposition of g(r + k) g(r) as an Lmodule. Covariant map k induces an Lequivariant linear map ~ kjV j : V j ! Pk(g(1))) with V j the dual of an irreducible constituent Vj of g(r + k) g(r) and Pk(g(1)) the space of polynomials on g(1) of degree k. We de ne differential operators from ~ kjV j (Y ). For Y 2 V j , let Ωk(Y ) denote the kth order differential operators that are constructed from ~ kjV j (Y ). We say that a list of differential operators D1; : : : ;Dn is the ΩkjV j system if it is 3 equivalent (in the sense of De nition 2.1.5) to a list of differential operators Ωk(Y 1 ); : : : ;Ωk(Y n ); (1.0.4) where fY 1 ; : : : ; Y n g is a basis for V j over C. By construction the ΩkjV j system consists of dimC(Vj) operators. It is not necessary for the ΩkjV j system to be conformally invariant; the conformal invariance of the operators (1.0.4) strongly depends on the complex parameter s for the line bundle Ls. Then we say that the ΩkjV j system has special value s0 if the system is conformally invariant on the line bundle Ls0 . The special values for the case that dim([n; n]) = 1 for q = l n are studied by BarchiniKableZierau in [1] and [2], and myself in [20]. In this thesis we consider a more general case; namely, q = l n is a maximal parabolic subalgebra and n satis es the condition that [n; [n; n]] = 0 and dimC([n; n]) > 1. We call such parabolic subalgebras maximal twostep nilpotent parabolic subal gebras of nonHeisenberg type. In this case we have r = 2 in (2.5.6). Therefore the Ωk systems for k 5 are zero. The main results of this thesis are Theorem 4.2.5 and Theorem 7.3.6, where the special values of the Ω1 system and Ω2 systems for the parabolic subalgebras are determined. We also classify the nonstandard homo morphisms between the generalized Verma modules that arise from our systems of differential operators. We may want to remark that, although the special value of s for the Ω1 system is easily found by computing the bracket [X;Ω1(Y i )], it is in general not easy to nd the special values for the Ω2 systems by a direct computation. (See Section 5 of [1].) In this thesis, to nd the special value for the Ω2jV j system, we use two reduction techniques to compute the special values. First, in order to show the equivariance condition (1.0.1) for Di = Ω2(Y i ) with Y i 2 V j , it is enough to compute [X;Ω2(Y i )] at the identity e. Furthermore, we show that it is even sufficient to compute only [Xh;Ω2(Y l )] at e, where Xh and Y l are a highest weight vector of g(1) g and a 4 lowest weight vector of V j , respectively. These two techniques signi cantly reduce the amount of computations. We now outline the contents of this thesis. In Chapter 2 we study conformally invariant systems of differential operators. We recapitulate Section 2 of [2] in Section 2.1. In Sections 2.2 and 2.3 we specialize the theory of conformally invariant systems to the situation that we are interested in. Two useful formulas on differential operators will be shown in Section 2.4. In Section 2.5, the general construction of the Ωk systems is given. Section 2.6 discusses two technical lemmas on the Ωk systems, and in Section 2.7, we describe a relationship between the Ωk systems and generalized Verma modules. The aim of Chapter 3 is to study the Zgrading g = ⊕r j=r g(j) on g and a maximal twostep nilpotent parabolic subalgebra q of nonHeisenberg type. We begin this chapter by classifying the kstep nilpotent parabolic subalgebras in Section 3.1. In Section 3.2 and Section 3.3, we study a maximal twostep nilpotent parabolic subalgebra q of nonHeisenberg type and the associated 2grading g = ⊕2 j=2 g(j) = z( n) g(1) l g(1) z(n) of g. In Chapter 4, we construct the Ω1 system and nd the special value of the system. In Section 4.1, we x normalizations for root vectors. The normalizations play an important role to construct the system. In Section 4.2 we show that the special value s1 of s for the Ω1 system is s1 = 0. This is done in Theorem 4.2.5. To build the Ω2 systems, we need to nd the irreducible constituents V of l z(n) so that ~ 2jV ̸= 0. In Chapters 5 and 6, we show preliminary results to nd such irreducible constituents. In Chapter 5 we decompose l z(n) into the direct sum of the irreducible constituents. We rst summarize our main decomposition results, Theorem 5.1.3, in Section 5.1. Section 5.2 contains preliminary results and technical lemmas that are used to prove the theorem. The proof for Theorem 5.1.3 is given in Section 5.3. In Chapter 6, by using the decomposition results, we determine 5 the candidates of the irreducible constituents V so that ~ 2jV ̸= 0. We call such constituents special. In Section 6.1 we de ne the special constituents. We then classify such constituents in Section 6.2. In Section 6.3 we collect the technical results on the special constituents, which are used to nd the special values for the Ω2 systems. In Chapter 7, we build the Ω2 systems and nd their special values. First, it is shown in Section 7.1 that the covariant maps 2 and the induced linear maps ~ 2jV for certain special constituents V are nonzero. We then construct the Ω2 systems in Section 7.2, and in Section 7.3, we nd their special values. This is done in Theorem 7.3.6. In Chapter 8, we determine whether or not the homomorphisms φΩk that are induced by the Ωk systems between appropriate generalized Verma modules are stan dard for k = 1; 2. In Section 8.1 we review the wellknown results on the standard map between generalized Verma modules. Technical results to determine the stan dardness of the maps φΩk are also shown in this section. We then determine the standardness of φΩ1 and φΩ2 in Section 8.2 and Section 8.3, respectively. In this thesis we also have the appendices. In Appendix A, as an Ωk system that is conformally invariant on the line bundle Ls0 induces the reducibility of a scalar generalized Verma module U(g) U(q) Cs0 , to support the results for the special values for the Ω2 systems, we show the reducibility points for the scalar generalized Verma modules for g exceptional algebras. To determine the reducibility we use a criterion due to Jantzen. (See Section A.2.) In Appendices B, C, and D, we collect miscellaneous useful data. Namely, Ap pendix B contains the Dynkin diagrams with the multiplicities of the simple roots in the highest root of g and extended Dynkin diagrams. Appendix C summarizes the useful data for the parabolic subalgebras under consideration such as the roots for l, g(1), and z(n). In Appendix D we include the lists of the positive roots for the exceptional algebras. 6 Finally, I would like to thank my advisor, Dr. Leticia Barchini, for introducing this topic for me and for her generous help. I would also like to thank Dr. Anthony Kable and Dr. Roger Zierau for their valuable comments on this work. 7 CHAPTER 2 Conformally Invariant Systems and the Ωk Systems The purpose of this chapter is to study conformally invariant systems of differential operators, that are the main objects of this thesis. In particular, we de ne systems of differential operators of order k, which we call the Ωk systems. 2.1 Conformally Invariant Systems The aim of this section is to introduce the de nition of conformally invariant systems. Suppose that V and W are nite dimensional complex vector spaces and C1(Rn; V ) is the space of smooth V valued functions on Rn. A linear map D : C1(Rn; V ) ! C1(Rn;W) is called a differential operator if it is of the form D h = Σ j j k T ( @ @x h ) (2.1.1) for some k 2 Z 0 and all h 2 C1(Rn; V ), where T are smooth functions from Rn to HomC(V;W), and multiindex notation is being used. Here, the dot denotes the action of differential operators on smooth functions. Now let M be a smooth manifold, and let prV : V ! M and prW : W ! M be smooth vector bundles over M of nite rank with prV and prW the bundle projections. For each p 2 M, there exists an open neighborhood U of p so that the local trivializations pr1 V (U) = U V and pr1 W (U) = U W hold. Then a linear map D from smooth sections of V to smooth sections of W is called a differential operator if in each local trivialization D is of the form of (2.1.1). The smallest integer k with j j k in (2.1.1), for which T ̸= 0, is called the order of D. We 8 denote by D(V) the space of differential operators on the smooth sections of V. Note that we regard smooth functions f on M as elements in D(V) by identifying them with the multiplication operator they induce. Let g0 be a real Lie algebra and X(M) be the space of smooth vector elds on M. De nition 2.1.2 [2, page 790] A smooth manifold M is called a g0manifold if there is an Rlinear map M : g0 ! C1(M) X(M) so that M([X; Y ]) = [ M(X); M(Y )] for all X; Y 2 g0. For each X 2 g0, we write M(X) = 0(X) + 1(X) with 0(X) 2 C1(M) and 1(X) 2 X(M). De nition 2.1.3 [2, page 791] Let M be a g0manifold. A vector bundle V ! M is called a g0bundle if there is an Rlinear map V : g0 ! D(V) that satis es the following properties: (B1) We have V([X; Y ]) = [ V(X); V(Y )] for all X; Y 2 g0. (B2) In D(V), [ V(X); f] = 1(X) f for all X 2 g0 and f 2 C1(M). Now we introduce the de nition of conformally invariant systems. De nition 2.1.4 [2, page 791] Let V ! M be a g0bundle. A conformally invari ant system on V with respect to V is a list of differential operators D1; : : : ;Dm 2 D(V) so that the following two conditions hold: (S1) At each point p 2 M, the list D1; : : : ;Dm is linearly independent over C. (S2) For each X 2 g0, there is a matrix C(X) in Mm m(C1(M)) so that [ V(X);Di] = Σm j=1 Cji(X)Dj in D(V). 9 The map C : g0 ! Mm m(C1(M)) is called the structure operator of the confor mally invariant system. If g is the complexi cation of g0 then gmanifolds and gbundles are de ned by extending the g0action Clinearly. De nition 2.1.5 [2, page 792] Two conformally invariant systems D1; : : : ;Dn and D′ 1; : : : ;D′ n are said to be equivalent if there is a matrix A 2 GL(n;C1(M)) so that D′ i = Σn j=1 AjiDj for 1 i n. De nition 2.1.6 [2, page 793] A conformally invariant system D1; : : : ;Dn is called reducible if there is an equivalent system D′ 1; : : : ;D′ n and an m < n such that the system D′ 1; : : : ;D′ m is conformally invariant. Otherwise we say that D1; : : : ;Dn is irreducible. The case that M is a homogeneous manifold is of our particular interest. In Section 2.2 and Section 2.3, we will specify the gmanifold and gbundle that we will work with. 2.2 A Specialization on a gmanifold and gbundle In this section we shall introduce the specializations on a smooth manifold M and a vector bundle V ! M, as in Section 5 of [2]. Let G be a complex, simple, connected, simplyconnected Lie group with Lie algebra g. Such G contains a maximal connected solvable subgroup B. Write b = h u for its Lie algebra with h the Cartan subalgebra and u the nilpotent subalgebra. Let q b be a parabolic subalgebra of g. We de ne Q = NG(q), a parabolic subgroup of G. It follows from Section 8.4 of [24] that Q is connected. Write Q = LN for the Levi decomposition of Q with L the Levi subgroup and N the nilpotent subgroup. 10 It is known, see Corollary 7.11 of [15], that the Levi subgroup L is the commuting product L = Z(L)◦Lss, where Z(L)◦ is the identity component of the center of L and Lss is the semisimple part of L. Let g0 be a real form of g and let G0 be the analytic subgroup of G with Lie algebra g0. De ne Q0 = NG0(q) Q, and write Q0 = L0N0. We will work on M = G0=Q0 for a class of maximal parabolic Q0 that will be speci ed in Chapter 3. Next, we need to specify a vector bundle V on M. To this end we recall the bijection between the standard parabolic subalgebras and the subsets of simple roots. Let Δ = Δ(g; h) be the set of roots of g with respect to h. We denote by Δ+ the positive system so that u = ⊕ 2Δ+ g with g the root spaces for . We write for the set of simple roots. Observe that the parabolic q contains the xed Borel subalgebra b. Therefore, it is of the form q = h ⊕ 2 g with Δ+ Δ. Each subset can be described in terms of a subset S of simple roots. Indeed, = Δ+ [ f 2 Δ j 2 span( nS)g; where nS is the complementary subset of S in . If ΔS = f 2 Δ j 2 span( nS)g then = ΔS [ (Δ+nΔS). Then q may be written as q = l n (2.2.1) with l = h ⊕ 2ΔS g and n = ⊕ 2Δ+nΔS g : (2.2.2) The subalgebras l and n are called the Levi factor and the nilpotent radical, respec tively. The Lie algebra l is reductive and n is a nilpotent ideal in q. Now we state the wellknown fact that there exists a onetoone correspondence between the standard parabolic subalgebras q and subsets of . 11 Theorem 2.2.3 There exists a onetoone correspondence between parabolic subal gebras q containing b and the subsets S of the set of simple roots . The parabolic subalgebra qS corresponding to the subset S is of the form (2.2.1) with (2.2.2). Since our parabolic Q0 will be maximal, by Theorem 2.2.3, there exists the cor responding simple root q 2 so that q = qf qg. Call q the fundamental weight of q. The weight q is orthogonal to any roots with g [l; l]. Hence it expo nentiates to a character q of L. As q takes real values on L0, for s 2 C, character s = j qjs is wellde ned on L0. Let C s be the onedimensional representation of L0 with character s. The representation s is extended to a representation of Q0 by making it trivial on N0. Then it deduces a line bundle Ls on M = G0=Q0 with ber C s . The group G0 acts on the space C1 (G0=Q0;C s) = fF 2 C1(G0;C s) j F(gq) = s(q1)F(g) for all q 2 Q0 and g 2 G0g by left translation. The action s of g0 on C1 (G0=Q0;C s) arising from this action is given by ( s(Y ) F)(g) = d dt F(exp(tY )g) t=0 for Y 2 g0. This action is extended Clinearly to g and then naturally to the universal enveloping algebra U(g). We use the same symbols for the extended actions. Let N 0 be the nilpotent subgroup opposite to N0. By the Bruhat theory, the subset N 0Q0 is open and dense in G0. Then the restriction map C1 (G0=Q0;C s) ! C1( N 0;C s) is an injection, where C1( N 0;C s) is the space of the smooth functions from N0 to C s . Then, for u 2 U(g) and F 2 C1 (G0=Q0;C s ), we let f = Fj N 0 and de ne the action of U(g) on the image of the restriction map by s(u) f = ( s(u) F ) j N 0 : (2.2.4) 12 The line bundle Ls ! G0=Q0 restricted to N 0 is the trivial bundle N 0 C s ! N 0. By slight abuse of notation, we refer to the trivial bundle over N 0 as Ls. Then in practice our manifold M will be M = N 0 and our vector bundle will be the trivial bundle. In the next section we shall show that N 0 and the trivial bundle Ls are a gmanifold and gbundle with the action s, respectively. 2.3 A gmanifold N 0 and gbundle Ls Here we prove that with the linear map s de ned in (2.2.4), (1) the manifold N 0 is a gmanifold, and (2) the trivial bundle Ls is a gbundle. Let n and q be the complexi cations of the Lie algebras of N 0 and Q0, respectively; we have the direct sum g = n q. For Y 2 g, write Y = Y n+Yq for the decomposition of Y in this direct sum. Similarly, write the Bruhat decomposition of g 2 N 0Q0 as g = n(g)q(g) with n(g) 2 N 0 and q(g) 2 Q0. For Y 2 g0, we have Y n = d dt n(exp(tY )) t=0; (2.3.1) and a similar equality holds for Yq. De ne a right action R of U( n) on C1( N 0;C s) by ( R(X) f ) ( n) = d dt f ( n exp(tX) ) t=0 (2.3.2) for X 2 n0 and f 2 C1( N 0;C s ). Observe that, by de nition, the differential d of is d = q. Proposition 2.3.3 We have ( s(Y ) f ) ( n) = s q ( (Ad( n1)Y )q ) f( n) ( R ( (Ad( n1)Y ) n ) f ) ( n) (2.3.4) for Y 2 g and f in the image of the restriction map C1 (G0=Q0;C s) ! C1( N 0;C s). 13 Proof. Suppose that f = Fj N 0 for some F 2 C1 (G0=Q0;C s ). If g1 n 2 N 0Q0 then we have (g f)( n) = F(g1 n) = s(q(g1 n)1)f( n(g1 n)): (2.3.5) Observe that if g is close enough to the identity then g1 n 2 N 0Q0 by the openness of N 0Q0. By replacing g by exp(tY ) in (2.3.5) with Y 2 g0 and differentiating at t = 0, we have ( s(Y ) f)( n) = d dt s( q(exp(tY ) n )1 )f( n(exp(tY ) n))jt=0 = d dt s( q(exp(tY ) n )1 )jt=0 f( n) + d dt f( n(exp(tY ) n))jt=0 = d dt s( q(exp(tAd( n1)Y ) )1 )jt=0 f( n) + d dt f( n n(exp(tAd( n1)Y )))jt=0 = s q ( (Ad( n1)Y )q ) f( n) ( R ( (Ad( n1)Y ) n ) f ) ( n): Note that the equality (2.3.1) is used from line three to line four. Now the proposed formula is obtained by extending the action Clinearly. Equation (2.3.4) implies that the representation s extends to a representation of U(g) on the whole space C1( N 0;C s ). Moreover, it also shows that for all Y 2 g, the linear map s(Y ) is in C1( N 0) X( N 0). Therefore, with this linear map s, N 0 is a gmanifold. Next, we show that the linear map s gives Ls the structure of a gbundle. As s is a representation of g, the condition (B1) of De nition 2.1.3 is trivial. Thus it suffices to show that the condition (B2) holds. Since Ls is the trivial bundle of N 0 with ber C s , the space of smooth sections of Ls is identi ed with C1( N 0;C s ). Proposition 2.3.6 In D(Ls) we have ( [ s(Y ); f] ) ( n) = ( R ( (Ad( n1)Y ) n ) f ) ( n) for Y 2 g and f 2 C1( N 0). In particular, the trivial bundle Ls with s is a gbundle. 14 Proof. Take h 2 C1( N 0;C s ). Since [ s(Y ); f] = s(Y )f f s(Y ) in D(Ls), the operator [ s(Y ); f] acts on h by ( [ s(Y ); f] h ) ( n) = ( s(Y ) (fh) ) ( n) f( n) ( s(Y ) h ) ( n): (2.3.7) It follows from Proposition 2.3.3 that the rst term evaluates to ( s(Y ) (fh) ) ( n) = s q ( (Ad( n1)Y )q ) f( n)h( n) ( R ( (Ad( n1)Y ) n ) (fh) ) ( n) (2.3.8) with ( R ( (Ad( n1)Y ) n ) (fh) ) ( n) = ( R ( (Ad( n1)Y ) n ) f ) ( n)h( n) + f( n) ( R ( (Ad( n1)Y ) n ) h ) ( n): Similarly, the second term evaluates to f( n) ( s(Y ) h ) ( n) = s q ( (Ad( n1)Y )q ) f( n)h( n) f( n) ( R ( (Ad( n1)Y ) n ) h ) ( n): (2.3.9) Now the proposed equality is obtained by substituting (2.3.8) and (2.3.9) into (2.3.7). In the next section we are going to construct systems of differential operators on Ls. The systems of operators will satisfy several properties of conformally invariant systems. To end this section we collect those properties here. De nition 2.3.10 [2, page 806] A conformally invariant system D1; : : : ;Dm on the line bundle Ls is called L0stable if there is a map c : L0 ! GL(n;C1( N 0)) such that l Di = Σm j=1 c(l)jiDj ; where the action l Di is given by (2.5.10). 15 It is known that there exists a semisimple element H0 2 l, so that ad(H0) has only integer eigenvalues on g with g(1) ̸= f0g, l = g(0), n = ⊕ j>0 g(j), and n = ⊕ j>0 g(j), where g(j) is the jeigenspace of ad(H0) (see for example [15, Section X.3]). De nition 2.3.11 [2, page 804] A conformally invariant system D1; : : : ;Dm is called homogeneous if C(H0) is a scalar matrix, where C is the structure operator of the conformally invariant system (see De nition 2.1.4). Proposition 2.3.12 [2, Proposition 17] Any irreducible conformally invariant sys tem is homogeneous. De ne D(Ls) n = fD 2 D(Ls) j [ s(X);D] = 0 for all X 2 ng: Observe that in the sense of [2, page 796], the gmanifold N 0 is straight with respect to the subalgebra n of g ([2, page 799]). Then we state the de nition of straight conformally invariant systems specialized to the present situation. For the general de nition see p.797 of [2]. De nition 2.3.13 We say that a conformally invariant system D1; : : : ;Dm is straight if Dj 2 D(Ls) n for j = 1; : : : ;m. In general, to show that a given list D1; : : : ;Dm of differential operators on N 0 is a conformally invariant system, we need check (S2) of De nition 2.1.4 at each point of N 0. Proposition 2.3.14 below shows that in the case D1; : : : ;Dm in D(Ls) n, it suffices to check the condition only at the identity e. Proposition 2.3.14 [2, Proposition 13] Let D1; : : : ;Dm be a list of operators in D(Ls) n. Suppose that the list is linearly independent at e and that there is a map b : g ! gl(m;C) such that ( [ s(Y );Di] f ) (e) = Σm j=1 b(Y )ji(Dj f)(e) 16 for all Y 2 g; f 2 C1( N 0;C s), and 1 i m. Then D1; : : : ;Dm is a conformally invariant system on Ls. The structure operator of the system is given by C(Y )( n) = b(Ad( n1)Y ) for all n 2 N 0 and Y 2 g. 2.4 Useful Formulas In this section we are going to show two formulas that will be helpful, when we study the conformal invariance of certain systems of differential operators on N 0 in Chapter 4 and Chapter 7. Proposition 2.4.1 For Y 2 g, X 2 n , and f 2 C1(N 0;C s), we have ( [ s(Y );R(X)] f ) ( n) = ( R([(Ad( n1)Y )q;X] n) f ) ( n) + s q ( [Ad( n1)Y;X]q ) f( n): Proof. Since [ s(Y );R(X)] = s(Y )R(X) R(X) s(Y ), it suffices to consider the contributions from each term. By Proposition 2.3.3, the contribution from s(Y )R(X) is ( ( s(Y )R(X)) f ) ( n) (2.4.2) = s q ( (Ad( n1)Y )q ) (R(X) f)( n) ( (R((Ad( n1)Y ) n)R(X)) f) ) ( n): To obtain the contribution from R(X) s(Y ), observe that ( R(X) s(Y ) f ) ( n) = d dt ( s(Y ) f ) ( n exp(tX))jt=0: By applying Proposition 2.3.3, differentiating with respect to t, and setting t = 0, the contribution from this term is ( R(X) s(Y ) f ) ( n) = s q ( [X; Ad( n1)Y ]q ) f( n) s q ( (Ad( n1)Y )q ) (R(X) f)( n) + ( R([X; Ad( n1)Y ] n) f ) ( n) ( (R(X)R((Ad( n1)Y ) n)) f ) ( n): (2.4.3) 17 Since R([X; (Ad( n1)Y ) n]) = R(X)R((Ad( n1)Y ) n) R((Ad( n1)Y ) n)R(X), it fol lows from (2.4.2) and (2.4.3) that ( [ s(Y );R(X)] f ) ( n) evaluates to ( [ s(Y );R(X)] f ) ( n) = (2.4.4) ( R([X; (Ad( n1)Y ) n]) f)( n) ( R([X; Ad( n1)Y ] n) f ) ( n) + s q([Ad( n1)Y;X]q ) f( n): As Ad( n1)Y = (Ad( n1)Y ) n + (Ad( n1)Y )q and X 2 n, we have [X; Ad( n1)Y ] n = [X; (Ad( n1)Y ) n] + [X; (Ad( n1)Y )q] n: Now the proposed formula follows from substituting this into the second term of the right hand side of (2.4.4). Proposition 2.4.5 For Y 2 g, X1;X2 2 n, and f 2 C1( N 0;C s), we have ( [ s(Y );R(X1)R(X2)] f ) ( n) = ( R([(Ad( n1)Y )q;X1] n)R(X2) f ) ( n) + ( R(X1)R([(Ad( n1)Y )q;X2] n) f ) ( n) + ( R([[Ad( n1)Y;X1]q;X2] n) f ) ( n) + s q([Ad( n1)Y;X1]q)(R(X2) f)( n) + s q([Ad( n1)Y;X2]q)(R(X1) f)( n) + s q([[Ad( n1)Y;X1];X2]q)f( n): Proof. Observe that [ s(Y );R(X1)R(X2)] is the sum of two terms [ s(Y );R(X1)R(X2)] = [ s(Y );R(X1)]R(X2) + R(X1)[ s(Y );R(X2)]: The contribution from the rst term is ( [ s(Y );R(X1)] (R(X2) f) ) ( n) = ( R([(Ad( n1)Y )q;X1] n) (R(X2) f) ) ( n) + s q ( [Ad( n1)Y;X1]q ) (R(X2) f)( n): (2.4.6) 18 The second term evaluates to ( R(X1)[ s(Y );R(X2)] f ) ( n) = d dt ( [ s(Y );R(X2)] f ) ( n exp(tX1))jt=0 = ( R([X1; Ad( n1)Y ]q;X2] n) f)( n) + ( R(X1)R([(Ad( n1)Y )q;X2] n) f)( n) s q ( [[X1; Ad( n1)Y ];X2]q ) f( n) + s q([Ad( n1)Y;X2]q ) (R(X1) f)( n): Now the proposed formula follows from adding this to (2.4.6). 2.5 The Ωk Systems The purpose of this section is to construct systems of differential operators in D(Ls) n in a systematic manner. We start with a Zgrading g = ⊕r j=r g(j) on g with g(1) ̸= 0. It is known that g(0) is reductive (see for instance [15, Corollary 10.17]). By construction, q = g(0) ⊕ j>0 g(j) is a parabolic subalgebra. Take L to be the analytic subgroup of G with Lie algebra g(0). Vinberg's Theorem ([15, Theorem 10.19]) shows that the adjoint action of L on g(1) has only nitely many orbits; in particular, L has an open orbit in g(1). Such a space is called prehomogeneous. In the theory of prehomogeneous vector spaces, it is natural to associate certain maps called covariant maps to a prehomogeneous vector space. To de ne our systems of differential operators, we use covariant maps that are associated to prehomogeneous vector space (L; Ad; g(1)). We denote the covariant maps by k and de ne them below. These maps can be thought to give symbols of a class of differential operators that we will study. We would like to acknowledge that the construction of k as in this thesis was suggested by Anthony Kable. De nition 2.5.1 Let g = ⊕r j=r g(j) be a graded complex simple Lie algebra with 19 g(1) ̸= 0. Then, for 0 k 2r, the map k on g(1) is de ned by k : g(1) ! g(r + k) g(r) X 7! 1 k! ad(X)k!0 with !0 = Σ j2Δ(g(r)) X j X j , where X j are root vectors for j and Δ(g(r)) is the set of roots so that g g(r). Here, we mean by ad(X)k!0 that X acts on the tensor product diagonally via the action ad( )k. Observe that since X 2 g(1) and [g(1); g(r)] = 0, we have ad(X)kX j = 0 for all j 2 Δ(g(r)). Therefore, ad(X)k!0 = Σ j ad(X)k(X j ) X j . When g(1) and g(r + k) g(r) are viewed as affine varieties, the maps k are indeed morphisms of varieties. We shall check in Lemma 2.5.4 that these maps are Lequivariant. This will show that k satisfy the de nition of covariant maps. To simplify a proof for Lemma 2.5.4, we rst show that !0 in De nition 2.5.1 is independent of a choice of a basis for g(r). Lemma 2.5.2 If Y1; : : : ; Ym is a basis for g(r) and Y 1 ; : : : ; Y m is the dual basis for g(r) with respect to the Killing form then !0 = Σm i=1(Yi Y i ). Proof. If Δ(g(r)) = f 1; : : : ; mg then each Yi may be expressed by Yi = Σm r=1 airX r for air 2 C. Let [air] be the change of basis matrix and set [bir] = [air]1. De ne Y i = Σm s=1 bsiX s for i = 1; : : : ;m. Since Σm s=1 aisbsj = ij and (X i ;X j ) = ij with ij the Kronecker delta, it follows that (Yi; Y j ) = ij . Thus fY 1 ; : : : ; Y m g is the dual basis of fY1; : : : ; Ymg. Hence, Σm i=1 (Y i Yi) = Σm r;s=1 (Σm i=1 bsiair ) (X s X r ) = Σm s=1 (X s X s): Corollary 2.5.3 Let g = ⊕r j=r g(j) be a graded complex simple Lie algebra with g(1) ̸= 0 and G be a complex analytic group with Lie algebra g. If L is the analytic 20 subgroup of G with Lie algebra g(0) and !0 is as in De nition 2.5.1 then, for all l 2 L, (Ad(l) Ad(l))!0 = !0: Proof. If g 2 L then fAd(l)X j j j 2 Δ(g(r))g forms a basis for g(r). It also holds that fAd(l)X j j j 2 Δ(g(r))g is the dual basis for g(r) with respect to the Killing form. Now the assertion follows from Lemma 2.5.2 Now we show that k are Lequivariant. Lemma 2.5.4 Let g = ⊕r j=r g(j) be a graded complex simple Lie algebra with g(1) ̸= 0 and G be a complex analytic group with Lie algebra g. If L is the ana lytic subgroup of G with Lie algebra g(0) then, for all l 2 L, X 2 g(1), and for 0 k 2r, we have k(Ad(l)X) = (Ad(l) Ad(l)) k(X): (2.5.5) Proof. For l 2 L, we have k(Ad(l)X) = 1 k! ad(Ad(l)(X))k!0 = 1 k! Σ j2Δ(z(n)) ad(Ad(l)(X))k(X j ) X j = 1 k! Σ j2Δ(z(n)) Ad(l) ( ad(X)k(Ad(l1)X j ) ) X j = (Ad(l) Ad(l)) ( 1 k! Σ j2Δ(z(n)) ad(X)k(Ad(l1)X j ) Ad(l1)(X j ) ) = (Ad(l) Ad(l)) ( 1 k! ad(X)k!0 ) = (Ad(l) Ad(l)) k(X): Note that Corollary 2.5.3 is applied from line four to line ve. Now we are going to build the systems of differential operators in D(Ls) n that 21 we study. It is useful to observe that k : g(1) ! g(r + k) g(r) = W are L equivariant polynomial maps of degree k. Here, by a polynomial map we mean a map for which each coordinate is a polynomial in g(1). Therefore the maps k can be thought of as elements in (Pk(g(1)) W)L, where Pk(g(1)) denotes the space of homogeneous polynomials on g(1) of degree k. Then the isomorphism (Pk(g(1)) W)L = HomL(W ;Pk(g(1))) yields the Lintertwining operators ~ k, that are given by ~ k(Y )(X) = Y ( k(X)); (2.5.6) where W is the dual module of W with respect to the Killing form. For each Y 2 W , we have ~ k(Y ) 2 Pk(g(1)) = Symk(g(1)). We de ne differential operators in D(Ls) n from ~ k(Y ). This is done as follows. Let : Symk(g(1)) ! U( n) be the symmetrization operator. Identify U( n) with D(Ls) n by making n act on C1( N 0;C s) via right differentiation R. Then we have a composition of linear maps W ! ~k Pk(g(1)) = Symk(g(1)) ,! U( n) R! D(Ls) n: For Y 2 W , we de ne a differential operator Ωk(Y ) 2 D(Ls) n by Ωk(Y ) = R ◦ ◦ ~ k(Y ): As we will work with irreducible systems we need to be a little more careful with our construction; in particular, we need to take an irreducible constituent of g(r + k) g(r) . Let g(r + k) g(r) = V1 Vm be the irreducible decomposition of g(r + k) g(r) as an Lmodule, and let g(r + k) g(r) = V 1 V m be the corresponding irreducible decomposition of g(r + k) g(r) , where g(j) are the dual spaces of g(j) with respect to the Killing form. For each irreducible 22 constituent V j of g(r + k) g(r) , there exists an Lintertwining operator ~ kjV j 2 HomL(V j ;Pk(g(1))) given as in (2.5.6). Then we de ne a linear operator ΩkjV j : V j ! D(Ls) n by ΩkjV j = R ◦ ◦ ~ kjV j : Since, for Y 2 V j , we have ΩkjV j (Y ) = Ωk(Y ) as a differential operator, we simply write Ωk(Y ) for the differential operator arising from Y 2 V j . De nition 2.5.7 Let g = ⊕r j=r g(j) be an rgraded complex simple Lie algebra with g(1) ̸= 0, and q = ⊕r j=0 g(j) be the parabolic subalgebra of g associated with the r grading. If V is an irreducible constituent of g(r + k) g(r) so that ~ kjV is not identically zero then a list of differential operators D1; : : : ;Dn 2 D(Ls) n is called the ΩkjV system if it is equivalent to a list of differential operators Ωk(Y 1 ); : : : ;Ωk(Y n ); (2.5.8) where fY 1 ; : : : ; Y n g is a basis for V over C. Each ΩkjW system is also simply referred to as an Ωk system. We want to remark that the construction of the Ωk systems might require additional modi cation to secure the conformal invariance. See Section 6 in [1] and Section 3 in [20] for the modi cation for the Ω3 systems of the Heisenberg parabolic subalgebra. It is important to notice that it is not necessary for the Ωk systems to be confor mally invariant; their conformal invariance strongly depends on the complex param eter s for the line bundle Ls. So, we give the following de nition. De nition 2.5.9 Let V be an irreducible constituent of g(r+k) g(r) . Then we say that the ΩkjV system has special value s0 if the system is conformally invariant on the line bundle Ls0 . 23 The goal of this thesis is to nd the special values of the Ω1 system and the Ω2 systems of a maximal twostep nilpotent parabolic subalgebra of nonHeisenberg type. This is done in Chapter 4 and Chapter 7. To nish this section we de ne an action of L0 on D(Ls) n so that the linear operator ΩkjV : V ! D(Ls) n is an L0intertwining operator. This will allow that the ΩkjV system is L0stable (see De nition 2.3.10). As on p.805 of [2], we rst de ne an action of L0 on C1( N 0;C s) by (l f)( n) = s(l)f(l1 nl): This action agrees with the action of L0 by the left translation on the image of the restriction map C1 (G0=Q0;C s) ! C1( N 0;C s ). In terms of this action we de ne an action of L0 on D(Ls) by (l D) f = l (D (l1 f)): (2.5.10) One can check that we have l R(u) = R(Ad(l)u) for l 2 L0 and u 2 U( n); in particular, this action stabilizes the subspace D(Ls) n. With the adjoint action of L0 on U( n), the linear isomorphism U( n) R! D(Ls) n is L0equvariant. It is clear that each map in V ~ kj V ! Pk(g(1)) = Symk(g(1)) ,! U( n) is L0equivariant with respect to the natural actions of L0 on each space, which are induced by the adjoint action of L0 on g. Therefore, with the L0action (2.5.10), the operator ΩkjV : V ! D(Ls) n is an L0intertwining operator. Now we summarize the properties of the ΩkjV system. Remark 2.5.11 It follows from the de nition and the observation above that the ΩkjV system satis es the following properties: 1. The ΩkjV system satis es the condition (S1) of De nition 2.1.4. 24 2. When the ΩkjV system is conformally invariant then it is an irreducible, straight, and L0stable system. By Proposition 2.3.12, it is also a homogeneous system. 2.6 Technical Lemmas The aim of this section is to show two technical lemmas that will be used in Section 7.3. For D 2 D(Ls), we denote by D n the linear functional f 7! (D f)( n) for f 2 C1( N 0;C s ). A simple observation shows that (D1D2) n = (D1) nD2 for D1;D2 2 D(Ls); in particular, if (D1) n = 0 then [D1;D2] n = (D2) nD1. Lemma 2.6.1 Suppose that V is an irreducible constituent of g(r + k) g(r) . Let X1;X2 2 g and Y 1 ; : : : ; Y n 2 V . If s(X1)e = 0 and if [ s(Xi);Ωk(Y t )]e 2 spanC fΩk(Y j )e j j = 1; : : : ng for all i = 1; 2 then [ s(X1); [ s(X2);Ωk(Y t )] ] e 2 spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg: (2.6.2) Proof. Observe that [ s(X1); [ s(X2);Ωk(Y t )]] is s(X1)[ s(X2);Ωk(Y t )] [ s(X2);Ωk(Y t )] s(X1): (2.6.3) Since, by assumption, we have s(X1)e = 0, the rst term is zero at e. By assumption, [ s(X2);Ωk(Y t )]e is a linear combination of Ωk(Y 1 )e; : : : ;Ωk(Y n )e over C. So it may be written as [ s(X2);Ωk(Y t )]e = Σn j=1 ajtΩk(Y j )e with ajt 2 C. Then, at the identity e, the second term in (2.6.3) evaluates to Σn j=1 ajtΩk(Y j )e s(X1): Since ( s(X1)Ωk(Y j ))e = s(X1)eΩk(Y j ) = 0, we obtain [ s(X1); [ s(X2);Ωk(Y t )]]e = Σn j=1 ajtΩk(Y j )e s(X1) = Σn j=1 ajt[ s(X1)Ωk(Y j )]e: 25 Now the proposed result follows from the assumption that [ s(X1);Ωk(Y t )]e is a linear combination of Ωk(Y j )e over C. We call ul = ⊕ Δ+(l) g and ul = ⊕ Δ+(l) g ; where Δ+(l) is the set of positive roots in l. Lemma 2.6.4 Suppose that g(1) is irreducible and that V is an irreducible con stituent of g(r + k) g(r) . Let Xh be a highest weight vector for g(1) and Y l be a lowest weight vector for V . If [ s(Xh);Ωk(Y l )]e = spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg (2.6.5) with fY 1 ; : : : ; Y n g a basis for V then, for any X 2 g(1) and Y 2 V , [ s(X);Ωk(Y )]e 2 spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg: Proof. Set E = spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg. We rst show that for each X 2 g(1), [ s(X);Ωk(Y l )]e 2 E: (2.6.6) Observe that since (L; g(1)) is assumed to be irreducible, the Lmodule g(1) is given by g(1) = U( ul)Xh. Then, as s is linear on g(1), it suffices to show that (2.6.6) holds when X = uk Xh with uk a monomial in U( ul). This is done by induction on the order of uk. Indeed, the proof is clear once we show that (2.6.6) holds for X = Z Xh = [ Z;Xh] with Z 2 ul. By the Jocobi identity, the commutator [ s([ Z;Xh]);Ωk(Y l )] is [ s([ Z;Xh]);Ωk(Y l )] = [ s( Z); [ s(Xh);Ωk(Y l )]] [ s(Xh); [ s( Z);Ωk(Y l )]]: (2.6.7) By the lequivariance of the operator Ωk : V ! D(Ls) n, it follows that [ s( Z);Ωk(Y l )] = Ωk([ Z; Y l ]): 26 Since Z 2 ul and Y l is a lowest weight vector, we have Ωk([ Z; Y l ]) = 0, and so is the second term of the right hand side of (2.6.7). Thus we have [ s([ Z;Xh]);Ωk(Y l )]e = [ s( Z); [ s(Xh);Ωk(Y l )]]e: (2.6.8) Now, by hypotheses and the lequivariance of Ωk, it follows that [ s(Xh);Ωk(Y l )]e; [ s( Z);Ωk(Y l )]e 2 E: As Z 2 ul, by Proposition 2.3.3, we have s( Z)e = 0. Thus, by Lemma 2.6.1, we obtain [ s( Z); [ s(Xh);Ωk(Y l )]]e 2 E, and so, by (2.6.8), [ s([ Z;Xh]);Ωk(Y l )]e 2 E. Next we show that for any X 2 g(1) and Y 2 V , [ s(X);Ωk(Y )]e 2 E: (2.6.9) Once again since V is irreducible, it is given by V = U(ul)Y l . As before, it is enough to show that (2.6.9) holds for Y = Z Y l with Z 2 ul. Since Ωk(Z Y l ) = [ s(Z);Ωk(Y l )], by the Jacobi identity, the commutator [ s(X);Ωk(Z Y l )] is [ s(X);Ωk(Z Y l )] = [ s(Z); [ s(X);Ωk(Y l )]] [[ s(Z); s(X)];Ωk(Y l )]: (2.6.10) We showed above that [ s(X);Ωk(Y l )]e 2 E. Since s(Z)e = 0 and [ s(Z);Ωk(Y l )]e 2 E, by Lemma 2.6.1, the rst term of the right hand side of (2.6.10) satis es [ s(Z); [ s(X);Ωk(Y l )]]e 2 E: Moreover, as [ s(Z); s(X)] = s([Z;X]) with [Z;X] 2 g(1), by what we have shown above, the second term satis es [[ s(Z); s(X)];Ωk(Y l )]e 2 E: Hence, [ s(X);Ωk(Z Y l )]e 2 E. 27 2.7 The Ωk Systems and Generalized Verma Modules To conclude this chapter, we show that conformally invariant Ωk systems induce nonzero U(g)homomorphisms between certain generalized Verma modules. The main idea is that conformally invariant Ωk systems yield nite dimensional simple lsubmodules of generalized Verma modules, on which n acts trivially. In general, to describe the relationship between conformally invariant systems on a g0bundle V ! M and generalized Verma modules, we realize generalized Verma modules as the space of smooth distributions on M supported at the identity. How ever, in our setting that the vector bundle V is a line bundle Ls, it is not necessary to use the realization. Thus, in this section, we are going to describe the relationship without using the realization. For more general theory on the relationship between conformally invariant systems and generalized Verma modules, see Sections 3, 5, and 6 of [2]. A generalized Verma module U(g) U(q) W is a U(g)module that is induced from a nite dimensional simple lmodule W on which n acts trivially. See Section A.1 for more details on generalized Verma modules. In this section we parametrize those modules as Mq[W] = U(g) U(q) W: We rst observe that the differential operators in D(Ls) n can be described in terms of elements of Mq[Cs q ], where Cs q is the qmodule derived from the Q0 representation ( s;C). By identifying Mq[Cs q ] as U( n) Cs q , the map Mq[Cs q ] ! U( n) given by u 1 7! u is an isomorphism of vector spaces. The composition Mq[Cs q ] ! U( n) R! D(Ls) n (2.7.1) is then a vectorspace isomorphism. Let W be an irreducible constituent of g(r + k) g(k) so that the L0 intertwining operator ΩkjW : W ! D(Ls) n is not identically zero. For Y 2 W , 28 if !k(Y ) = !kjW (Y ) denotes the element in U( n) that corresponds to Ωk(Y ) = ΩkjW (Y ) in D(Ls) n via right differentiation R in (2.7.1) then the linear operator !kjW : W ! U( n) is Lequivariant. Indeed, for l 2 L and Y 2 W , we have !k(l Y ) = Ad(l)!k(Y ); where the action l Y is the standard action of L on W , which is induced from the adjoint action of L on W. De ne Mq[W]n = fv 2 Mq[W] j X v = 0 for all X 2 ng: The following result is the specialization of Theorem 19 in [2] to the present situation. Theorem 2.7.2 If D = D1; : : : ;Dm is a straight L0stable homogeneous conformally invariant system on the line bundle Ls, and if !j denotes the element in U( n) that corresponds to Dj for j = 1; : : : ;m via right differentiation R then the space F(D) = spanC f!j 1 j j = 1; : : : ;mg is an Linvariant subspace of Mq[Cs q ]n. If the ΩkjW system is ΩkjW = Ωk(Y 1 ); : : : ;Ωk(Y m); where fY 1 ; : : : ; Y m g is a basis of W , then the space F(ΩkjW ) is given by F(ΩkjW ) = spanC f!k(Y j ) 1 j j = 1; : : : ;mg Mq[Cs q ]: Corollary 2.7.3 If the ΩkjW system is conformally invariant on the line bundle Ls0 then F(ΩkjW ) is an Linvariant subspace of Mq[Cs0 q ]n. Proof. By Remark 2.5.11, if the ΩkjW system is conformally invariant then it is a straight, L0stable, and homogeneous system. Now this corollary follows from Theorem 2.7.2. 29 Now suppose that the ΩkjW system is conformally invariant over Ls0 . Then, by Corollary 2.7.3, it follows that F(ΩkjW ) is an Linvariant subspace of Mq[Cs0 q ]n. On the other hand, there exists a vector space isomorphism F(ΩkjW ) ! W Cs q ; (2.7.4) that is given by !k(Y j ) 1 7! Y j 1. It is clear that the vector space isomorphism is Lequivariant with respect to the standard action of L on the tensor products F(ΩkjW ) U( n) Cs q and W Cs q . In particular, since W is an irreducible Lmodule, if W has highest weight then F(ΩkjW ) is the irreducible Lmodule with highest weight s0 q. 1 Moreover, as F(ΩkjW ) Mq[Cs0 q ]n, the nilradical n acts on F(ΩkjW ) trivially. Therefore the inclusion map 2 HomL ( F(ΩkjW );Mq[Cs0 q ] ) induces a nonzero U(g)homomorphism φΩk 2 HomU(g);L ( Mq[F(ΩkjW )];Mq[Cs0 q ] ) of generalized Verma modules, that is given by Mq[F(ΩkjW )] φ!Ωk Mq[Cs0 q ] (2.7.5) u ( !k(Y ) 1) 7! u ( !k(Y ) 1): If F(ΩkjW ) = Cs0 q then the map in (2.7.5) is just the identity map. However, Proposition 2.7.6 below shows that it does not happen. Proposition 2.7.6 Let W be an irreducible constituent of g(r + k) g(r) with k = 1; : : : ; 2r, so that ΩkjW : W ! D(Ls) n is not identically zero. If the ΩkjW system is conformally invariant on the line bundle Ls0 then F(ΩkjW ) ̸= Cs0 q Proof. Observe that if is the highest weight for W then F(ΩkjW ) has highest weight s0 q. If F(ΩkjW ) = Cs0 q then = 0, and so the irreducible constituent W g(r+k) g(r) would have highest weight 0. It is known that if is the highest weight for g(r) then the highest weight of any irreducible constituent of g(r+k) g(r) 1See Section 3.2 for the details of what we mean by a highest weight of a nite dimensional representation of reductive group L. 30 is of the form + with some weight for g(r+k) (see for instance [21, Proposition 3.2]). Thus, the highest weight 0 for W must be of the form 0 = + ( ). However, cannot be a weight for g(r + k) for any k = 1; : : : ; 2r, since only g(r) has weight . Therefore F(ΩkjW ) ̸= Cs0 q . Corollary 2.7.7 Under the same hypotheses for Proposition 2.7.6, the generalized Verma module Mq[Cs0 q ] is reducible. Proof. If is the highest weight for W then, by the proof for Proposition 2.7.6, it follows that F(ΩkjW ) ̸= Cs0 q . Now this corollary follows from (2.7.5). 31 CHAPTER 3 Parabolic Subalgebras and Zgradings It has been observed in Section 2.5 that the Zgrading g = ⊕r j=r g(j) on g and the parabolic subalgebra q play a role to construct the Ωk systems. In this chapter we study those in detail for q a maximal twostep nilpotent parabolic subalgebra of nonHeisenberg type. The Ω1 system and the Ω2 systems of those parabolics will be studied in Chapter 4 and Chapter 7, respectively. 3.1 kstep Nilpotent Parabolic Subalgebras We shall later construct the Ω1 system and the Ω2 systems of a maximal twostep nilpotent parabolic q. To do so, in this section we classify the kstep nilpotent parabolic subalgebras q by the subsets of simple roots. This is done in Proposition 3.1.4. Let r be any nonzero Lie algebra. Put r0 = r, r1 = [r; r], and rk = [r; rk1] for k 2 Z>0. We call rk the kth step of r for k 2 Z 0. The Lie algebra r is called nilpotent if rk = 0 for some k, and it is called kstep nilpotent if rk1 ̸= 0 and rk = 0. In particular, if [r; r] = 0 then r is called abelian, and if dim([r; r]) = 1 then r is called Heisenberg. Note that r is Heisenberg if and only if its center z(r) is one dimensional. If the nilpotent radical n of a parabolic subalgebra q = l n is kstep nilpotent (resp. abelian or Heisenberg) then we say that q is a kstep nilpotent (resp. abelian or Heisenberg) parabolic. If = Σ 2 m 2 Σ 2 Z then we say that jm j are the multiplicities of in . Proposition 3.1.4 below determines kstep nilpotent parabolic subalgebras 32 qS by the sum of the multiplicities of the simple roots of S in the highest root. Although it is a wellknown fact, we include a proof in this thesis, since we couldn't nd one in the literature. To prove the proposition it is convenient to show two technical lemmas, namely, Lemma 3.1.2 and Lemma 3.1.3. In Lemma 3.1.2 and Lemma 3.1.3, the subalgebras l and n are assumed to be the Levi factor and the nilpotent radical of qS with S = f i1 ; : : : ; ir g, respectively. Remark 3.1.1 It is easily shown by the Jacobi identity and the induction on k that we have [l; nk] nk for each k. In particular, if + 2 Δ with 2 Δ(l) and 2 Δ(nk) then + 2 Δ(nk), where Δ(l) and Δ(nk) are the subsets of roots that contribute to l and nk, respectively. Lemma 3.1.2 Suppose that is a root in Δ and let mij be the multiplicity of ij in . If Σr j=1mij = k then 2 Δ(nk1). Proof. For 2 Δ, it is well known that there exists an ordered set O = f 1; : : : ; sg of simple roots so that = Σs t=1 t having the property that each ordered partial sum is a root (see for instance [9, Corollary 10.2A]). Note that some of the roots in O belong to S and others are in (l) = Δ(l) \ . We prove this lemma by induction on the sum, Σr j=1mij , of the multiplicities of ij in S. When Σr j=1mij = 1, we have O \ S = f hg for some h 2 S Δ(n). Write = Σh t=1 t. If h = 1 then = h 2 Δ(n) = Δ(n0). If h ̸= 1 then since each partial sum is a root, we have Σh1 t=1 t 2 Δ(l). Since [l; n0] n0, it follows that = Σh t=1 t = Σh1 t=1 t + h 2 Δ(n0): Since each sum Σd t=1 t for d h is a root and all t for t > h are in Δ(l), by Remark 3.1.1, we conclude that = + h+1 + + s 2 Δ(n0): 33 Now we assume that the proposed statement holds for k1 Σr j=1mij 1. Let Σr j=1mij = k. There are two cases, s 2 S or s 2 (l). If s 2 S then the sum of the multiplicities of the simple roots in S contributing to s is equal to k 1. By induction hypothesis, we have s 2 Δ(nk2). Therefore, = ( s) + s 2 Δ([n; nk2]) = Δ(nk1). When s 2 (l), let l be the largest root in the order of O so that l 2 S. Then the sum of the multiplicities of the simple roots from S in the root Σl t=1 t is equal to k. Assuming as before, we conclude that Σl t=1 t 2 Δ(nk1). Now, once again, since each sum Σd t=1 t for d l is a root and all t for t > l are in Δ(l), by Remark 3.1.1, we conclude that = Σl t=1 t + l+1 + + s 2 Δ(nk1): Lemma 3.1.3 If 2 Δ(nk) and mij are the multiplicities of ij in then Σr j=1mij k + 1. Proof. We prove it by induction on k. Observe that if 2 Δ(n) = Δ+nΔ(l) then there exists ij 2 S so that the multiplicity of ij in is nonzero, because we would have 2 Δ(l), otherwise. Thus the case k = 0 is clear. We then assume that this holds for k = l. Let 2 Δ(nl+1). Since nl+1 = [nl; n], the root may be written as = ′ + ′′ with ′ 2 Δ(nl) and ′′ 2 Δ(n). Denoting by mij ( ) the multiplicities of ij in , we have Σr j=1 mij ( ) = Σr j=1 mij ( ′ + ′′) = Σr j=1 mij ( ′) + Σr j=1 mij ( ′′) (l + 1) + 1 = l + 2: By induction the lemma follows. We remark that if the highest root is = Σ 2 m then for any root = Σ 2 n , it follows that n m for all 2 . 34 Proposition 3.1.4 Let g be a complex simple Lie algebra with highest root , and qS = l n be the parabolic subalgebra of g that is parametrized by S with S = f i1 ; : : : ; ir g . Then n is kstep nilpotent if and only if k = mi1 +mi2 + +mir , where mij are the multiplicities of ij in . Proof. First we show that if k = Σr j=1mij then n is kstep nilpotent. If k = Σr j=1mij then, by Lemma 3.1.2, we have 2 Δ(nk1); in particular, nk1 ̸= 0. If nk ̸= 0 then there would exist 2 Δ(nk). If nij are the multiplicities of ij in then, by Lemma 3.1.3, it follows that Σr j=1 nij k + 1 > k: This contradicts the remark above. Therefore nk = 0, and so n is kstep nilpotent. Conversely, suppose that n is kstep nilpotent. If Σr j=1mij = l then, as we showed above, n is lstep nilpotent. Hence, l = k. To nish this section we introduce subdiagrams of Dynkin diagrams that associate to parabolics qS and classi cation types of them. First, Theorem 2.2.3 shows that there exists a bijection between the standard parabolics qS and the subsets S of simple roots. This allows us to associate qS to subdiagrams of Dynkin diagrams. The subdiagrams that associates to qS are obtained by deleting the nodes of the Dynkin diagram of g that correspond to the simple roots in S, and the edges in incident on them. We call such subdiagrams deleted Dynkin diagrams. With the multiplicities of simple roots in the highest root of g in hand, by Proposition 3.1.4, we can also see the number of steps of nilradical n of qS from the deleted Dynkin diagram. Example 3.1.5 below describes the deleted Dynkin diagram of a given parabolic qS and how we read the diagram. For simplicity, we depict deleted Dynkin diagrams by crossing out the deleted nodes. 35 Example 3.1.5 Take g = sl(6;C). The set of simple roots is = f 1; 2; 3; 4; 5g with Dynkin diagram ◦ 1 ◦ 2 ◦ 3 ◦ 4 ◦ 5 : Choose S = f 2; 4g. Then the deleted Dynkin diagram of parabolic subalgebra qS corresponding to the subset S is ◦ 1 2 ◦ 3 4 ◦ 5 : Moreover, Figure B.2 in Appendix B shows that the multiplicity of each simple root in the highest root of g of type An is 1, so this parabolic qS is a twostep nilpotent parabolic. In later sections we often refer to parabolic subalgebras qS by their corresponding subset S of simple roots. To this end, we are going to de ne classi cation types of parabolics qS. In De nition 3.1.6 below, we mean by classi cation type T of g type An, Bn, Cn, Dn, E6, E7, E8, F4, or G2. De nition 3.1.6 If g is a complex simple Lie algebra of classi cation type T and S is a subset of of simple roots then we say that a parabolic subalgebra qS of g is of type T (S), or type T (i1; : : : ; ik) if S = f i1 ; : : : ; ik g. For example, the parabolic subalgebra qS in Example 3.1.5 is of type A5(2; 4). Any maximal parabolic subalgebra is of type T (i) for some i 2 . In this thesis we use the Bourbaki conventions [4] for the labels of the simple roots (see Figure B.1 in Appendix B for the labels). 3.2 Maximal TwoStep Nilpotent Parabolic q of NonHeisenberg type The aim of this section is to study the 2grading g = ⊕2 j=2 g(j) on g, that is induced from a maximal twostep nilpotent parabolic subalgebra q of nonHeisenberg type. 36 Assume that g has rank greater than one and that q is a simple root, so that the parabolic subalgebra q = qf qg = l n parameterized by q is a maximal twostep nilpotent parabolic with dim([n; n]) > 1. Let ⟨ ; ⟩ be the inner product induced on h corresponding to the Killing form . Write jj jj2 = ⟨ ; ⟩ for 2 Δ. The coroot of is _ = 2 =⟨ ; ⟩. Recall from Section 2.2 that q denotes the fundamental weight for q. As Δ(l) = f 2 Δ j 2 span( nf qg)g and Δ(n) = Δ+nΔ(l), we have ⟨ q; ⟩ 8>>< >>: = 0 if 2 Δ(l) > 0 if 2 Δ(n) : Observe that if H q 2 h is de ned by (H;H q) = q(H) for all H 2 h and if Hq = 2 jj qjj2H q (3.2.1) then (Hq) is the multiplicity of q in . In particular, it follows from Proposition 3.1.4 that for 2 Δ+, (Hq) only can assume the values of 0, 1, or 2. Therefore, if g(j) denotes the jeigenspace of ad(Hq) then the action of ad(Hq) on g induces a 2grading g = g(2) g(1) g(0) g(1) g(2) with parabolic subalgebra q = g(0) g(1) g(2): Here we have l = g(0) and n = g(1) g(2). The subalgebra n, the opposite of n, is given by n = g(1) g(2): Observe that L acts on each of the subspaces g(j) via the adjoint representation. The goal of this section is to show that g(j) are irreducible Lmodules for j ̸= 0. 37 Via the Killing form, g(1) and g(2) are dual to g(1) and g(2), respectively. Thus, we will show that g(1) and g(2) are Lirreducible; hence, so are true for g(1) and g(2). The following proposition is well known. However, since the argument used in the proof will be referred in the proof for Corollary 3.2.3 below, we give a proof. Proposition 3.2.2 Assume that g is a graded complex semisimple Lie algebra with g = ⊕ j g(j), and let q = g(0) ⊕ j>0 g(j) with g(1) ̸= 0. Then g(1) is g(0)irreducible if and only if q is a maximal parabolic. Proof. We rst show that if q is not maximal then g(1) is not g(0)irreducible. Under this assumption there are at least two distinct simple roots in nΔ(g(0)), say 1 and 2. Let X 1 and X 2 be root vectors for 1 and 2, respectively. If U(g(0)) denotes the universal enveloping algebra of g(0) then U(g(0))X 1 and U(g(0))X 2 are two g(0)submodules of g(1). Since 1 and 2 are simple, U(g(0))X 1 ̸= U(g(0))X 2 . Hence g(1) is reducible. To prove the converse, as g(0) = z(g(0)) g(0)ss and the center z(g(0)) acts by scalars on g(1), it suffices to show that g(1) is an irreducible g(0)ssmodule. As in [9, Corollary 10.2A] we write 2 Δ+ as = i1 + + in with ij 2 (not necessarily distinct) in such a way that each partial sum i1+ + ij is a root. If q is maximal then there exists unique simple root 2 nΔ(g(0)). Each root 2 Δ(g(1)) is of the form = i1 + + ik + + im + + in; where the sum i1 + + ik q is a root with ij 2 Δ(g(0)). Let X q and X be root vectors for q and , respectively. If Xj is a root vector for ij then 0 ̸= ad(Xn)ad(Xn1) ad(Xm+1)ad(Xm)ad(X q)X 38 is a nonzero element in (U(g(0)ss)X ) \ g . Since 2 Δ(g(1)) is arbitrary, it is followed that g(1) = U(g(0)ss)X . We quote the Theorem of the Highest Weight to conclude that g(1) is g(0)ssirreducible with lowest weight . Let l = z(l) lss be the decomposition of l, that corresponds to L = Z(L)◦Lss with Z(L)◦ the identity component of the center of L and Lss the semisimple part of L. We say that a weight 2 h is a highest weight of a nite dimensional Lmodule V if jhss is a highest weight of V as an Lssmodule, where hss = h \ lss. A lowest weight of a nite dimensional Lmodule is similarly de ned. Corollary 3.2.3 If q = g(0) g(1) g(2) is the maximal twostep nilpotent parabolic of nonHeisenberg type determined by q then g(1) is the irreducible Lmodule with lowest weight q. Proof. Observe that since a root vector for q is an element of g(1), we have g(1) ̸= ∅. As Ad(L) preserves g(1), Proposition 3.2.2 implies that g(1) is Lirreducible. Next we show that g(2) is the irreducible Lmodule with highest weight . Since the argument of the proof works for general rgrading g = ⊕r j=r g(j), we give the proof in the general setting. Proposition 3.2.4 Assume that g = ⊕r j=r g(j) is a graded complex simple Lie algebra with n = ⊕r j=1 g(j). If the positive system Δ+ is chosen so that Δ+ = Δ+(g(0)) [ Δ(n) and is the highest root of g with respect to Δ+ then g(r) is the irreducible g(0)module with highest weight . Proof. As g is simple and is the highest root with respect to Δ+, g = U(g)X = U( n)(U(g(0))X ): Observe that since X 2 g(r) and g(r) is g(0)stable, we have U(g(0))X g(r). On 39 the other hand, as n = ⊕1 j=r g(j), it follows that U( n)g(r) ⊕r1 j=r g(j): As g = ⊕r j=r g(j), this shows that U(g(0))X g(r). Corollary 3.2.5 If q = g(0) g(1) g(2) is the maximal twostep nilpotent parabolic of nonHeisenberg type determined by q then g(2) is the irreducible Lmodule with highest weight . Proof. Observe that is the highest root of g for Δ+ = Δ+(l)[Δ(n). Now, as Ad(L) preserves g(2), Proposition 3.2.4 implies that g(2) is Lirreducible. To conclude this section we show that z(n) = g(2) and z( n) = g(2), where z(n) and z( n) are the centers of n and n, respectively. Because of the identi cation of g(j) with g(j) via the Killing form, it suffices to show that z(n) = g(2). The following technical lemma will simplify the expositions. Lemma 3.2.6 If q = g(0) g(1) g(2) is a maximal twostep nilpotent parabolic of nonHeisenberg type with n = g(1) g(2) then z(n) \ g(1) = f0g. Proof. One can easily check that z(n) is an lmodule by using the Jacobi identity and the fact that n is an lmodule. Therefore the intersection z(n) \ g(1) is an l submodule of g(1). The irreducibility of g(1) from Corollary 3.2.3 then forces that z(n) \ g(1) = f0g or g(1). However, the second is impossible; otherwise, we would have [n; n] = [g(1); g(1)] = 0; contrary to [n; n] ̸= 0. Therefore, z(n) \ g(1) = f0g. 40 Lemma 3.2.7 If q = g(0) g(1) g(2) is a maximal twostep nilpotent parabolic of nonHeisenberg type with n = g(1) g(2) then z(n) = g(2). Proof. Since g(2) z(n), it suffices to show the other inclusion. Take X 2 z(n). Since n = g(1) g(2), there exist Xj 2 g(j) for j = 1; 2 so that X = X1 + X2. Since X;X2 2 z(n), we have for any Y 2 n, [Y;X1] = [Y;X1] + [Y;X2] = [Y;X] = 0: Thus X1 2 z(n) \ g(1). Lemma 3.2.6 then concludes that X1 = 0, and so we have X = X2 2 g(2). Since X 2 z(n) is arbitrary, this yields that z(n) g(2). Now, since l = g(0), g(2) = z(n) and g(2) = z( n), we write the 2grading g = ⊕2 j=2 g(j) as g = z( n) g(1) l g(1) z(n) (3.2.8) with parabolic subalgebra q = l g(1) z(n): (3.2.9) 3.3 The Simple Subalgebras l and ln The purpose of this section is to study the structure of the Levi subalgebra l = z(l) lss. The material of this section will play a role in Chapter 5 and Chapter 6 when we decompose l z(n) into Lirreducible subspaces. The center z(l) is of the form z(l) = ∩ 2 (l) ker( ). Since g has rank greater than one and (l) = nf qg, z(l) is nonzero and onedimensional. It is clear from (3.2.1) that Hq is an element of z(l). Therefore we have z(l) = CHq. Next we consider the structure of lss. Observe that the Dynkin diagram of g can be extended by attaching the lowest root to the diagram. If g is not of type An then 41 there is exactly one simple root, that is connected to in the extended diagram (see Figure B.3 in Appendix B). Let denote such a unique simple root. It is easy to see that qf g is the Heisenberg parabolic of g; that is, the twostep nilpotent parabolic with dim([n; n]) = 1. Hence, if qf qg is a maximal twostep nilpotent parabolic with dim([n; n]) > 1 then 2 (l) = nf qg. If we delete the node corresponding to q then we obtain one, two, or three subgraphs with one subgraph containing . This implies that the subalgebra lss is either simple or the direct sum of two or three simple subalgebras with only one simple subalgebra containing the root space g for . The three subgraphs occur only when q is of type Dn(n 2). So, if q is not of type Dn(n 2) then there are at most two subgraphs. In this case we denote by l (resp. ln ) the simple subalgebra of l whose subgraph in the deleted Dynkin diagram contains (resp. does not contain) the node for . Thus the Levi subalgebra l may decompose into l = CHq l ln : (3.3.1) Then, for the rest of this chapter, we assume that q is not of type Dn(n2), so that the Levi subalgebra l can be expressed as (3.3.1). Recall from De nition 3.1.6 that if g is of type T then we say that the parabolic subalgebra q determined by i 2 is of type T (i). Then the parabolic subalgebras q under consideration are given as follows: Bn(i) (3 i n); Cn(i) (2 i n 1); Dn(i) (3 i n 3); (3.3.2) and E6(3); E6(5); E7(2); E7(6); E8(1); F4(4): (3.3.3) Note that in type An the nilradical n of any maximal parabolic subalgebra is abelian. Write (l ) = f 2 j 2 Δ(l )g and (ln ) = f 2 j 2 Δ(ln )g. Example 3.3.4 below exhibits the subgraphs for l and ln of q of type B5(3) with (l ) and 42 (ln ). One can nd those data in Appendix C for each maximal parabolic subalgebra in (3.3.2) or (3.3.3). Example 3.3.4 Let q be the parabolic subalgebra of type B5(3) with deleted Dynkin diagram ◦ 1 ◦ 2 3 ◦ 4 +3 ◦ 5 : Figure B.3 in Appendix B shows that = 2. Therefore, the subgraph for l is ◦ 1 ◦ 2 and that for ln is ◦ 4 +3 ◦ 5 with (l ) = f 1; 2g and (ln ) = f 4; 5g. Remark 3.3.5 It is clear from the extended Dynkin diagrams that ⟨ ; ⟩ > 0 and ⟨ ; ⟩ = 0 for any other simple roots . In particular, ⟨ ; ⟩ = 0 for all 2 (ln ). 3.4 Technical Facts on the Highest Weights for l , ln , g(1), and z(n) In this section we summarize technical lemmas on the Lhighest weights for l , ln , g(1), and z(n). These technical facts will be used in later computations. Proposition 3.2.4 shows that z(n) has highest weight , which is the highest root of g. We denote by , n , and the highest weights for l , ln , and g(1), respectively. In Appendix C we give the explicit values for these highest weights for each of the parabolic subalgebras under consideration. We remark that all these highest weights are indeed roots in Δ+. Observe that the highest weights and n of l and ln , respectively, are also the highest roots of l and ln as simple algebras; in particular, the multiplicities of 2 (l ) (resp. 2 (ln )) in (resp. n ) are all strictly positive. 43 Lemma 3.4.1 If q is the simple root that determines q = l g(1) z(n) then + q and n + q are roots. Proof. We only prove that + q 2 Δ; the other assertion that n + q 2 Δ can be proven similarly. It suffices to show that ⟨ ; q⟩ < 0, since both and q are roots. For 2 we observe that ⟨ ; q⟩ < 0 if is adjacent to q in the Dynkin diagram and ⟨ ; q⟩ = 0 otherwise. An observation on the deleted Dynkin diagrams shows that there exists a unique simple root k in (l ) that is adjacent to q. Since is the highest root for l as a simple algebra, the multiplicity of k in is strictly positive. Thus ⟨ ; q⟩ < 0. Lemma 3.4.2 If , n , , and are the highest weights of l , ln , g(1), and z(n), respectively, then the following hold: (1) 2 Δ, but n =2 Δ. (2) 2 Δ. (3) ; n 2 Δ. Proof. To prove n =2 Δ, we recall a wellknown fact that if n and m are the largest nonnegative integers so that n n 2 Δ and + m n 2 Δ, respectively, then ⟨ ; _ n ⟩ is given by ⟨ ; _ n ⟩ = n m (see for instance [9, Section 9.4]). Observe that the roots in Δ(ln ) are orthogonal to ; in particular, ⟨ ; _ n ⟩ = 0. Thus, we have n = m. As n 2 Δ+ and is the highest root, + n =2 Δ. Therefore, n = m = 0, which concludes that n is not a root. To prove 2 Δ, it suffices to show that ⟨ ; ⟩ > 0, since both and are roots. Write in terms of simple roots in (l ). Observe that each 2 (l ) has positive multiplicity m in . As is orthogonal to for any 2 (l )nf g, we have ⟨ ; ⟩ = m ⟨ ; ⟩ > 0. To prove the assertion (2), we show that ⟨ ; ⟩ > 0. Since, for simple and ̸= , we have ⟨ ; ⟩ = 0 and ⟨ ; ⟩ > 0, it suffices to show that the multiplicity 44 n of in is n > 0. Observe that the root = Σ 2 belongs to Δ(g(1)). The multiplicity of in is one. As g(1) is an irreducible Lmodule with highest weight , the root is of the form = Σ 2 (l) c with c nonnegative integers. Therefore = + Σ 2 (l) c , and so n = 1 + c > 0. Next we show that n 2 Δ. The other assertion in (3) is proven in a similar manner. It suffices to show that ⟨ ; n ⟩ > 0. We write as = Σ 2 (l ) m ϖ + Σ 2 (ln ) n eϖ with m ; n 2 Z 0; (3.4.3) where ϖ and eϖ are the fundamental weights of 2 (l ) and 2 (ln ), re spectively. The root n is an integer combination of simple roots in (ln ) of the form n = Σ 2 (ln ) m with m 2 Z>0: Then ⟨ϖ ; n ⟩ = 0 for all 2 (l ), and ⟨ eϖ ; n ⟩ > 0 for all 2 (ln ). It follows from Lemma 3.4.1 that ln acts on g(1) nontrivially. Thus, there exists ′ 2 (ln ) so that n ′ ̸= 0 in (3.4.3), and so we obtain ⟨ ; n ⟩ n ′m ′ > 0. When g is not simply laced then there are two root lengths in Δ. A root is called long or short accordingly. The following technical lemma will simplify arguments concerning the long roots later. We regard any root as a long root, when g is simply laced. Lemma 3.4.4 Suppose that 2 Δ is a long root. For any 2 Δ, the following hold. (1) If 2 Δ then ⟨ ; _⟩ = 1. (2) If + 2 Δ then ⟨ ; _⟩ = 1. (3) If 2 Δ then ∓ =2 Δ. (4) 2 =2 Δ. 45 Proof. Assume that 2 Δ. Since is a long root, we have 1 jj jj2=jj jj2 > 0. Thus, 1 jj jj2 jj jj2 ⟨ ; _⟩ + 1 > 0; which implies that 0 < jj jj2 jj jj2 ⟨ ; _⟩ < 1 + jj jj2 jj jj2 2: Therefore ⟨ ; _⟩ = 1. Part (2) may be shown similarly, and (3) and (4) follow from (1) and (2) with the fact that ⟨ ; _⟩ = p ; q ; , where p ; = maxfj 2 Z 0 j j 2 Δg and q ; = maxfj 2 Z 0 j + j 2 Δg. Lemma 3.4.5 If , n , , and are the highest weights of l , ln , g(1), and z(n), respectively, then the following hold: (1) + n 2 Δ. (2) n =2 Δ. (3) If is a long root then =2 Δ. Proof. Lemma 3.4.2 shows that 2 Δ. Then in order to prove (1), it is enough to show that ⟨ n ; ⟩ < 0. It follows from Remark 3.3.5 that ⟨ n ; ⟩ = 0. On the other hand, we have ⟨ n ; ⟩ > 0 by the proof for (3) of Lemma 3.4.2. Therefore, ⟨ n ; ⟩ = ⟨ n ; ⟩ ⟨ n ; ⟩ < 0: When n is a long root of g, the assertion (2) follows from (1) and Lemma 3.4.4. The data in Appendix C shows that n is a long root unless q is of type Bn(n 1). If q is of type Bn(n 1) then we have = "1 + "2, = "1 + "n, and n = "n. Thus n =2 Δ. To show (3), observe that, by Lemma 3.4.2, we have ; 2 Δ. Since is assumed to be a long root, it follows from Lemma 3.4.4 that ⟨ ; _ ⟩ = ⟨ ; _ ⟩ = 1. 46 Therefore ⟨ ; _ ⟩ = 0, which forces that jj jj2 = jj jj2 + jj jj2: (3.4.6) Since is a root, we have jj jj ̸= 0. As is assumed to be a long root, (3.4.6) implies that ( ) =2 Δ. Remark 3.4.7 Direct observation shows that is a long root, unless q is of type Cn(i). If q is of type Cn(i) then the data in Appendix C shows = 2"1, = "1+"i+1, and = "1 "i. Thus + =2 Δ, but 2 Δ. 47 CHAPTER 4 The Ω1 System The aim of this chapter is to determine the complex parameter s1 2 C for the line bundle Ls so that the Ω1 system of a maximal twostep nilpotent parabolic q of nonHeisenberg type is conformally invariant on Ls1 . The special value is given in Theorem 4.2.5. 4.1 Normalizations The purpose of this section is to x normalizations for root vectors. In the next section we are going to construct the Ω1 system and determine its special value of s. To do so, it is essential to set up convenient normalizations. If ; 2 Δ then de ne p ; = maxfj 2 Z 0 j j 2 Δg and q ; = maxfj 2 Z 0 j + j 2 Δg: (4.1.1) In particular, we have ⟨ ; _⟩ = p ; q ; : (4.1.2) It is known that we can choose X 2 g and H 2 h for each 2 Δ in such a way that the following conditions hold (see for instance [7, Sections III.4 and III.5]). The reader may want to notice that our normalizations are different from those used in [1]. (H1) For each 2 Δ+, fX ;X ;H g is an sl(2;C) triple; in particular, [X ;X ] = H : 48 (H2) For each ; 2 Δ+, [H ;X ] = (H )X . (H3) For 2 Δ we have (X ;X ) = 1. (H4) For ; 2 Δ we have (H ) = ⟨ ; ⟩. (H5) For ; 2 Δ with + ̸= 0, there is a constant N ; so that [X ;X ] = N ; X + if + 2 Δ, N ; = 0 if + =2 Δ: (H6) If 1; 2; 3 2 Δ+ with 1 + 2 + 3 = 0 then N 1; 2 = N 2; 3 = N 3; 1 : (H7) If ; 2 Δ and + 2 Δ then N ; N ; = q ; (1 + p ; ) 2 (H ): In particular, N ; is nonzero if + 2 Δ. We call the constants N ; structure constants. 4.2 The Ω1 System In this section we shall build the Ω1 system and determine its special value. As we have observed in Section 2.5, we use the covariant map 1 and the associated L intertwining operators ~ 1jV , where V are irreducible constituents of g(1) g(2) . By De nition 2.5.1, the covariant map 1 is given by 1 : g(1) ! g(1) z(n) X 7! ad(X)!0 49 with !0 = Σ j2Δ(z(n)) X j X j . It is clear that 1 is not identically zero. Indeed, if X = X with the highest weight for g(1) then 1(X ) = ad(X )!0 = Σ Δ (z(n)) N ; jX j X j with Δ (z(n)) = f j 2 Δ(z(n)) j j 2 Δg. By Lemma 3.4.2, we have 2 Δ with the highest weight for z(n), so Δ (z(n)) ̸= ∅. Since the vectors X j X j for j 2 Δ (z(n)) are linearly independent, we have 1(X ) ̸= 0. For each irreducible constituent V of g(1) z(n) , there exists an associated Lintertwining operator ~ 1jV 2 HomL(V ;P1(g(1))) so that, for all Y 2 V , ~ 1jV (Y )(X) = Y ( 1(X)): Observe that the duality for V is de ned with respect to the Killing form . More over, via the Killing form , we have g(1) z(n) = g(1) z( n). Thus, if Y = X X t with 2 Δ(g(1)) and t 2 Δ(z(n)) then Y ( 1(X)) is given by Y ( 1(X)) = Σ j2Δ(z(n)) (X ; ad(X)X j ) (X t ;X j ); (4.2.1) as 1(X) = Σ j2Δ(z(n)) ad(X)X j X j . Now we wish to determine all the irreducible constituents V of g(1) z( n), so that ~ 1jV are not identically zero. Observe that P1(g(1)) = Sym1(g(1)) = g(1) and that g(1) is an irreducible Lmodule, as q is a maximal parabolic. Thus, if ~ 1jV is not identically zero then V = g(1). Proposition 4.2.2 below shows that the converse also holds. Proposition 4.2.2 Let V be an irreducible constituent of g(1) z( n). Then ~ 1jV is not identically zero if and only if V = g(1). 50 Proof. First observe that g(1) is an irreducible constituent of g(1) z( n). Indeed, since 1 is linear, we have 1(g(1)) = g(1) as an Lmodule; in particular, g(1) is an irreducible constituent of g(1) z(n). Therefore g(1) = g(1) is an irreducible constituent of g(1) z( n) = (g(1) z(n)) . To prove ~ 1jg(1) is a nonzero map, it suffices to show that ~ 1jg(1)(Y ) ̸= 0 for some Y 2 g(1) g(1) z( n). To do so, consider a map 1 : g(1) ! g(1) z( n) X 7! ad( X ) !0 with !0 = Σ t2Δ(z(n)) X t X t . This is a nonzero Lintertwining operator. Thus 1(g(1)) = g(1) as an Lmodule, and 1(X ) is a weight vector with weight for all 2 Δ(g(1)). As g(1) has highest weight , the lowest weight for g(1) is . Now we set c = Σ t2Δ (z(n)) N ; tN ; t with Δ (z(n)) = f t 2 Δ(z(n)) j t 2 Δg. By Lemma 3.4.2, it follows that 2 Δ; in particular, Δ (z(n)) ̸= ∅. The normalization (H7) in Section 4.1 shows that N ; tN ; t < 0 for all t 2 Δ (z(n)). Therefore c ̸= 0. Then de ne Y l 2 g(1) by means of Y l = 1 c 1(X ) = 1 c Σ t2Δ (z(n)) N ; tX t X t : We claim that ~ 1jg(1)(Y l )(X) ̸= 0. By (4.2.1), the polynomial ~ 1jg(1)(Y l )(X) is ~ 1jg(1)(Y l )(X) = Y l ( 1(X)) = 1 c Σ t2Δ (z(n)) j2Δ(z(n)) N ; t (X t ; ad(X)X j ) (X t ;X j ) = 1 c Σ t2Δ (z(n)) N ; t (X t ; ad(X)X t): 51 Write X = Σ 2Δ(g(1)) X , where 2 n is the coordinate dual to X with respect to the Killing form . Then, ~ 1jg(1)(Y l )(X) = 1 c Σ t2Δ (z(n)) N ; t (X t ; ad(X)X t) = 1 c Σ 2Δ(g(1)) t2Δ (z(n)) N ; t (X t ; ad(X )X t) = 1 c Σ t2Δ (z(n)) N ; tN ; t = = (X;X ): (4.2.3) Hence ~ 1jg(1)(Y l )(X) ̸= 0. Since only g(1) contributes to the construction of the Ω1 systems, we simply refer to the Ω1 system as the Ω1jg(1) system. As we observed in Section 2.5, the operator Ω1jg(1) : g(1) ! D(Ls) n is obtained via the composition of maps g(1) ~ 1j !g(1) P1(g(1)) ! g(1) ,! U( n) R! D(Ls) n: By (4.2.3), we have ~ 1jg(1)(Y l )(X) = (X;X ). Therefore, Ω1(Y l ) = R(X ): Now, for all 2 Δ(g(1)), set Y = 1(X ): Then, as Y l = (1=c ) 1(X ), we have Ω1(Y ) = c R(X ): 52 Since both Ω1jg(1) and 1 are L0intertwining operators and g(1) = U(l)X , for any 2 Δ(g(1)), we obtain Ω1(Y ) = c R(X ) (4.2.4) with some constant c . Then, for Δ(g(1)) = f 1; : : : ; mg, the Ω1 system is given by R(X 1); : : : ;R(X m): The following theorem shows that the Ω1 system is conformally invariant on L0. Theorem 4.2.5 Let g be a complex simple Lie algebra, and let q be a maximal two step nilpotent parabolic subalgebra of nonHeisenberg type. Then the Ω1 system is conformally invariant on Ls if and only if s = 0. Proof. By Remark 2.5.11, we only need to show that the condition (S2) in De nition 2.1.4 holds if and only if s = 0. By Theorem 2.4.1, ( [ s(Y );R(X j )] f ) ( n) = ( R([(Ad( n1)Y )q;X j ] n) f ) ( n) + s q ( [Ad( n1)Y;X j ]q ) f( n) for any Y 2 g and any f 2 C1(N 0;C s ). Hence, the condition (S2) holds if and only if s = 0. 53 CHAPTER 5 Irreducible Decomposition of l z(n) Our next goal is to construct the Ω2 systems and to nd their special values. To do so, we need to detect the irreducible constituents V of l z(n) so that ~ 2jV is not identically zero. (see Section 2.5 for the general construction of the Ωk systems). In this chapter and the next one, we shall show preliminary results to nd such irreducible constituents. 5.1 Irreducible Decomposition of l z(n) We continue with q = l g(1) z(n) a maximal twostep nilpotent parabolic subalgebra of nonHeisenberg type listed in (3.3.2) or (3.3.3), and Q = LN = NG(q). The Levi subgroup L acts on l z(n) g g via the standard action on the tensor product induced by the adjoint representation on l and z(n). As L is complex reductive, this action is completely reducible. Since l = z(l) l ln with z(l) = CHq, we have l z(n) = ( CHq z(n) ) ( l z(n) ) ( ln z(n) ) : (5.1.1) It is clear that CHq z(n) = z(n) = g(2) as an Lmodule. Thus, by Corollary 3.2.5, CHq z(n) is Lirreducible. It is also easy to show that ln z(n) is Lirreducible. Let L (resp. Ln ) be the analytic subgroup of L with Lie algebra l (resp. ln ). As in Section 3.2, we call a weight for a nite dimensional Lmodule V a highest weight for V if the restriction jhss onto hss is a highest weight for V as an Lssmodule. Proposition 5.1.2 Suppose that ln ̸= 0. If n and are the highest weights of ln and z(n), respectively, then ln z(n) is the irreducible Lmodule with highest weight 54 n + . Proof. First we observe that Ln acts trivially on z(n). By Corollary 3.2.5, we have z(n) = g(2) = U(lss)X . By the observation made in Remark 3.3.5, it follows that ? for all 2 Δ(ln ). Thus z(n) = U(l )X . Hence Ln acts trivially; in particular, the irreducible Lmodule z(n) is L irreducible. On the other hand, it is clear that L acts on ln trivially. Therefore the representation (L; Ad Ad; ln z(n)) is equivalent to (L Ln ; Ad^ Ad; ln z(n)), where ^ denotes the outer tensor product. Since ln and z(n) have highest weight n and , respectively, the lemma follows. Now we focus on the decomposition of l z(n) into irreducible Lsubmodules. As noted in the proof for Lemma 5.1.2, the subgroup Ln acts trivially on l z(n). Hence we study l z(n) as an L module. For 2 h with ⟨ ; _⟩ 2 Z 0 for all 2 (l ), we will denote by V ( ) the irreducible constituent with highest weight jh , where h = h\l . For classical algebra g, we use the standard realization of the roots "i, the dual basis of the standard orthonormal basis for Rn. Theorem 5.1.3 The Lmodule l z(n) is reducible. If V ( ) denotes the irreducible representation of L with highest weight jh then the irreducible decomposition of l z(n) is given as follows. 1. Bn(i); 3 i n : l z(n) = 8>>< >>: V ( + ) V ( ) V ( + ("1 + "3)) if i = 3 V ( + ) V ( ) V ( + ("1 + "i)) V ( + ("2 + "3)) if 4 i n 2. Cn(i); 2 i n 1 : l z(n) 55 = 8>>< >>: V ( + ) V ( ) V ( + 2"2) if i = 2 V ( + ) V ( ) V ( + ("2 + "i)) V ( + ("1 + "2)) if 3 i n 1 3. Dn(i); 3 i n 3 : l z(n) = 8>>< >>: V ( + ) V ( ) V ( + ("1 + "3)) if i = 3 V ( + ) V ( ) V ( + ("1 + "i)) V ( + ("2 + "3)) if 4 i n 3 4. All exceptional cases (E6(3), E6(5), E7(2), E7(6), E8(1), F4(4)): l z(n) = V ( + ) V ( ) V ( + 0); where 0 is the following root contributing to z(n): E6(3) : 0 = 1 + 2 + 2 3 + 3 4 + 2 5 + 6 E6(5) : 0 = 1 + 2 + 2 3 + 3 4 + 2 5 + 6 E7(2) : 0 = 1 + 2 2 + 3 3 + 4 4 + 3 5 + 2 6 + 7 E7(6) : 0 = 1 + 2 2 + 2 3 + 4 4 + 3 5 + 2 6 + 7 E8(1) : 0 = 2 1 + 3 2 + 4 3 + 6 4 + 5 5 + 4 6 + 2 7 + 8 F4(4) : 0 = 1 + 2 2 + 4 3 + 2 4. 5.2 Technical Results on l z(n) In general, the study of tensor product decomposition of irreducible nite dimen sional representations is complicated. Techniques from representation theory and algebraic geometry have been used to study the problem (See for instance [21]). In our setting l = V ( ) and z(n) = V ( ), the standard techniques suffice to decompose 56 V ( ) V ( ) under L action. We have already observed that this action is com pletely reducible. The goal is to nd all the constituents and their multiplicities. To this end, it is enough to study V ( ) V ( ) as an l module. Our main technique is to analyze the character formula for l z(n) = V ( ) V ( ) as an l module. We will freely use the standard notions of dominant weights and regular weights. When we say that is Δ(l )dominant (resp. Δ(l )regular), we mean that ⟨ ; ⟩ 0 (resp. ⟨ ; ⟩ ̸= 0) for all 2 Δ+(l ). For V ( ), the nite dimensional l module with highest weight jh , and a weight 2 h , we denote by m ( ) the multiplicity jh in V ( ); that is, the dimension of the weight space V ( ) jh in V ( ). A weight is either Δ(l )regular or not. If is Δ(l )regular then no nontrivial element w in theWeyl group W(l ) of l xes . Otherwise, there is w ̸= 1 in W(l ) so that w = . Hence, if is a Δ(l )regular weight then there is a unique w 2 W(l ) so that w is Δ(l )dominant. We will write d( ) = w . De ne sgn( ) = 8>>< >>: 0 if some w ̸= 1 in W(l ) xes (1)l(w ) otherwise, where w 2 W(l ) so that w = d( ); where l(w ) is the length of w . We denote by (l ) half the sum of positive roots in Δ+(l ). Then if (resp. ′) is the character for V ( ) (resp. V ( ′)) then the character formula for the character ′ for the l module V ( ) V ( ′) is ′ = Σ ′′2Δ(V ( )) m ( ′′)sgn( ′′ + ′ + (l )) d( ′′+ ′+ (l )) (l ); (5.2.1) where Δ(V ( )) is the set of the weights for V ( ). This character formula is due to Klimyk [14, Corollary]. Among the standard facts, we use the following to analyze (5.2.1): (I) The constituent V ( + ′) occurs exactly once in V ( ) V ( ′). Moreover, if v 57 and v ′ are highest weight vectors of V ( ) and V ( ′), respectively, then v v ′ is a highest weight vector of V ( ) V ( ′). (II) If ′′ is the highest weight of some irreducible constituent of V ( ) V ( ′) then ′′ is of the form ′′ = + for some weight of V ( ′). (III) If all weights of V ( ) have multiplicity one then each irreducible constituent of V ( ) V ( ′) has multiplicity one. The unique irreducible constituent V ( + ′) is called the Cartan component of V ( ) V ( ′) (see for instance [21, page 1230]). In our setting l z(n) = V ( ) V ( ), the weights and are roots. By Fact (I) the highest weights of the irreducible constituents of l z(n) are of the form + j with j 2 Δ(z(n)). The character formula (5.2.1) is particularly simple when (l ) consists solely of long roots. We obtain a couple of results under this assumption. Lemma 5.2.2 Suppose that (l ) consists solely of long roots of g. If + j is not Δ(l )dominant then sgn( + j + (l )) = 0. Proof. We show that there exists 2 (l ) so that s xes + j + (l ). Since ⟨ (l ); _⟩ = 1 for all 2 (l ), it suffices to show that ⟨ + j ; _⟩ = 1 for some 2 (l ). Under our hypothesis + j is not Δ(l )dominant. Hence there exists 2 (l ) so that ⟨ + j ; _⟩ < 0. On the other hand, since is the highest weight of l , it follows that ⟨ ; _⟩ 0. We have ⟨ j ; _⟩ < ⟨ ; _⟩ 0; (5.2.3) and j + 2 Δ. Since (l ) contains only long roots, Lemma 3.4.4 shows that ⟨ j ; _⟩ = 1. Then (5.2.3) forces ⟨ ; _⟩ = 0, since ⟨ ; _⟩ is an integer. Therefore ⟨ + j ; _⟩ = 1. 58 Remark 5.2.4 If + j is Δ(l )dominant then + j + (l ) is Δ(l )dominant and Δ(l )regular. Hence, we have sgn( + j + (l )) = 1. Proposition 5.2.5 Suppose that (l ) consists solely of long roots of g. Then V ( + j) is an irreducible constituent of l z(n) if and only if + j is Δ(l )dominant. Proof. One of the directions is obvious. We then show that V ( + j) is an irreducible constituent if + j is Δ(l )dominant. By Klimyk's character formula, the character is of the form = Σ j2Δ(z(n)) m ( j)sgn( + j + (l )) d( + j+ (l )) (l ): (5.2.6) Since the weights of z(n) are roots of g, they have multiplicity one. Thus m ( j) = 1 for all j 2 Δ(z(n)). Moreover, Lemma 5.2.2 and Remark 5.2.4 show that sgn( + j + (l )) = 8>>< >>: 1 if + j is Δ(l )dominant 0 otherwise: Thus (5.2.6) is reduced to = Σ + j ; (5.2.7) where the sum runs over all j 2 Δ(z(n)) so that + j is Δ(l )dominant. Now the proposed assertion follows. Corollary 5.2.8 If (l ) consists solely of long roots of g then V ( ) occurs in the decomposition of l z(n) into irreducibles. Proof. By Lemma 3.4.2, we have 2 Δ(z(n)). Thus there exists j 2 Δ(z(n)) so that + j = . Since is Δ(l )dominant, the corollary follows from Proposition 5.2.5. Remark 5.2.9 Theorem 5.1.3 shows that V ( ) in fact occurs in l z(n) in every case. 59 5.3 Proof of Theorem 5.1.3 In the previous section we have shown that the character formula (5.2.1) is simple, when (l ) consists solely of long roots. Then in order to prove Theorem 5.1.3, we consider two cases, namely, Case 1: (l ) consists solely of long roots. Case 2: (l ) contains at least one short root. When g is simply laced, we regard any roots as long roots. Direct observation shows that the parabolic subalgebras q in (3.3.2) and (3.3.3) are then classi ed as follows: Case 1: Bn(i), Dn(i), E6(3), E6(5), E7(2), E7(6), E8(1) Case 2: Cn(i), F4(4) We start by proving Theorem 5.1.3 for parabolic subalgebras q in Case 1. Proof. [Proof for Theorem 5.1.3 for Case 1] Let be the set of all roots j 2 Δ(z(n)) so that + j is Δ(l )dominant. It follows from Fact (III) and Proposition 5.2.5 that the character is of the form = Σ j2 + j : (5.3.1) Moreover, Fact (I) and Corollary 5.2.8 show that V ( + ) and V ( ) occur in the decomposition. Therefore (5.3.1) might be expressed as = + + + Σ j2nf ; g + j : It remains to identify the roots in nf ; g. This is done in a case by case fashion. We include the computation for type E6(3). Other cases may be handled similarly. 60 The parabolic subalgebra q of type E6(3) corresponds to the deleted Dynkin dia gram ◦2 ◦ 1 3 ◦ 4 ◦ 5 ◦ 6: The subgraph corresponding to l is ◦ 2 ◦ 4 ◦ 5 ◦ 6: So the simple subalgebra l is isomorphic to sl(5;C). Write the fundamental weights of sl(5;C) corresponding to 2, 4, 5, 6 as ϖ1, ϖ2, ϖ3, ϖ4, respectively. The l module z(n) has highest weight . As ⟨ ; i⟩ = i;2 with i;2 the Kronecker delta for all i = 2; 4; 5; 6, we have z(n) = V (ϖ1). Thus, the adjoint representation l on z(n) is equivalent to the standard representation of sl(5;C) on C5. We then identify the weights of the adjoint action of l on z(n) with those of the standard action of sl(5;C) on C5; that is, Δ(z(n)) = f"1; "2; "3; "4; "5g: In terms of the fundamental weights we have "1 = ϖ1; "2 = ϖ1 + ϖ2; "3 = ϖ2 + ϖ3; "4 = ϖ3 + ϖ4; "5 = ϖ4: The highest weight of l is = ϖ1 + ϖ4. Therefore, the weights j 2 Δ(z(n)) that make + j Δ(l )dominant are j = ϖ1, ϖ4, or ϖ1 + ϖ2. Here, we have + ϖ1 = + , + (ϖ4) = ϖ1 = , and + (ϖ1 + ϖ2) = + 0 with 0 the root in Δ(z(n)) listed in Theorem 5.1.3. We next show Theorem 5.1.3 for parabolic subalgebras q in Case 2, namely, Cn(i) for 2 i n 1, and F4(4). Proof. [Proof for Theorem 5.1.3 for Case 2] The character formula of the tensor 61 product (5.2.6) is of the form = Σ j2Δ(z(n)) sgn( + j + (l )) d( + j+ (l )) (l ): (5.3.2) Here, we use the fact that m ( j) = 1 for j roots in z(n). Our strategy is to rst nd all j 2 Δ(z(n)) so that + j is Δ(l )dominant. We then consider the contributions from roots j with + j not Δ(l )dominant. The case Cn(i) for 2 i n 1 is demonstrated rst. Later, we handle the F4(4) case. Let q be of type Cn(i) for 2 i n 1. The deleted Dynkin diagram is ◦ 1 : : : ◦ i1 i ◦ i+1 ◦ n1 ks ◦ n and the subgraph corresponding to l is ◦ 1 ◦ 2 ◦ 3 : : : ◦ i1: (5.3.3) The data in Appendix C shows that Δ+(l ) = f"j "k j 1 j < k ig and Δ(z(n)) = f"j + "k j 1 j < k ig [ f2"j j 1 j ig: We have = "1 "i and = 2"1. If is the set of all j 2 Δ(z(n)) so that + j is Δ(l )dominant then, by Remark 5.2.4, the character may be written as = Σ j2 + j + Σ j2Δ(z(n))n sgn( + j + (l )) d( + j+ (l )) (l ): (5.3.4) One can see by direct computation that = 8>>>>>>< >>>>>>: f ; "1 + "2; 2"2g if i = 2 f ; "1 + "2; "1 + "3; "2 + "3g if i = 3 f ; "1 + "2; "1 + "i; "2 + "3; "2 + "ig if 4 i n 1 : 62 When i = 2, we have = Δ(z(n)), and so, is = Σ j2 + j = + + +("1+"2) + +(2"2): Since = "1 "2, we have + ("1 + "2) = 2"1 = . When i = 3, it follows that Δ(z(n))n = f2"2; 2"3g. Since we have s"1"2( + 2"2 + (l )) = + 2"2 + (l ) and s"2"3( + 2"3 + (l )) = + 2"3 + (l ), both weights are not Δ(l )regular and do not contribute to the character. Therefore, when i = 3, = Σ j2 + j = + + +("1+"2) + +("1+"3) + +("2+"3): Since = "1 "3, we have + ("1 + "3) = 2"1 = . If 4 i n 1 then j 2 Δ(z(n))n is "1 + "k for 3 k i 1; "2 + "k for 4 k i 1; "r + "k for 3 r < k i, or 2"r for 2 r i: An observation shows that, for each j 2 Δ(z(n))n with j ̸= 2"3, there exists w 2 W(l ) with w ̸= 1 so that w xes + j + (l ). Indeed, it is clear from (5.3.3) that l is of type Ai1. Thus (l ) is given by (l ) = Σi s=1 (i (2s 1) 2 ) "s: (5.3.5) If w = 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: s"k1"k when j = "1 + "k; "2 + "k s"r1"r when j = "r + "k s"1"2 when j = 2"2 s"r2"r when j = 2"r for 4 r i 1 s"i1"i when j = 2"i 63 then w( + j + (l )) = + j + (l ). Therefore sgn( + j + (l )) = 0 for such j . Now suppose that j = 2"3. We rst show that + 2"3 + (l ) is Δ(l )regular. By (5.3.5), we have + 2"3 + (l ) = (i + 1 2 ) "1 + (i 3 2 ) "2 + (i 1 2 ) "3 + Σi1 s=4 (i (2s 1) 2 ) "s + ( i + 1 2 ) "i: (5.3.6) The coefficients of "s and "t with s ̸= t in (5.3.6) are different. Since roots in Δ+(l ) are of the form "s "t with s < t, this shows that the weight + 2"3 + (l ) is Δ(l )regular. The re ection s"2"3 conjugates +2"3 + (l ) to the Δ(l )dominant weight s"2"3( + 2"3 + (l )) = + ("2 + "3) + (l ): Thus sgn( + j + (l )) = 1 and d( + j + (l )) = +("2 +"3)+ (l ); we have sgn( + j + (l )) d( + j+ (l )) (l ) = +("2+"3): Hence, = Σ j2 + j + Σ j2Δ(z(n))n sgn( + j + (l )) d( + j+ (l )) (l ) = Σ j2 + j +("2+"3) (5.3.7) with = f ; "1 + "2; "1 + "i; "2 + "3; "2 + "ig for 4 i n 1. Then we obtain = Σ j2 + j +("2+"3) = + + +("1+"2) + +("1+"i) + +("2+"3) + +("2+"i) +("2+"3) = + + +("1+"2) + +("1+"i) + +("2+"i): Since = "1 "i, we have + ("1 + "i) = 2"1 = . 64 Next we consider the case that q is of type F4(4). The deleted Dynkin diagram is ◦ 1 ◦ 2 +3 ◦ 3 4 and the subgraph corresponding to l is ◦ 1 ◦ 2 +3 ◦ 3: The simple subalgebra l is isomorphic to so(7;C). If we write the fundamental weights of l = so(7;C) corresponding to 1, 2, 3 as ϖ1, ϖ2, ϖ3, respectively, then the highest weights for l and for z(n) are written in terms of the fundamental weights as = ϖ2 and = ϖ1; we have l = V (ϖ2) and z(n) = V (ϖ1). Therefore the adjoint action of l on itself (resp. on z(n)) is equivalent to the standard action of so(7;C) on ^2C7 (resp. on C7). We then identify the l module l z(n) as the so(7;C)module (^2C7) (C7), and consider the irreducible decomposition of (^2C7) (C7). Let Δ+ be the standard choice of a positive system of so(7;C) and be half the sum of the positive roots; that is, Δ+ = f"1 "2; "2 "3; "1 "3g [ f"1; "2; "3g and = 5 2 "1 + 3 2 "2 + 1 2 "3: If = f 2 Δ(C7) j ϖ2 + is dominantg with Δ(C7) the set of weights for C7 then the character ϖ2 ϖ1 for (^2C7) (C7) = 65 V (ϖ2) V (ϖ1) is ϖ2 ϖ1 = Σ 2Δ(C7) mϖ1( )sgn(ϖ2 + + ) d(ϖ2+ + ) = Σ 2Δ(C7) sgn(ϖ2 + + ) d(ϖ2+ + ) = Σ 2 ϖ2+ + Σ 2Δ(C7)n sgn(ϖ2 + + ) d(ϖ2+ + ) : We need determine the contributions from 2 Δ(C7)n. The weights for C7 under the standard action of so(7;C) are Δ(C7) = f "1; "2; "3; 0g: In terms of the fundamental weights ϖ1, ϖ2, and ϖ3, we have "1 = ϖ1; "2 = ϖ1 + ϖ2; "3 = ϖ2 + 2ϖ3: Therefore, the weights for C7 may be written in terms of the fundamental weights as Δ(C7) = f ϖ1; (ϖ1 + ϖ2); (ϖ2 + 2ϖ3); 0g: If is a weight for C7 so that ϖ2 + is Δ(l )dominant then must be = ϖ1;ϖ2 ϖ2;ϖ2 + 2ϖ3; or 0: (5.3.8) Thus, Δ(C7)n = fϖ1;ϖ1 + ϖ2;ϖ2 2ϖ3g = f"1; "2;"3g: Observe that when = "1 or "2, there exists a Weyl group element w 2 W of so(7;C) that xes ϖ2+ + . Indeed, for either case = "1 or "2, the root re ection s"1"2 xes ϖ2 + + , as ϖ2 = "1 + "2. Thus sgn(ϖ2 + + ) = 0 when = "1 or "2. On the other hand, when = "3, we have ϖ2 "3 + = 7 2 "1 + 5 2 "2 1 2 "3: (5.3.9) 66 The coefficients of "s and "t with s ̸= t in (5.3.9) are different. Since roots in Δ+ are of the form "s "t with s < t or "s, this shows that the weight ϖ2 "3 + is Δ(l )regular. The re ection s"3 conjugates ϖ2"3+ to the Δ(l )dominant weight s"3(ϖ2 "3 + ) = 7 2 "1 + 5 2 "2 + 1 2 "3: Thus sgn(ϖ2 "3 + ) = 1 and d(ϖ2 "3 + ) = "1 + "2 = ϖ2; we have sgn(ϖ2 "3 + ) d(ϖ2"3+ ) = ϖ2 : Hence, ϖ2 ϖ1 = Σ 2 ϖ2+ + Σ 2Δ(C7)n sgn(ϖ2 + + ) d(ϖ2+ + ) = Σ 2 ϖ2+ ϖ2 : By (5.3.8), we have = fϖ1;ϖ2 ϖ2;ϖ2 + 2ϖ3; 0g. Therefore, ϖ2 ϖ1 = Σ 2 ϖ2+ ϖ2 = ϖ2+ϖ1 + ϖ2+(ϖ1ϖ2) + ϖ2+(ϖ2+2ϖ3): We have ϖ2+ϖ1 = + , ϖ2+(ϖ1ϖ2) = ϖ1 = , and ϖ2+(ϖ2+2ϖ3) = + 0 with 0 the root in Δ(z(n)) in Theorem 5.1.3. This completes the proof. 67 CHAPTER 6 Special Constituents of l z(n) In this chapter, by using the decomposition results in Chapter 5, we shall determine the candidates of the irreducible constituents of l z(n) that will contribute to the Ω2 systems; that is, the irreducible constituents V ( ) so that ~ 2jV ( ) are not identically zero. 6.1 Special Constituents Given V ( ), an irreducible constituent in l z(n), we build an Lintertwining map ~ 2jV ( ) 2 HomL(V ( ) ;P2(g(1))) with V ( ) the dual of V ( ) with respect to the Killing form . From ~ 2jV ( ) , we construct operator Ω2jV ( ) : V ( ) ! D(Ls) n. To do so, it is necessary to determine which irreducible constituents V ( ) have property that ~ 2jV ( ) ̸= 0. We start by observing the vector space isomorphism P2(g(1)) = Sym2(g(1)) . With the natural Laction on P2(g(1)) and Sym2(g(1)) , this vector space isomor phism is Lequivariant. Thus, if ~ 2 V ( ) is a nonzero map then V ( ) is an irreducible constituent of Sym2(g(1)) g(1) g(1); in particular, by Fact (II) in Section 5.2, is of the form = + ϵ for some ϵ 2 Δ(g(1)), where is the highest weight of g(1). One can see from the decompositions in Theorem 5.1.3 that V ( ) is an irreducible constituent of l z(n) for any q under consideration. By Lemma 3.4.2, we have = + ϵ for some ϵ 2 Δ(g(1)). Now we claim that ~ 2jV ( ) is identically zero. It is 68 wellknown that g(1) g(1) = Sym2(g(1)) ^2(g(1)) (6.1.1) as an Lmodule. Since each weight for g(1) is a root of g, by Fact (III) in Section 5.2, the Lmodule decomposition (6.1.1) is multiplicity free. Proposition 6.1.2 The Lmodule V ( ) is an irreducible constituent of ^2(g(1)). Proof. De ne a linear map φ : z(n) ! ^2(g(1)) by means of φ(W) = Σ 2Δ(g(1)) ad(W)X ^ X : By using an argument similar to that for Lemma 2.5.4, one can show that φ is L equivariant. Then, since z(n) = V ( ) as an irreducible Lmodule, it suffices to show that φ is a nonzero map. Write Δ (g(1)) = f 2 Δ(g(1)) j 2 Δg. By Lemma 3.4.2, we have 2 Δ. Hence Δ (g(1)) ̸= ∅. By writing ′ = for 2 Δ (g(1)), φ(X ) is given by φ(X ) = Σ 2Δ(g(1)) ad(X )X ^ X = Σ 2Δ (g(1)) N ; X ′ ^ X : Observe that for each 2 Δ (g(1)), we have 2 Δ (g(1)). Moreover, by Property (H6) of our normalizations in Section 4.1, it follows that N ; ′ = N ; . Therefore, N ; X ′ ^ X + N ; ′X ^ X ′ = 2N ; X ′ ^ X : (6.1.3) Since N ; ̸= 0 for 2 Δ (g(1)), equation (6.1.3) is nonzero. On the other hand, if 2 Δ (g(1)) and 2 Δ (g(1)) is so that ̸= ; ′ then X ′ ^ X and X ^ X are linearly independent. Hence, φ(X ) ̸= 0. De nition 6.1.4 An irreducible constituent V ( ) of l z(n) is called special if ̸= and there exists ϵ 2 Δ(g(1)) so that = +ϵ, where and are the highest weights for g(1) and z(n), respectively. 69 Proposition 6.1.5 Let V ( ) be an irreducible constituent of l z(n). Then ~ 2 V ( ) is not identically zero only if V ( ) is a special constituent of l z(n). Proof. At the beginning of this section we observed that if ~ 2jV ( ) ̸= 0 then must be of the form = + ϵ for some ϵ 2 Δ(g(1)). Then V ( ) is either a special constituent or V ( ) (by Lemma 3.4.2, satis es the form). However, by Proposition 6.1.2, it follows that ~ 2jV ( ) is identically zero. Therefore, V ( ) must be a special constituent. We will show in Chapter 7 that the converse of Proposition 6.1.5 also holds for certain special constituents (see Proposition 7.1.6). 6.2 Types of Special Constituents The aim of this section is to determine and classify all the special constituents of l z(n). Such a classi cation will play a role in the explicit construction of the Ω2 systems. We use the decomposition results in Chapter 5 for the rest of this chapter. The parabolic subalgebra q under consideration is assumed to be one in (3.3.2) or (3.3.3). Since l z(n) = (CHq z(n)) (lss z(n)) and CHq z(n) = V ( ), it suffices to consider lss z(n) = (l z(n)) (ln z(n)). We start by observing that, by Proposition 5.1.2, ln z(n) = V ( n + ). Proposition 6.2.1 Suppose that ln ̸= 0. Then the irreducible constituent V ( n + ) is special. Proof. We need to show that n + = + for some 2 Δ(g(1)). This is precisely the statement (1) of Lemma 3.4.5. We next investigate the Cartan component V ( + ) of l z(n) = V ( ) V ( ). 70 Lemma 6.2.2 The Cartan component V ( + ) of l z(n) is not special. Proof. Lemma 3.4.5 and Remark 3.4.7 show that + =2 Δ(g(1)), which implies that + ̸= + for all 2 Δ(g(1)). We determine all the special constituents of l z(n) in two steps. First we assume that g is a classical algebra, and then consider the case that g is an exceptional algebra. For classical cases the parabolic subalgebras q under consideration are of type Bn(i) (3 i n), Cn(i) (2 i n 1), or Dn(i) (3 i n 3). It will be convenient to write 2 Δ(g(1)) in terms of the fundamental weights of l and ln . It is clear from the deleted Dynkin diagrams that, for each of the cases, (l ) and (ln ) are given by (l ) = f r j 1 r i 1g and (ln ) = f i+s j 1 s n ig; where j are the simple roots with the standard numbering. By using the standard realizations of roots, we have r = "r "r+1 for 1 r i 1, i+s = "i+s "i+s+1 for 1 s n i 1, and n = 8>>>>>>< >>>>>>: "n if g is of type Bn 2"n if g is of type Cn "n1 + "n if g is of type Dn. The data in Appendix C shows that Δ(g(1)) is Δ(g(1)) = 8>>< >>: f"j "k j 1 j i and i + 1 k ng [ f"j j 1 j ig if q is of type Bn(i) f"j "k j 1 j i and i + 1 k ng if q is of type Cn(i) or Dn(i). 71 Since we have two simple algebras l and ln , we use the notation ϖr for the funda mental weights of r 2 (l ) and ~ϖs for those of i+s 2 (ln ). Direct computation then shows that each 2 Δ(g(1)) is exactly one of the following form: = 8>>>>>>< >>>>>>: ϖ1 + Σni s=1 ~ms ~ϖs; (ϖr + ϖr+1) + Σni s=1 ~ms ~ϖs with 1 r i 2, or ϖi1 + Σni s=1 ~ms ~ϖs (6.2.3) for some ~ms 2 Z. Proposition 6.2.4 Let V ( ) be an irreducible constituent of l z(n). 1. If q is of type Bn(i) (3 i n) or Dn(i) (3 i n3) then V ( ) is a special constituent if and only if = 2"1. 2. If q is of type Cn(i) (2 i n 1) then V ( ) is a special constituent if and only if = "1 + "2. Proof. Suppose that q is of type Bn(i), Cn(i), or Dn(i). By De nition 6.1.4, we need to nd all of the form = + for some 2 Δ(g(1)). Here , the highest weight for g(1), is = 8>>< >>: "1 + "i+1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) "1 if q is of type Bn(n): We write in terms of the fundamental weights of l and ln ; that is, = 8>>< >>: ϖ1 + ~ϖ1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) ϖ1 if q is of type Bn(n); (6.2.5) where ϖ1 and ~ϖ1 are the fundamental weights of 1 = "1 "2 and i+1 = "i+1 "i+2, respectively. As ln acts trivially on both l and z(n), the highest weight for a 72 constituent V ( ) l z(n) is of the form = Σi1 j=1 njϖj for nj 2 Z 0. (6.2.6) If there exists 2 Δ(g(1)) so that = + then (6.2.5) and (6.2.6) imply that = is of the form = 8>>< >>: (n1 1)ϖ1 + Σi1 j=2 njϖj ~ϖ1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) (n1 1)ϖ1 + Σi1 j=2 njϖj if q is of type Bn(n) (6.2.7) for nj 2 Z 0. On the other hand, we observed that the root must be one of the forms in (6.2.3). Then observation shows that if satis es both (6.2.3) and (6.2.7) then must be = 8>>< >>: ϖ1 ~ϖ1 or (ϖ1 + ϖ2) ~ϖ1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) ϖ1 or (ϖ1 + ϖ2) if q is of type Bn(n): Therefore = + is = 2ϖ1 or ϖ2, which shows that = 2"1 or "1 + "2. As = "1 "i for q of type Bn(i), Cn(i), or Dn(i), Theorem 5.1.3 shows that both V (2"1) and V ("1+"2) occur in l z(n). Now the assertions follow from the fact that the highest root of g is = "1 + "2 if g is of type Bn or Dn, and = 2"1 if g is of type Cn. If g is an exceptional algebra then the parabolic subalgebras q under consideration are E6(3);E6(5);E7(2);E7(6);E8(1); and F4(4): (6.2.8) Lemma 6.2.9 If q is of exceptional type as in (6.2.8) then V ( + 0) in Theorem 5.1.3 is a special constituent. Proof. This is done by a direct computation. The roots ϵ in Δ(g(1)) so that + 0 = + ϵ are given in Table 6.4 below. 73 Proposition 6.2.10 There exists a unique special constituent in l z(n). Proof. If q is of classical type then this proposition follows from Proposition 6.2.4. For q of exceptional type, by Theorem 5.1.3, the tensor product l z(n) decomposes into l z(n) = V ( + ) V ( ) V ( + 0) with 0 2 Δ(n) as in Theorem 5.1.3. Then Lemma 6.2.2 and Lemma 6.2.9 show that V ( + 0) is the unique special constituent. Since the weight ϵ 2 Δ(g(1)) so that + ϵ is the highest weight of a special constituent will play a role later, we introduce the notation related to ϵ. De nition 6.2.11 We denote by ϵ the root contributing to g(1) so that V ( + ϵ ) is the special constituent of l z(n). Similarly, we denote by ϵn the root for g(1) so that V ( + ϵn ) = ln z(n). In Table 6.1, Table 6.2, Table 6.3, and Table 6.4 we summarize the results of this section. Table 6.1 and Table 6.2 contain the highest weight of each special constituent occurring in l z(n) for each parabolic q of classical algebras and exceptional algebras. Table 6.3 and Table 6.4 list the roots , ϵ , and ϵn for each q. A dash indicates that no special constituent exists for the case. Table 6.1: Highest Weights for Special Constituents (Classical Cases) Type V ( + ϵ ) V ( + ϵn ) Bn(i); 3 i n 2 2"1 "1 + "2 + "i+1 + "i+2 Bn(n 1) 2"1 "1 + "2 + "n Bn(n) 2"1 Cn(i); 2 i n 1 "1 + "2 2"1 + 2"i+1 Dn(i); 3 i n 3 2"1 "1 + "2 + "i+1 + "i+2 74 Table 6.2: Highest Weights for Special Constituents (Exceptional Cases) Type V ( + ϵ ) V ( + ϵn ) E6(3) 1 + 2 2 + 2 3 + 4 4 + 3 5 + 2 6 2 1 + 2 2 + 2 3 + 3 4 + 2 5 + 6 E6(5) 2 1 + 2 2 + 3 3 + 4 4 + 2 5 + 6 1 + 2 2 + 2 3 + 3 4 + 2 5 + 2 6 E7(2) 2 1 + 2 2 + 4 3 + 5 4 + 4 5 + 3 6 + 2 7 E7(6) 2 1 + 3 2 + 4 3 + 6 4 + 4 5 + 2 6 + 7 2 1 + 2 2 + 3 3 + 4 4 + 3 5 + 2 6 + 2 7 E8(1) 2 1 + 4 2 + 5 3 + 8 4 + 7 5 + 6 6 + 4 7 + 2 8 F4(4) 2 1 + 4 2 + 6 3 + 2 4 Table 6.3: The Roots , ϵ , and ϵn (Classical Cases) Type ϵ ϵn Bn(i); 3 i n 2 "1 + "i+1 "1 "i+1 "2 + "i+2 Bn(n 1) "1 + "n "1 "n "2 Bn(n) "1 "1 Cn(i); 2 i n 1 "1 + "i+1 "2 "i+1 "1 + "i+1 Dn(i); 3 i n 3 "1 + "i+1 "1 "i+1 "2 + "i+2 75 Table 6.4: The Roots , ϵ , and ϵn (Exceptional Cases) Type ϵ ϵn E6(3) 2 + 3 + 2 4 + 5 + 6 1 + 2 + 3 + 4 E6(5) 1 + 2 + 3 + 2 4 + 5 2 + 4 + 5 + 6 E7(2) 1 + 2 + 2 3 + 2 4 + 5 + 6 + 7 E7(6) 1 + 2 + 2 3 + 3 4 + 2 5 + 6 1 + 3 + 4 + 5 + 6 + 7 E8(1) 1 + 2 + 2 3 + 3 4 + 3 5 + 3 6 + 2 7 + 8 F4(4) 1 + 2 2 + 3 3 + 4 with E6(3) : = 1 + 2 + 3 + 2 4 + 2 5 + 6 E6(5) : = 1 + 2 + 2 3 + 2 4 + 5 + 6 E7(2) : = 1 + 2 + 2 3 + 3 4 + 3 5 + 2 6 + 7 E7(6) : = 1 + 2 2 + 2 3 + 3 4 + 2 5 + 6 + 7 E8(1) : = 1 + 3 2 + 3 3 + 5 4 + 4 5 + 3 6 + 2 7 + 8 F4(4) : = 1 + 2 2 + 3 3 + 4 76 By Proposition 6.1.5, only special constituents could contribute to the construc tion of the Ω2 systems. Next we want to show that ~ 2jV ̸= 0 when V is a special constituent. An observation on the highest weights for the special constituents will simplify the argument. We classify them by their highest weights and call them type 1a, type 1b, type 2, and type 3. De nition 6.2.12 We say that a special constituent V ( ) of l z(n) is of 1. type 1a if = + ϵ is not a root with ϵ ̸= and both and ϵ are long roots, 2. type 1b if = + ϵ is not a root with ϵ ̸= and either or ϵ is a short root, 3. type 2 if = + ϵ = 2 is not a root, or 4. type 3 if = + ϵ is a root, where is the highest weight for g(1) and ϵ = ϵ or ϵn is the root in Δ(g(1)) de ned in De nition 6.2.11. Example 6.2.13 The following are examples of each type of special constituents: 1. type 1a: V ( + ϵ ) for type Bn(n 1) ( + ϵ = ("1 + "n) + ("1 "n) ) 2. type 1b: V ( + ϵn ) for type Bn(n 1) ( + ϵn = ("1 + "n) + ("2) ) 3. type 2: V ( + ϵn ) for type Cn(i) ( + ϵn = 2("1 + "i+1) = 2 ) 4. type 3: V ( + ϵ ) for type Cn(i) ( + ϵ = "1 + "2 ) Table 6.5 summarizes the types of special constituents for each parabolic subagle bra q. One may want to observe that almost all the special constituents are of type 1a. We regard any roots as long roots, if g is simply laced. A dash indicates that no special constituent exists in the case. 77 Table 6.5: Types of Special Constituents Type V ( + ϵ ) V ( + ϵn ) Bn(i); 3 i n 2 Type 1a Type 1a Bn(n 1) Type 1a Type 1b Bn(n) Type 2 Cn(i); 2 i n 1 Type 3 Type 2 Dn(i); 3 i n 3 Type 1a Type 1a E6(3) Type 1a Type 1a E6(5) Type 1a Type 1a E7(2) Type 1a E7(6) Type 1a Type 1a E8(1) Type 1a F4(4) Type 2 Remark 6.2.14 It is observed from Table 6.3 and Table 6.4 that we have ϵ =2 Δ, unless V ( + ϵ) is of type 3. Remark 6.2.15 Table 6.5 shows that when V ( + ϵ) is a special constituent of type 1a, the parabolic subalgebra q is of type Bn(i) (3 i n 1), Dn(i), E6(3), E6(5), E7(2), E7(6), or E8(1). The data in Appendix C shows that when q is of type Bn(i) for 3 i n 1, the simple root q = "i "i+1 that parametrizes q is a long root and Δ(z(n)) contains solely long roots. Since we regard any roots as long roots for g simply laced, it follows that when V ( + ϵ) is of type 1a, the simple root q and any root j 2 Δ(z(n)) are all long roots. 78 6.3 Technical Results In this section we collect technical results on the special constituents, so that certain arguments will go smoothly in Chapter 7. The weight vectors X and the structure constants N ; are normalized as in Section 4.1. Lemma 6.3.1 Let V ( +ϵ) be a special constituent l z(n) of type 1a, and 2 Δ+(l). If ϵ + 2 Δ then 2 Δ. Proof. We show that ⟨ ; ⟩ > 0. Since + ϵ is the highest weight of an irreducible lmodule, it is Δ(l)dominant. Thus, ⟨ + ϵ; ⟩ = ⟨ ; ⟩ + ⟨ϵ; ⟩ 0: (6.3.2) Observe that, as + ϵ is of type 1a, ϵ is a long root of g. Since + ϵ is assumed to be a root, Lemma 3.4.4 implies that ⟨ ; ϵ_⟩ = 1; in particular, ⟨ϵ; ⟩ < 0. Now, by (6.3.2), we have ⟨ ; ⟩ ⟨ϵ; ⟩ > 0: Lemma 6.3.3 Let V ( + ϵ) be a special constituent of l z(n) of type 1a. Then, for 2 Δ+(l) with + ϵ 2 Δ, we have ad(X )ad(X +ϵ)X j = 0 for all j 2 Δ(z(n)). Proof. If ( + ϵ) j =2 Δ then there is nothing to prove. So we assume that ( + ϵ) j 2 Δ and + ( + ϵ) j 2 Δ. Since + ϵ is assumed to be of type 1a, the root is long. Lemma 3.4.4 then implies that ⟨( + ϵ) j ; _⟩ = 1: (6.3.4) 79 By Remark 6.2.14, we have ⟨ϵ; _⟩ = 0. Thus (6.3.4) becomes ⟨ ; _⟩ ⟨ j ; _⟩ = 1: (6.3.5) Since is the highest weight for g(1), j 2 Δ(z(n)), and 2 Δ+(l), neither + nor j + is a root. Then, as is a long root, (6.3.5) holds if and only if ⟨ ; _⟩ = 0 and ⟨ j ; _⟩ = 1. On the other hand, since + ϵ is a root by hypothesis and by Lemma 6.3.1, is a root. In particular, by Lemma 3.4.4, ⟨ ; _⟩ = 1. Now we have ⟨ ; _⟩ = 1 and ⟨ ; _⟩ = 0, which is a contradiction. For any ad(h)invariant subspace W g and any weight 2 h , we write Δ (W) = f 2 Δ(W) j 2 Δg: In Chapter 7, we will construct the Ω2jV ( +ϵ) systems and nd their special values, when V ( +ϵ) is of either type 1a or type 2. When we do so, the roots 2 Δ +ϵ(g(1)) and j 2 Δ +ϵ(z(n)) will play a role. Therefore, for the rest of this section, we shall show several technical results about those roots, so that certain argument will become simple. First of all, we need check that Δ +ϵ(g(1)) and Δ +ϵ(z(n)) are not empty. It is clear that Δ +ϵ(g(1)) ̸= ∅, since , ϵ 2 Δ +ϵ(g(1)). Moreover, Lemma 6.3.6 below shows that when V ( + ϵ) is of type 2, we have Δ +ϵ(g(1)) = f g. Lemma 6.3.6 If V ( +ϵ) is a special constituent of l z(n) of type 2 then Δ +ϵ(g(1)) = f g. Proof. First we claim that has the maximum height among the roots 2 Δ(g(1)). As g(1) is the irreducible Lmodule with highest weight , any root 2 Δ(g(1)) is of the form = Σ 2 (l) n with n 2 Z 0. Then if ht( ) and ht( ) denote the 80 heights of and , respectively, then ht( ) = ht( ) + Σ 2 (l) n ht( ): Now as V ( + ϵ) is of type 2, by de nition, we have + ϵ = 2 . If 2 Δ2 (g(1)) then 2 2 Δ(g(1)). In particular, the height ht(2 ) satis es ht( ) ht(2 ). If = Σ 2 (l) n with n 2 Z 0 then ht( ) ht(2 ) = 2ht( ) ht( ) = 2ht( ) ht( ) + Σ 2 (l) n = ht( ) + Σ 2 (l) n : This forces that Σ 2 (l) n = 0. Therefore = . Lemma 6.3.7 If V ( + ϵ) is a special constituent of l z(n) then Δ +ϵ(z(n)) ̸= ∅. Proof. By Fact (II) in Section 5.2, the highest weight + ϵ of V ( + ϵ) l z(n) is of the form + ϵ = 8>>< >>: + ′ if V ( + ϵ) l z(n) n + ′′ if V ( + ϵ) = ln z(n) for some ′; ′′ 2 Δ(z(n)), where and n are the highest weights for l and ln , respectively. Then we have ′; ′′ 2 Δ +ϵ(z(n)). The following simple technical lemma will simplify an argument in later proofs. Lemma 6.3.8 Let ; ; 2 Δ with , ̸= . If + =2 Δ and + 2 Δ then the following hold: (1) ; 2 Δ, and (2) N ; N ; = N ; N ; . Proof. For the rst assertion, we show that 2 Δ. Suppose that =2 Δ, so ⟨ ; ⟩ 0. By hypothesis, we have ⟨ ; ⟩ 0. Thus it follows that ⟨ + ; ⟩ = ⟨ ; ⟩ + ⟨ ; ⟩ ⟨ ; ⟩ > 0: 81 Therefore, = ( + ) is a root. Now let X , X , and X be the root vectors of , , and , respectively, normalized as in Section 4.1. Since 2 Δ, we have N ; ̸= 0 (see Property (H7) in Section 4.1). Moreover, the conditions that ; + 2 Δ imply that N ; ̸= 0. On the other hand, we have [X ;X ] = 0 by assumption, and [X ;X ] = 0 by hypothesis. So it follows from the Jacobi identity that 0 = [X ; [X ;X ]] = [X ; [X ;X ]] = N ; N ; X + ̸= 0; which is absurd. Therefore 2 Δ. Since it may be shown similarly that 2 Δ, we omit the proof. Observe that the condition + =2 Δ implies that ad(X )ad(X ) = ad(X )ad(X ) by the Jacobi identity. Therefore, ad(X )ad(X )X = ad(X )ad(X )X , which implies that N ; N ; = N ; N ; : Lemma 6.3.9 Let W be any ad(h)invariant subspace of g with Δ +ϵ(W)nf ; ϵg ̸= ∅. If V ( + ϵ) is a special constituent of l z(n) of type 1a, type 1b, or type 2 then, for any 2 Δ +ϵ(W)nf ; ϵg, we have , ϵ 2 Δ. Proof. If V ( + ϵ) is of type 1a, type 1b, or type 2 then, by de nition, + ϵ is not a root. Then this lemma simply follows from Lemma 6.3.8 Remark 6.3.10 A direct observation shows that if V ( + ϵ) is a special constituent of type 1a then Δ +ϵ(g(1))nf ; ϵg ̸= ∅. Lemma 6.3.11 If V ( + ϵ) is a special constituent of l z(n) of type 1a then, for any 2 Δ +ϵ(g(1)) and any j 2 Δ +ϵ(z(n)), we have j 2 Δ. Proof. By Lemma 6.3.8, we have j ; j ϵ 2 Δ. So, let ̸= ; ϵ. We show that ⟨ j ; ⟩ > 0. Observe that since 2 Δ(g(1)) and j 2 Δ(z(n)), we have j + =2 Δ. 82 Thus ⟨ j ; ⟩ 0. Since 2 Δ +ϵ(g(1))nf ; ϵg and j 2 Δ +ϵ(z(n)), by Lemma 6.3.9, we have , ϵ j 2 Δ. Then we rst claim that if ⟨ j ; ⟩ = 0 then ( ) + (ϵ j) 2 Δ. Since V ( + ϵ) is assumed to be of type 1a, both and ϵ are long roots. Thus, by Lemma 3.4.4, ⟨ j ; _⟩ = ⟨ ; ϵ_⟩ = 1; in particular, ⟨ j ; ⟩, ⟨ ; ϵ⟩ > 0. By Remark 6.2.14, we have ⟨ ; ϵ⟩ = 0. Then, ⟨ ; ϵ j⟩ = ⟨ ; j⟩ ⟨ ; ϵ⟩ < 0: Therefore, as ; ϵ j 2 Δ, it follows that ( ) + (ϵ j) 2 Δ. On the other hand, since ⟨ ; ϵ⟩ = 0 and ⟨ j ; ⟩ is assumed to be 0, we have jj( ) + (ϵ j)jj2 = jj jj2 + jj jj2 + jjϵjj2 + jj j jj2 2⟨ ; ⟩ 2⟨ ; ϵ⟩ 2⟨ j ; ⟩ 2⟨ j ; ϵ⟩: For = ; j and = ; ϵ, by Lemma 3.4.4, we have ⟨ ; _⟩ = 2⟨ ; ⟩=jj jj2 = 1, as and ϵ are long roots. Therefore, 2⟨ ; ⟩ = jj jj2, and so, jj( ) + (ϵ j)jj2 = jj jj2 + jj j jj2 jj jj2 jjϵjj2: Since and ϵ are assumed to be long roots, this shows that jj( )+(ϵ j)jj2 0, which contradicts that ( ) + (ϵ j) is a root. Hence, ⟨ j ; ⟩ > 0. Lemma 6.3.12 If V ( + ϵ) is a special constituent of l z(n) of type 1a or type 2 then, for any j 2 Δ +ϵ(z(n)), Δ +ϵ(g(1)) Δ j (g(1)): In particular, Δ j (g(1)) ̸= ∅ for any j 2 Δ +ϵ(z(n)). Proof. If V ( + ϵ) is of type 1a then the assertion follows from Lemma 6.3.11. If V ( + ϵ) is of type 2 then Lemma 6.3.6 implies that Δ +ϵ(g(1)) = f g. Now this lemma follows from Lemma 6.3.9 by taking = j 2 Δ +ϵ(z(n)). 83 If V ( + ϵ) is a special constituent of l z(n) then, for 2 Δ, we write ( ) = ( + ϵ) : Lemma 6.3.13 If V ( + ϵ) is a special constituent of l z(n) of type 1a or type 2 then, for any j 2 Δ +ϵ(z(n)), Δ ( j )(g(1)) ̸= ∅: Proof. Since j 2 Δ +ϵ(z(n)), we have ( + ϵ) j 2 Δ. As V ( + ϵ) is assumed to be of type 1a or type 2, by de nition, it follows that + ϵ =2 Δ. Thus, by Lemma 6.3.8, we have j 2 Δ and ϵ j 2 Δ. Then, ( j) = ( + ϵ) j = ϵ j 2 Δ; that is, 2 Δ ( j )(g(1)). Lemma 6.3.14 If V ( + ϵ) is a special constituent of type 1a or type 2 then Σ j2Δ +ϵ(z(n)) N ;ϵ jN ; jϵNϵ; jNϵ; j > 0; where N ; are the structure constants for ; 2 Δ, de ned in Section 4.1. Proof. It follows from Property (H7) of our normalizations in Section 4.1 that N ;ϵ jN ; jϵ = q ;ϵ j (1 + p ;ϵ j ) 2 jj jj2 and Nϵ; jNϵ; j = qϵ; j (1 + pϵ; j ) 2 jjϵjj2: In particular, by (4.1.1) in Section 4.1, we have N ;ϵ jN ; jϵ 0 and Nϵ; jNϵ; j 0. By Lemma 6.3.7 and Lemma 6.3.9, Δ +ϵ(z(n)) ̸= ∅ and j ϵ 2 Δ for any j 2 Δ +ϵ(z(n)). Therefore, for all j 2 Δ +ϵ(z(n)), we have N ;ϵ jN ; jϵNϵ; jNϵ; j > 0: 84 Lemma 6.3.15 If V ( + ϵ) is a special constituent of type 1a then, for any 2 Δ +ϵ(g(1))nf ; ϵg and any j 2 Δ +ϵ(z(n)), [X j ;X ] = [X ( j );X ] = 0: Proof. We show that j + and ( j)+ are neither zero nor roots. First of all, if j + = 0 then j = 2 Δ(l), which contradicts that j 2 Δ(z(n)). Next, if ( j) + = 0 then since ( j) + = ϵ + j , we would have + ϵ = j 2 Δ. On the other hand, as V ( + ϵ) is assumed to be of type 1a, ϵ is a long root. As 2 Δ +ϵ(g(1))nf ; ϵg, by Lemma 6.3.9, we have ϵ 2 Δ. Then, by Lemma 3.4.4, it follows that + ϵ ̸2 Δ, which is a contradiction. To show j + is not a root, observe that, by Lemma 3.4.4, we have ⟨ j + ; _⟩ = 1 + 1 2 = 2: Thus, if j + 2 Δ then ( j + ) + 2 would be a root. However, since is a long root, it is impossible. The fact that ( j) + =2 Δ can be shown in a similar manner. Lemma 6.3.16 If V ( + ϵ) is a special constituent of type 1a then, for any 2 Δ +ϵ(g(1))nf ; ϵg and any j 2 Δ +ϵ(z(n)), p j; = 0 and q j; = 1; where p ; and q ; are the constants de ned in (4.1.1) in Section 4.1. In particular, we have N j; N( j );( ) = jj j jj2 2 : (6.3.17) Proof. Observe that, by Lemma 6.3.11, ( ) + ( j) = j is a root. As V ( +ϵ) is assumed to be of type 1a, is a long root. By Remark 6.2.15, the root j is also a long root. Therefore j is a long root. Now the rst part of the lemma follows immediately from Lemma 3.4.4, and the second follows from Property (H7) in our normalizations in Section 4.1. 85 Lemma 6.3.18 If V ( +ϵ) is a special constituent of type 1a or type 2 then, for any 2 Δ +ϵ(g(1)) and any j 2 Δ +ϵ(z(n)), N ; jN ( j ); ( ) = N ( ); jN ( j ); : Proof. Observe that, by Property (H3) in Section 4.1, we have (X ;
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Title  Conformally Invariant Systems of Differential Operators Associated to Twostep Nilpotent Maximal Parabolics of Nonheisenberg Type 
Date  20120501 
Author  Kubo, Toshihisa 
Keywords  generalized Verma modules, invariant differential operators, prehomogeneous vector spaces 
Department  Mathematics 
Document Type  
Full Text Type  Open Access 
Abstract  The main work of this thesis concerns systems of differential operators that are equivariant under an action of a Lie algebra. We call such systems conformally invariant. The main goal of this thesis is to construct such systems of operators for a homogeneous manifold G_0/Q_0 with G_0 a Lie group and Q_0 a maximal twostep nilpotent parabolic subgroup. We use the invariant theory of a prehomogeneous vector space to build such systems. We determined the complex parameters for the line bundles L_{s} on which our systems of differential operators are conformally invariant. The systems that we construct yield explicit homomorphisms between appropriate generalized Verma modules. We also determine whether or not these homomorphisms are standard. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  CONFORMALLY INVARIANT SYSTEMS OF DIFFERENTIAL OPERATORS ASSOCIATED TO TWOSTEP NILPOTENT MAXIMAL PARABOLICS OF NONHEISENBERG TYPE By TOSHIHISA KUBO Bachelor of Science in Mathematics University of Central Oklahoma Edmond, Oklahoma, USA 2003 Master of Science in Mathematics Oklahoma State University Stillwater, Oklahoma, USA 2005 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 2012 COPYRIGHT c⃝ By TOSHIHISA KUBO May, 2012 CONFORMALLY INVARIANT SYSTEMS OF DIFFERENTIAL OPERATORS ASSOCIATED TO TWOSTEP NILPOTENT MAXIMAL PARABOLICS OF NONHEISENBERG TYPE Dissertation Approved: Dr. Leticia Barchini Dissertation Advisor Dr. Anthony C. Kable Dr. Roger Zierau Dr. K.S. Babu Dr. Sheryl A. Tucker Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Conformally Invariant Systems and the Ωk Systems 8 2.1 Conformally Invariant Systems . . . . . . . . . . . . . . . . . . . . . . 8 2.2 A Specialization on a gmanifold and gbundle . . . . . . . . . . . . . 10 2.3 A gmanifold N 0 and gbundle Ls . . . . . . . . . . . . . . . . . . . 13 2.4 Useful Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 The Ωk Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 The Ωk Systems and Generalized Verma Modules . . . . . . . . . . . 28 3 Parabolic Subalgebras and Zgradings 32 3.1 kstep Nilpotent Parabolic Subalgebras . . . . . . . . . . . . . . . . . 32 3.2 Maximal TwoStep Nilpotent Parabolic q of NonHeisenberg type . . 36 3.3 The Simple Subalgebras l and ln . . . . . . . . . . . . . . . . . . . . 41 3.4 Technical Facts on the Highest Weights for l , ln , g(1), and z(n) . . . 43 4 The Ω1 System 48 4.1 Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 The Ω1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 Irreducible Decomposition of l z(n) 54 5.1 Irreducible Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 54 iv 5.2 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Special Constituents of l z(n) 68 6.1 Special Constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.2 Types of Special Constituents . . . . . . . . . . . . . . . . . . . . . . 70 6.3 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 The Ω2 Systems 89 7.1 Covariant Map 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2 The Ω2jV ( +ϵ) Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3 Special Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8 The Homomorphisms between Generalized Verma Modules induced by the Ω1 System and Ω2 Systems 106 8.1 The Standard Map between Generalized Verma Modules . . . . . . . 107 8.2 The Homomorphism φΩ1 induced by the Ω1 System . . . . . . . . . . 111 8.3 The Homomorphisms φΩ2 induced by the Ω2 Systems . . . . . . . . . 112 8.3.1 The Type 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.3.2 The Positive Integer Special Value Case . . . . . . . . . . . . 115 8.3.3 The V ( + ϵ ) Case for Bn(i) for 3 i n 1 . . . . . . . . 122 BIBLIOGRAPHY 138 A Reducibility Points 141 A.1 Verma modules and Generalized Verma Modules . . . . . . . . . . . . 141 A.2 Jantzen's Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.3 Necessary Conditions of the Reducibility of Mq( t) . . . . . . . . . . 149 A.4 Reducibility Criteria for SimplyLaced Case . . . . . . . . . . . . . . 151 A.5 Reducibility Points of Mq( t) for Exceptional Algebras . . . . . . . . 156 v A.5.1 E6(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.5.2 E7(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 A.5.3 E7(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 A.5.4 E8(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.5.5 F4(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.6 The Special Values and The Reducibility Points . . . . . . . . . . . . 173 B Dynkin Diagrams and Extended Dynkin Diagrams 176 C Basic Data 183 D Lists of Positive Roots for Exceptional Algebras 198 vi LIST OF TABLES Table Page 6.1 Highest Weights for Special Constituents (Classical Cases) . . . . . . 74 6.2 Highest Weights for Special Constituents (Exceptional Cases) . . . . 75 6.3 The Roots , ϵ , and ϵn (Classical Cases) . . . . . . . . . . . . . . . 75 6.4 The Roots , ϵ , and ϵn (Exceptional Cases) . . . . . . . . . . . . . 76 6.5 Types of Special Constituents . . . . . . . . . . . . . . . . . . . . . . 78 7.1 Line Bundles with Special Values . . . . . . . . . . . . . . . . . . . . 104 7.2 The Generalized Verma Modules corresponding to L(s0 q) in Table 7.1 105 8.1 The Homomorphism φΩ2 for the NonHeisenberg Case . . . . . . . . . 137 A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 vii LIST OF FIGURES Figure Page B.1 The Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.2 The Dynkin diagrams with the multiplicities of the simple roots in the highest root of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.3 The extended Dynkin diagrams with the highest root of g . . . . . 181 viii CHAPTER 1 Introduction The main work of this thesis concerns systems of differential operators that are equiv ariant under an action of a Lie algebra. We call such systems conformally invariant. To explain the meaning of the equivariance condition, suppose that V ! M is a vec tor bundle over a smooth manifold M and g is a Lie algebra of rstorder differential operators that act on sections of V. A linearly independent list D1; : : : ;Dn of linear differential operators on sections of V is called a conformally invariant system if, for each X 2 g, there are smooth functions CX ij (m) on M so that, for all 1 i n, and sections f of V, we have ( [X;Di] f ) (m) = Σn j=1 CX ji (m)(Dj f)(m); (1.0.1) where [X;Dj ] = XDjDjX, and the dot denotes the action of differential operators on smooth functions. (See De nition 2.1.4 for the precise de nition.) A typical example for a conformally invariant system of one differential operator is the wave operator □ = @2 @x21 + @2 @x22 + @2 @x23 @2 @x24 on the Minkowski space R3;1. If X is an element of g = so(4; 2) acting as a rstorder differential operators on sections of an appropriate line bundle over R3;1 then there is a smooth function CX on R3;1 so that [X;□] = CX□: An important consequence of the de nition (1.0.1) is that the common kernel of the operators in the conformally invariant system D1; : : : ;Dn is invariant under a Lie algebra action. The representation theoretic question of understanding the common 1 kernel as a gmodule is an open question (except for a small number of very special examples). The notion of conformally invariant systems generalizes that of quasiinvariant differential operators introduced by Kostant in [19] and is related to a work of Huang ([8]). It is also compatible with the de nition given by Ehrenpreis in [6]. Confor mally invariant systems are explicitly or implicitly presented in the work of Davidson EnrightStanke ([5]), Kable ([12], [13]), Kobayashi rsted ([16], [17], [18]), Wallach ([25]), among others. Much of the published work is for the case that M = G=Q with Q = LN, N abelian. The systematic study of conformally invariant systems started with the work of BarchiniKableZierau in [1] and [2] Although the theory of conformally invariant systems can be viewed as a geometric analytic theory, it is closely related to algebraic objects such as generalized Verma modules. It has been shown in [2] that a conformally invariant system yields a ho momorphism between certain generalized Verma modules. The classi cation of non standard homomorphisms between generalized Verma modules is an open problem. The main goal of this thesis is to build systems of differential operators that satisfy the condition (1.0.1), when M is a homogeneous manifold G=Q with Q a maximal twostep nilpotent parabolic subgroup. This is to construct systems D1; : : : ;Dn acting on sections of bundles Vs ! G=Q over G=Q in a systematic manner and to determine the bundles Vs on which the systems are conformally invariant. The method that we use is different from one used by BarchiniKableZierau in [1]. The systems that we build yield explicit homomorphisms between appropriate generalized Verma modules. We show that the most of those homomorphisms are nonstandard. To describe our work more precisely, let G be a complex, simple, connected, simplyconnected Lie group with Lie algebra g. It is known that g has a Zgrading g = ⊕r j=r g(j) so that q = g(0) ⊕ j>0 g(j) = l n is a parabolic subalgebra of g. Let Q = NG(q) = LN. For a real form g0 of g, de ne G0 to be an analytic subgroup 2 of G with Lie algebra g. Set Q0 = NG0(q). Our manifold is M = G0=Q0 and we consider a line bundle Ls ! G0=Q0 for each s 2 C. It is known, by the Bruhat theory, that G0=Q0 admits an open dense submanifold N 0Q0=Q0. We restrict our bundle to this submanifold. The systems that we study act on sections of the restricted bundle. To build systems of differential operators we observe that L acts by the adjoint representation on g(1) with a unique open orbit. This makes g(1) a prehomogeneous vector space. Our construction is based on the invariant theory of a prehomogeneous vector space. It is natural to associate Lequivariant polynomial maps called covariant maps to the prehomogeneous vector space (L; Ad; g(1)). To de ne our systems of differential operators, we use covariant maps that are associated to g(1). We denote the covariant maps by k. Each k can be thought of as giving the symbols of the differential operators that we study. For 0 k 2r, the maps k are de ned by k : g(1) ! g(r + k) g(r) (1.0.2) X 7! 1 k! ad(X)k!0; where !0 is a certain element in g(r + k) g(r). (See De nition 2.5.1.) Let g(r + k) g(r) = V1 Vm (1.0.3) be the irreducible decomposition of g(r + k) g(r) as an Lmodule. Covariant map k induces an Lequivariant linear map ~ kjV j : V j ! Pk(g(1))) with V j the dual of an irreducible constituent Vj of g(r + k) g(r) and Pk(g(1)) the space of polynomials on g(1) of degree k. We de ne differential operators from ~ kjV j (Y ). For Y 2 V j , let Ωk(Y ) denote the kth order differential operators that are constructed from ~ kjV j (Y ). We say that a list of differential operators D1; : : : ;Dn is the ΩkjV j system if it is 3 equivalent (in the sense of De nition 2.1.5) to a list of differential operators Ωk(Y 1 ); : : : ;Ωk(Y n ); (1.0.4) where fY 1 ; : : : ; Y n g is a basis for V j over C. By construction the ΩkjV j system consists of dimC(Vj) operators. It is not necessary for the ΩkjV j system to be conformally invariant; the conformal invariance of the operators (1.0.4) strongly depends on the complex parameter s for the line bundle Ls. Then we say that the ΩkjV j system has special value s0 if the system is conformally invariant on the line bundle Ls0 . The special values for the case that dim([n; n]) = 1 for q = l n are studied by BarchiniKableZierau in [1] and [2], and myself in [20]. In this thesis we consider a more general case; namely, q = l n is a maximal parabolic subalgebra and n satis es the condition that [n; [n; n]] = 0 and dimC([n; n]) > 1. We call such parabolic subalgebras maximal twostep nilpotent parabolic subal gebras of nonHeisenberg type. In this case we have r = 2 in (2.5.6). Therefore the Ωk systems for k 5 are zero. The main results of this thesis are Theorem 4.2.5 and Theorem 7.3.6, where the special values of the Ω1 system and Ω2 systems for the parabolic subalgebras are determined. We also classify the nonstandard homo morphisms between the generalized Verma modules that arise from our systems of differential operators. We may want to remark that, although the special value of s for the Ω1 system is easily found by computing the bracket [X;Ω1(Y i )], it is in general not easy to nd the special values for the Ω2 systems by a direct computation. (See Section 5 of [1].) In this thesis, to nd the special value for the Ω2jV j system, we use two reduction techniques to compute the special values. First, in order to show the equivariance condition (1.0.1) for Di = Ω2(Y i ) with Y i 2 V j , it is enough to compute [X;Ω2(Y i )] at the identity e. Furthermore, we show that it is even sufficient to compute only [Xh;Ω2(Y l )] at e, where Xh and Y l are a highest weight vector of g(1) g and a 4 lowest weight vector of V j , respectively. These two techniques signi cantly reduce the amount of computations. We now outline the contents of this thesis. In Chapter 2 we study conformally invariant systems of differential operators. We recapitulate Section 2 of [2] in Section 2.1. In Sections 2.2 and 2.3 we specialize the theory of conformally invariant systems to the situation that we are interested in. Two useful formulas on differential operators will be shown in Section 2.4. In Section 2.5, the general construction of the Ωk systems is given. Section 2.6 discusses two technical lemmas on the Ωk systems, and in Section 2.7, we describe a relationship between the Ωk systems and generalized Verma modules. The aim of Chapter 3 is to study the Zgrading g = ⊕r j=r g(j) on g and a maximal twostep nilpotent parabolic subalgebra q of nonHeisenberg type. We begin this chapter by classifying the kstep nilpotent parabolic subalgebras in Section 3.1. In Section 3.2 and Section 3.3, we study a maximal twostep nilpotent parabolic subalgebra q of nonHeisenberg type and the associated 2grading g = ⊕2 j=2 g(j) = z( n) g(1) l g(1) z(n) of g. In Chapter 4, we construct the Ω1 system and nd the special value of the system. In Section 4.1, we x normalizations for root vectors. The normalizations play an important role to construct the system. In Section 4.2 we show that the special value s1 of s for the Ω1 system is s1 = 0. This is done in Theorem 4.2.5. To build the Ω2 systems, we need to nd the irreducible constituents V of l z(n) so that ~ 2jV ̸= 0. In Chapters 5 and 6, we show preliminary results to nd such irreducible constituents. In Chapter 5 we decompose l z(n) into the direct sum of the irreducible constituents. We rst summarize our main decomposition results, Theorem 5.1.3, in Section 5.1. Section 5.2 contains preliminary results and technical lemmas that are used to prove the theorem. The proof for Theorem 5.1.3 is given in Section 5.3. In Chapter 6, by using the decomposition results, we determine 5 the candidates of the irreducible constituents V so that ~ 2jV ̸= 0. We call such constituents special. In Section 6.1 we de ne the special constituents. We then classify such constituents in Section 6.2. In Section 6.3 we collect the technical results on the special constituents, which are used to nd the special values for the Ω2 systems. In Chapter 7, we build the Ω2 systems and nd their special values. First, it is shown in Section 7.1 that the covariant maps 2 and the induced linear maps ~ 2jV for certain special constituents V are nonzero. We then construct the Ω2 systems in Section 7.2, and in Section 7.3, we nd their special values. This is done in Theorem 7.3.6. In Chapter 8, we determine whether or not the homomorphisms φΩk that are induced by the Ωk systems between appropriate generalized Verma modules are stan dard for k = 1; 2. In Section 8.1 we review the wellknown results on the standard map between generalized Verma modules. Technical results to determine the stan dardness of the maps φΩk are also shown in this section. We then determine the standardness of φΩ1 and φΩ2 in Section 8.2 and Section 8.3, respectively. In this thesis we also have the appendices. In Appendix A, as an Ωk system that is conformally invariant on the line bundle Ls0 induces the reducibility of a scalar generalized Verma module U(g) U(q) Cs0 , to support the results for the special values for the Ω2 systems, we show the reducibility points for the scalar generalized Verma modules for g exceptional algebras. To determine the reducibility we use a criterion due to Jantzen. (See Section A.2.) In Appendices B, C, and D, we collect miscellaneous useful data. Namely, Ap pendix B contains the Dynkin diagrams with the multiplicities of the simple roots in the highest root of g and extended Dynkin diagrams. Appendix C summarizes the useful data for the parabolic subalgebras under consideration such as the roots for l, g(1), and z(n). In Appendix D we include the lists of the positive roots for the exceptional algebras. 6 Finally, I would like to thank my advisor, Dr. Leticia Barchini, for introducing this topic for me and for her generous help. I would also like to thank Dr. Anthony Kable and Dr. Roger Zierau for their valuable comments on this work. 7 CHAPTER 2 Conformally Invariant Systems and the Ωk Systems The purpose of this chapter is to study conformally invariant systems of differential operators, that are the main objects of this thesis. In particular, we de ne systems of differential operators of order k, which we call the Ωk systems. 2.1 Conformally Invariant Systems The aim of this section is to introduce the de nition of conformally invariant systems. Suppose that V and W are nite dimensional complex vector spaces and C1(Rn; V ) is the space of smooth V valued functions on Rn. A linear map D : C1(Rn; V ) ! C1(Rn;W) is called a differential operator if it is of the form D h = Σ j j k T ( @ @x h ) (2.1.1) for some k 2 Z 0 and all h 2 C1(Rn; V ), where T are smooth functions from Rn to HomC(V;W), and multiindex notation is being used. Here, the dot denotes the action of differential operators on smooth functions. Now let M be a smooth manifold, and let prV : V ! M and prW : W ! M be smooth vector bundles over M of nite rank with prV and prW the bundle projections. For each p 2 M, there exists an open neighborhood U of p so that the local trivializations pr1 V (U) = U V and pr1 W (U) = U W hold. Then a linear map D from smooth sections of V to smooth sections of W is called a differential operator if in each local trivialization D is of the form of (2.1.1). The smallest integer k with j j k in (2.1.1), for which T ̸= 0, is called the order of D. We 8 denote by D(V) the space of differential operators on the smooth sections of V. Note that we regard smooth functions f on M as elements in D(V) by identifying them with the multiplication operator they induce. Let g0 be a real Lie algebra and X(M) be the space of smooth vector elds on M. De nition 2.1.2 [2, page 790] A smooth manifold M is called a g0manifold if there is an Rlinear map M : g0 ! C1(M) X(M) so that M([X; Y ]) = [ M(X); M(Y )] for all X; Y 2 g0. For each X 2 g0, we write M(X) = 0(X) + 1(X) with 0(X) 2 C1(M) and 1(X) 2 X(M). De nition 2.1.3 [2, page 791] Let M be a g0manifold. A vector bundle V ! M is called a g0bundle if there is an Rlinear map V : g0 ! D(V) that satis es the following properties: (B1) We have V([X; Y ]) = [ V(X); V(Y )] for all X; Y 2 g0. (B2) In D(V), [ V(X); f] = 1(X) f for all X 2 g0 and f 2 C1(M). Now we introduce the de nition of conformally invariant systems. De nition 2.1.4 [2, page 791] Let V ! M be a g0bundle. A conformally invari ant system on V with respect to V is a list of differential operators D1; : : : ;Dm 2 D(V) so that the following two conditions hold: (S1) At each point p 2 M, the list D1; : : : ;Dm is linearly independent over C. (S2) For each X 2 g0, there is a matrix C(X) in Mm m(C1(M)) so that [ V(X);Di] = Σm j=1 Cji(X)Dj in D(V). 9 The map C : g0 ! Mm m(C1(M)) is called the structure operator of the confor mally invariant system. If g is the complexi cation of g0 then gmanifolds and gbundles are de ned by extending the g0action Clinearly. De nition 2.1.5 [2, page 792] Two conformally invariant systems D1; : : : ;Dn and D′ 1; : : : ;D′ n are said to be equivalent if there is a matrix A 2 GL(n;C1(M)) so that D′ i = Σn j=1 AjiDj for 1 i n. De nition 2.1.6 [2, page 793] A conformally invariant system D1; : : : ;Dn is called reducible if there is an equivalent system D′ 1; : : : ;D′ n and an m < n such that the system D′ 1; : : : ;D′ m is conformally invariant. Otherwise we say that D1; : : : ;Dn is irreducible. The case that M is a homogeneous manifold is of our particular interest. In Section 2.2 and Section 2.3, we will specify the gmanifold and gbundle that we will work with. 2.2 A Specialization on a gmanifold and gbundle In this section we shall introduce the specializations on a smooth manifold M and a vector bundle V ! M, as in Section 5 of [2]. Let G be a complex, simple, connected, simplyconnected Lie group with Lie algebra g. Such G contains a maximal connected solvable subgroup B. Write b = h u for its Lie algebra with h the Cartan subalgebra and u the nilpotent subalgebra. Let q b be a parabolic subalgebra of g. We de ne Q = NG(q), a parabolic subgroup of G. It follows from Section 8.4 of [24] that Q is connected. Write Q = LN for the Levi decomposition of Q with L the Levi subgroup and N the nilpotent subgroup. 10 It is known, see Corollary 7.11 of [15], that the Levi subgroup L is the commuting product L = Z(L)◦Lss, where Z(L)◦ is the identity component of the center of L and Lss is the semisimple part of L. Let g0 be a real form of g and let G0 be the analytic subgroup of G with Lie algebra g0. De ne Q0 = NG0(q) Q, and write Q0 = L0N0. We will work on M = G0=Q0 for a class of maximal parabolic Q0 that will be speci ed in Chapter 3. Next, we need to specify a vector bundle V on M. To this end we recall the bijection between the standard parabolic subalgebras and the subsets of simple roots. Let Δ = Δ(g; h) be the set of roots of g with respect to h. We denote by Δ+ the positive system so that u = ⊕ 2Δ+ g with g the root spaces for . We write for the set of simple roots. Observe that the parabolic q contains the xed Borel subalgebra b. Therefore, it is of the form q = h ⊕ 2 g with Δ+ Δ. Each subset can be described in terms of a subset S of simple roots. Indeed, = Δ+ [ f 2 Δ j 2 span( nS)g; where nS is the complementary subset of S in . If ΔS = f 2 Δ j 2 span( nS)g then = ΔS [ (Δ+nΔS). Then q may be written as q = l n (2.2.1) with l = h ⊕ 2ΔS g and n = ⊕ 2Δ+nΔS g : (2.2.2) The subalgebras l and n are called the Levi factor and the nilpotent radical, respec tively. The Lie algebra l is reductive and n is a nilpotent ideal in q. Now we state the wellknown fact that there exists a onetoone correspondence between the standard parabolic subalgebras q and subsets of . 11 Theorem 2.2.3 There exists a onetoone correspondence between parabolic subal gebras q containing b and the subsets S of the set of simple roots . The parabolic subalgebra qS corresponding to the subset S is of the form (2.2.1) with (2.2.2). Since our parabolic Q0 will be maximal, by Theorem 2.2.3, there exists the cor responding simple root q 2 so that q = qf qg. Call q the fundamental weight of q. The weight q is orthogonal to any roots with g [l; l]. Hence it expo nentiates to a character q of L. As q takes real values on L0, for s 2 C, character s = j qjs is wellde ned on L0. Let C s be the onedimensional representation of L0 with character s. The representation s is extended to a representation of Q0 by making it trivial on N0. Then it deduces a line bundle Ls on M = G0=Q0 with ber C s . The group G0 acts on the space C1 (G0=Q0;C s) = fF 2 C1(G0;C s) j F(gq) = s(q1)F(g) for all q 2 Q0 and g 2 G0g by left translation. The action s of g0 on C1 (G0=Q0;C s) arising from this action is given by ( s(Y ) F)(g) = d dt F(exp(tY )g) t=0 for Y 2 g0. This action is extended Clinearly to g and then naturally to the universal enveloping algebra U(g). We use the same symbols for the extended actions. Let N 0 be the nilpotent subgroup opposite to N0. By the Bruhat theory, the subset N 0Q0 is open and dense in G0. Then the restriction map C1 (G0=Q0;C s) ! C1( N 0;C s) is an injection, where C1( N 0;C s) is the space of the smooth functions from N0 to C s . Then, for u 2 U(g) and F 2 C1 (G0=Q0;C s ), we let f = Fj N 0 and de ne the action of U(g) on the image of the restriction map by s(u) f = ( s(u) F ) j N 0 : (2.2.4) 12 The line bundle Ls ! G0=Q0 restricted to N 0 is the trivial bundle N 0 C s ! N 0. By slight abuse of notation, we refer to the trivial bundle over N 0 as Ls. Then in practice our manifold M will be M = N 0 and our vector bundle will be the trivial bundle. In the next section we shall show that N 0 and the trivial bundle Ls are a gmanifold and gbundle with the action s, respectively. 2.3 A gmanifold N 0 and gbundle Ls Here we prove that with the linear map s de ned in (2.2.4), (1) the manifold N 0 is a gmanifold, and (2) the trivial bundle Ls is a gbundle. Let n and q be the complexi cations of the Lie algebras of N 0 and Q0, respectively; we have the direct sum g = n q. For Y 2 g, write Y = Y n+Yq for the decomposition of Y in this direct sum. Similarly, write the Bruhat decomposition of g 2 N 0Q0 as g = n(g)q(g) with n(g) 2 N 0 and q(g) 2 Q0. For Y 2 g0, we have Y n = d dt n(exp(tY )) t=0; (2.3.1) and a similar equality holds for Yq. De ne a right action R of U( n) on C1( N 0;C s) by ( R(X) f ) ( n) = d dt f ( n exp(tX) ) t=0 (2.3.2) for X 2 n0 and f 2 C1( N 0;C s ). Observe that, by de nition, the differential d of is d = q. Proposition 2.3.3 We have ( s(Y ) f ) ( n) = s q ( (Ad( n1)Y )q ) f( n) ( R ( (Ad( n1)Y ) n ) f ) ( n) (2.3.4) for Y 2 g and f in the image of the restriction map C1 (G0=Q0;C s) ! C1( N 0;C s). 13 Proof. Suppose that f = Fj N 0 for some F 2 C1 (G0=Q0;C s ). If g1 n 2 N 0Q0 then we have (g f)( n) = F(g1 n) = s(q(g1 n)1)f( n(g1 n)): (2.3.5) Observe that if g is close enough to the identity then g1 n 2 N 0Q0 by the openness of N 0Q0. By replacing g by exp(tY ) in (2.3.5) with Y 2 g0 and differentiating at t = 0, we have ( s(Y ) f)( n) = d dt s( q(exp(tY ) n )1 )f( n(exp(tY ) n))jt=0 = d dt s( q(exp(tY ) n )1 )jt=0 f( n) + d dt f( n(exp(tY ) n))jt=0 = d dt s( q(exp(tAd( n1)Y ) )1 )jt=0 f( n) + d dt f( n n(exp(tAd( n1)Y )))jt=0 = s q ( (Ad( n1)Y )q ) f( n) ( R ( (Ad( n1)Y ) n ) f ) ( n): Note that the equality (2.3.1) is used from line three to line four. Now the proposed formula is obtained by extending the action Clinearly. Equation (2.3.4) implies that the representation s extends to a representation of U(g) on the whole space C1( N 0;C s ). Moreover, it also shows that for all Y 2 g, the linear map s(Y ) is in C1( N 0) X( N 0). Therefore, with this linear map s, N 0 is a gmanifold. Next, we show that the linear map s gives Ls the structure of a gbundle. As s is a representation of g, the condition (B1) of De nition 2.1.3 is trivial. Thus it suffices to show that the condition (B2) holds. Since Ls is the trivial bundle of N 0 with ber C s , the space of smooth sections of Ls is identi ed with C1( N 0;C s ). Proposition 2.3.6 In D(Ls) we have ( [ s(Y ); f] ) ( n) = ( R ( (Ad( n1)Y ) n ) f ) ( n) for Y 2 g and f 2 C1( N 0). In particular, the trivial bundle Ls with s is a gbundle. 14 Proof. Take h 2 C1( N 0;C s ). Since [ s(Y ); f] = s(Y )f f s(Y ) in D(Ls), the operator [ s(Y ); f] acts on h by ( [ s(Y ); f] h ) ( n) = ( s(Y ) (fh) ) ( n) f( n) ( s(Y ) h ) ( n): (2.3.7) It follows from Proposition 2.3.3 that the rst term evaluates to ( s(Y ) (fh) ) ( n) = s q ( (Ad( n1)Y )q ) f( n)h( n) ( R ( (Ad( n1)Y ) n ) (fh) ) ( n) (2.3.8) with ( R ( (Ad( n1)Y ) n ) (fh) ) ( n) = ( R ( (Ad( n1)Y ) n ) f ) ( n)h( n) + f( n) ( R ( (Ad( n1)Y ) n ) h ) ( n): Similarly, the second term evaluates to f( n) ( s(Y ) h ) ( n) = s q ( (Ad( n1)Y )q ) f( n)h( n) f( n) ( R ( (Ad( n1)Y ) n ) h ) ( n): (2.3.9) Now the proposed equality is obtained by substituting (2.3.8) and (2.3.9) into (2.3.7). In the next section we are going to construct systems of differential operators on Ls. The systems of operators will satisfy several properties of conformally invariant systems. To end this section we collect those properties here. De nition 2.3.10 [2, page 806] A conformally invariant system D1; : : : ;Dm on the line bundle Ls is called L0stable if there is a map c : L0 ! GL(n;C1( N 0)) such that l Di = Σm j=1 c(l)jiDj ; where the action l Di is given by (2.5.10). 15 It is known that there exists a semisimple element H0 2 l, so that ad(H0) has only integer eigenvalues on g with g(1) ̸= f0g, l = g(0), n = ⊕ j>0 g(j), and n = ⊕ j>0 g(j), where g(j) is the jeigenspace of ad(H0) (see for example [15, Section X.3]). De nition 2.3.11 [2, page 804] A conformally invariant system D1; : : : ;Dm is called homogeneous if C(H0) is a scalar matrix, where C is the structure operator of the conformally invariant system (see De nition 2.1.4). Proposition 2.3.12 [2, Proposition 17] Any irreducible conformally invariant sys tem is homogeneous. De ne D(Ls) n = fD 2 D(Ls) j [ s(X);D] = 0 for all X 2 ng: Observe that in the sense of [2, page 796], the gmanifold N 0 is straight with respect to the subalgebra n of g ([2, page 799]). Then we state the de nition of straight conformally invariant systems specialized to the present situation. For the general de nition see p.797 of [2]. De nition 2.3.13 We say that a conformally invariant system D1; : : : ;Dm is straight if Dj 2 D(Ls) n for j = 1; : : : ;m. In general, to show that a given list D1; : : : ;Dm of differential operators on N 0 is a conformally invariant system, we need check (S2) of De nition 2.1.4 at each point of N 0. Proposition 2.3.14 below shows that in the case D1; : : : ;Dm in D(Ls) n, it suffices to check the condition only at the identity e. Proposition 2.3.14 [2, Proposition 13] Let D1; : : : ;Dm be a list of operators in D(Ls) n. Suppose that the list is linearly independent at e and that there is a map b : g ! gl(m;C) such that ( [ s(Y );Di] f ) (e) = Σm j=1 b(Y )ji(Dj f)(e) 16 for all Y 2 g; f 2 C1( N 0;C s), and 1 i m. Then D1; : : : ;Dm is a conformally invariant system on Ls. The structure operator of the system is given by C(Y )( n) = b(Ad( n1)Y ) for all n 2 N 0 and Y 2 g. 2.4 Useful Formulas In this section we are going to show two formulas that will be helpful, when we study the conformal invariance of certain systems of differential operators on N 0 in Chapter 4 and Chapter 7. Proposition 2.4.1 For Y 2 g, X 2 n , and f 2 C1(N 0;C s), we have ( [ s(Y );R(X)] f ) ( n) = ( R([(Ad( n1)Y )q;X] n) f ) ( n) + s q ( [Ad( n1)Y;X]q ) f( n): Proof. Since [ s(Y );R(X)] = s(Y )R(X) R(X) s(Y ), it suffices to consider the contributions from each term. By Proposition 2.3.3, the contribution from s(Y )R(X) is ( ( s(Y )R(X)) f ) ( n) (2.4.2) = s q ( (Ad( n1)Y )q ) (R(X) f)( n) ( (R((Ad( n1)Y ) n)R(X)) f) ) ( n): To obtain the contribution from R(X) s(Y ), observe that ( R(X) s(Y ) f ) ( n) = d dt ( s(Y ) f ) ( n exp(tX))jt=0: By applying Proposition 2.3.3, differentiating with respect to t, and setting t = 0, the contribution from this term is ( R(X) s(Y ) f ) ( n) = s q ( [X; Ad( n1)Y ]q ) f( n) s q ( (Ad( n1)Y )q ) (R(X) f)( n) + ( R([X; Ad( n1)Y ] n) f ) ( n) ( (R(X)R((Ad( n1)Y ) n)) f ) ( n): (2.4.3) 17 Since R([X; (Ad( n1)Y ) n]) = R(X)R((Ad( n1)Y ) n) R((Ad( n1)Y ) n)R(X), it fol lows from (2.4.2) and (2.4.3) that ( [ s(Y );R(X)] f ) ( n) evaluates to ( [ s(Y );R(X)] f ) ( n) = (2.4.4) ( R([X; (Ad( n1)Y ) n]) f)( n) ( R([X; Ad( n1)Y ] n) f ) ( n) + s q([Ad( n1)Y;X]q ) f( n): As Ad( n1)Y = (Ad( n1)Y ) n + (Ad( n1)Y )q and X 2 n, we have [X; Ad( n1)Y ] n = [X; (Ad( n1)Y ) n] + [X; (Ad( n1)Y )q] n: Now the proposed formula follows from substituting this into the second term of the right hand side of (2.4.4). Proposition 2.4.5 For Y 2 g, X1;X2 2 n, and f 2 C1( N 0;C s), we have ( [ s(Y );R(X1)R(X2)] f ) ( n) = ( R([(Ad( n1)Y )q;X1] n)R(X2) f ) ( n) + ( R(X1)R([(Ad( n1)Y )q;X2] n) f ) ( n) + ( R([[Ad( n1)Y;X1]q;X2] n) f ) ( n) + s q([Ad( n1)Y;X1]q)(R(X2) f)( n) + s q([Ad( n1)Y;X2]q)(R(X1) f)( n) + s q([[Ad( n1)Y;X1];X2]q)f( n): Proof. Observe that [ s(Y );R(X1)R(X2)] is the sum of two terms [ s(Y );R(X1)R(X2)] = [ s(Y );R(X1)]R(X2) + R(X1)[ s(Y );R(X2)]: The contribution from the rst term is ( [ s(Y );R(X1)] (R(X2) f) ) ( n) = ( R([(Ad( n1)Y )q;X1] n) (R(X2) f) ) ( n) + s q ( [Ad( n1)Y;X1]q ) (R(X2) f)( n): (2.4.6) 18 The second term evaluates to ( R(X1)[ s(Y );R(X2)] f ) ( n) = d dt ( [ s(Y );R(X2)] f ) ( n exp(tX1))jt=0 = ( R([X1; Ad( n1)Y ]q;X2] n) f)( n) + ( R(X1)R([(Ad( n1)Y )q;X2] n) f)( n) s q ( [[X1; Ad( n1)Y ];X2]q ) f( n) + s q([Ad( n1)Y;X2]q ) (R(X1) f)( n): Now the proposed formula follows from adding this to (2.4.6). 2.5 The Ωk Systems The purpose of this section is to construct systems of differential operators in D(Ls) n in a systematic manner. We start with a Zgrading g = ⊕r j=r g(j) on g with g(1) ̸= 0. It is known that g(0) is reductive (see for instance [15, Corollary 10.17]). By construction, q = g(0) ⊕ j>0 g(j) is a parabolic subalgebra. Take L to be the analytic subgroup of G with Lie algebra g(0). Vinberg's Theorem ([15, Theorem 10.19]) shows that the adjoint action of L on g(1) has only nitely many orbits; in particular, L has an open orbit in g(1). Such a space is called prehomogeneous. In the theory of prehomogeneous vector spaces, it is natural to associate certain maps called covariant maps to a prehomogeneous vector space. To de ne our systems of differential operators, we use covariant maps that are associated to prehomogeneous vector space (L; Ad; g(1)). We denote the covariant maps by k and de ne them below. These maps can be thought to give symbols of a class of differential operators that we will study. We would like to acknowledge that the construction of k as in this thesis was suggested by Anthony Kable. De nition 2.5.1 Let g = ⊕r j=r g(j) be a graded complex simple Lie algebra with 19 g(1) ̸= 0. Then, for 0 k 2r, the map k on g(1) is de ned by k : g(1) ! g(r + k) g(r) X 7! 1 k! ad(X)k!0 with !0 = Σ j2Δ(g(r)) X j X j , where X j are root vectors for j and Δ(g(r)) is the set of roots so that g g(r). Here, we mean by ad(X)k!0 that X acts on the tensor product diagonally via the action ad( )k. Observe that since X 2 g(1) and [g(1); g(r)] = 0, we have ad(X)kX j = 0 for all j 2 Δ(g(r)). Therefore, ad(X)k!0 = Σ j ad(X)k(X j ) X j . When g(1) and g(r + k) g(r) are viewed as affine varieties, the maps k are indeed morphisms of varieties. We shall check in Lemma 2.5.4 that these maps are Lequivariant. This will show that k satisfy the de nition of covariant maps. To simplify a proof for Lemma 2.5.4, we rst show that !0 in De nition 2.5.1 is independent of a choice of a basis for g(r). Lemma 2.5.2 If Y1; : : : ; Ym is a basis for g(r) and Y 1 ; : : : ; Y m is the dual basis for g(r) with respect to the Killing form then !0 = Σm i=1(Yi Y i ). Proof. If Δ(g(r)) = f 1; : : : ; mg then each Yi may be expressed by Yi = Σm r=1 airX r for air 2 C. Let [air] be the change of basis matrix and set [bir] = [air]1. De ne Y i = Σm s=1 bsiX s for i = 1; : : : ;m. Since Σm s=1 aisbsj = ij and (X i ;X j ) = ij with ij the Kronecker delta, it follows that (Yi; Y j ) = ij . Thus fY 1 ; : : : ; Y m g is the dual basis of fY1; : : : ; Ymg. Hence, Σm i=1 (Y i Yi) = Σm r;s=1 (Σm i=1 bsiair ) (X s X r ) = Σm s=1 (X s X s): Corollary 2.5.3 Let g = ⊕r j=r g(j) be a graded complex simple Lie algebra with g(1) ̸= 0 and G be a complex analytic group with Lie algebra g. If L is the analytic 20 subgroup of G with Lie algebra g(0) and !0 is as in De nition 2.5.1 then, for all l 2 L, (Ad(l) Ad(l))!0 = !0: Proof. If g 2 L then fAd(l)X j j j 2 Δ(g(r))g forms a basis for g(r). It also holds that fAd(l)X j j j 2 Δ(g(r))g is the dual basis for g(r) with respect to the Killing form. Now the assertion follows from Lemma 2.5.2 Now we show that k are Lequivariant. Lemma 2.5.4 Let g = ⊕r j=r g(j) be a graded complex simple Lie algebra with g(1) ̸= 0 and G be a complex analytic group with Lie algebra g. If L is the ana lytic subgroup of G with Lie algebra g(0) then, for all l 2 L, X 2 g(1), and for 0 k 2r, we have k(Ad(l)X) = (Ad(l) Ad(l)) k(X): (2.5.5) Proof. For l 2 L, we have k(Ad(l)X) = 1 k! ad(Ad(l)(X))k!0 = 1 k! Σ j2Δ(z(n)) ad(Ad(l)(X))k(X j ) X j = 1 k! Σ j2Δ(z(n)) Ad(l) ( ad(X)k(Ad(l1)X j ) ) X j = (Ad(l) Ad(l)) ( 1 k! Σ j2Δ(z(n)) ad(X)k(Ad(l1)X j ) Ad(l1)(X j ) ) = (Ad(l) Ad(l)) ( 1 k! ad(X)k!0 ) = (Ad(l) Ad(l)) k(X): Note that Corollary 2.5.3 is applied from line four to line ve. Now we are going to build the systems of differential operators in D(Ls) n that 21 we study. It is useful to observe that k : g(1) ! g(r + k) g(r) = W are L equivariant polynomial maps of degree k. Here, by a polynomial map we mean a map for which each coordinate is a polynomial in g(1). Therefore the maps k can be thought of as elements in (Pk(g(1)) W)L, where Pk(g(1)) denotes the space of homogeneous polynomials on g(1) of degree k. Then the isomorphism (Pk(g(1)) W)L = HomL(W ;Pk(g(1))) yields the Lintertwining operators ~ k, that are given by ~ k(Y )(X) = Y ( k(X)); (2.5.6) where W is the dual module of W with respect to the Killing form. For each Y 2 W , we have ~ k(Y ) 2 Pk(g(1)) = Symk(g(1)). We de ne differential operators in D(Ls) n from ~ k(Y ). This is done as follows. Let : Symk(g(1)) ! U( n) be the symmetrization operator. Identify U( n) with D(Ls) n by making n act on C1( N 0;C s) via right differentiation R. Then we have a composition of linear maps W ! ~k Pk(g(1)) = Symk(g(1)) ,! U( n) R! D(Ls) n: For Y 2 W , we de ne a differential operator Ωk(Y ) 2 D(Ls) n by Ωk(Y ) = R ◦ ◦ ~ k(Y ): As we will work with irreducible systems we need to be a little more careful with our construction; in particular, we need to take an irreducible constituent of g(r + k) g(r) . Let g(r + k) g(r) = V1 Vm be the irreducible decomposition of g(r + k) g(r) as an Lmodule, and let g(r + k) g(r) = V 1 V m be the corresponding irreducible decomposition of g(r + k) g(r) , where g(j) are the dual spaces of g(j) with respect to the Killing form. For each irreducible 22 constituent V j of g(r + k) g(r) , there exists an Lintertwining operator ~ kjV j 2 HomL(V j ;Pk(g(1))) given as in (2.5.6). Then we de ne a linear operator ΩkjV j : V j ! D(Ls) n by ΩkjV j = R ◦ ◦ ~ kjV j : Since, for Y 2 V j , we have ΩkjV j (Y ) = Ωk(Y ) as a differential operator, we simply write Ωk(Y ) for the differential operator arising from Y 2 V j . De nition 2.5.7 Let g = ⊕r j=r g(j) be an rgraded complex simple Lie algebra with g(1) ̸= 0, and q = ⊕r j=0 g(j) be the parabolic subalgebra of g associated with the r grading. If V is an irreducible constituent of g(r + k) g(r) so that ~ kjV is not identically zero then a list of differential operators D1; : : : ;Dn 2 D(Ls) n is called the ΩkjV system if it is equivalent to a list of differential operators Ωk(Y 1 ); : : : ;Ωk(Y n ); (2.5.8) where fY 1 ; : : : ; Y n g is a basis for V over C. Each ΩkjW system is also simply referred to as an Ωk system. We want to remark that the construction of the Ωk systems might require additional modi cation to secure the conformal invariance. See Section 6 in [1] and Section 3 in [20] for the modi cation for the Ω3 systems of the Heisenberg parabolic subalgebra. It is important to notice that it is not necessary for the Ωk systems to be confor mally invariant; their conformal invariance strongly depends on the complex param eter s for the line bundle Ls. So, we give the following de nition. De nition 2.5.9 Let V be an irreducible constituent of g(r+k) g(r) . Then we say that the ΩkjV system has special value s0 if the system is conformally invariant on the line bundle Ls0 . 23 The goal of this thesis is to nd the special values of the Ω1 system and the Ω2 systems of a maximal twostep nilpotent parabolic subalgebra of nonHeisenberg type. This is done in Chapter 4 and Chapter 7. To nish this section we de ne an action of L0 on D(Ls) n so that the linear operator ΩkjV : V ! D(Ls) n is an L0intertwining operator. This will allow that the ΩkjV system is L0stable (see De nition 2.3.10). As on p.805 of [2], we rst de ne an action of L0 on C1( N 0;C s) by (l f)( n) = s(l)f(l1 nl): This action agrees with the action of L0 by the left translation on the image of the restriction map C1 (G0=Q0;C s) ! C1( N 0;C s ). In terms of this action we de ne an action of L0 on D(Ls) by (l D) f = l (D (l1 f)): (2.5.10) One can check that we have l R(u) = R(Ad(l)u) for l 2 L0 and u 2 U( n); in particular, this action stabilizes the subspace D(Ls) n. With the adjoint action of L0 on U( n), the linear isomorphism U( n) R! D(Ls) n is L0equvariant. It is clear that each map in V ~ kj V ! Pk(g(1)) = Symk(g(1)) ,! U( n) is L0equivariant with respect to the natural actions of L0 on each space, which are induced by the adjoint action of L0 on g. Therefore, with the L0action (2.5.10), the operator ΩkjV : V ! D(Ls) n is an L0intertwining operator. Now we summarize the properties of the ΩkjV system. Remark 2.5.11 It follows from the de nition and the observation above that the ΩkjV system satis es the following properties: 1. The ΩkjV system satis es the condition (S1) of De nition 2.1.4. 24 2. When the ΩkjV system is conformally invariant then it is an irreducible, straight, and L0stable system. By Proposition 2.3.12, it is also a homogeneous system. 2.6 Technical Lemmas The aim of this section is to show two technical lemmas that will be used in Section 7.3. For D 2 D(Ls), we denote by D n the linear functional f 7! (D f)( n) for f 2 C1( N 0;C s ). A simple observation shows that (D1D2) n = (D1) nD2 for D1;D2 2 D(Ls); in particular, if (D1) n = 0 then [D1;D2] n = (D2) nD1. Lemma 2.6.1 Suppose that V is an irreducible constituent of g(r + k) g(r) . Let X1;X2 2 g and Y 1 ; : : : ; Y n 2 V . If s(X1)e = 0 and if [ s(Xi);Ωk(Y t )]e 2 spanC fΩk(Y j )e j j = 1; : : : ng for all i = 1; 2 then [ s(X1); [ s(X2);Ωk(Y t )] ] e 2 spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg: (2.6.2) Proof. Observe that [ s(X1); [ s(X2);Ωk(Y t )]] is s(X1)[ s(X2);Ωk(Y t )] [ s(X2);Ωk(Y t )] s(X1): (2.6.3) Since, by assumption, we have s(X1)e = 0, the rst term is zero at e. By assumption, [ s(X2);Ωk(Y t )]e is a linear combination of Ωk(Y 1 )e; : : : ;Ωk(Y n )e over C. So it may be written as [ s(X2);Ωk(Y t )]e = Σn j=1 ajtΩk(Y j )e with ajt 2 C. Then, at the identity e, the second term in (2.6.3) evaluates to Σn j=1 ajtΩk(Y j )e s(X1): Since ( s(X1)Ωk(Y j ))e = s(X1)eΩk(Y j ) = 0, we obtain [ s(X1); [ s(X2);Ωk(Y t )]]e = Σn j=1 ajtΩk(Y j )e s(X1) = Σn j=1 ajt[ s(X1)Ωk(Y j )]e: 25 Now the proposed result follows from the assumption that [ s(X1);Ωk(Y t )]e is a linear combination of Ωk(Y j )e over C. We call ul = ⊕ Δ+(l) g and ul = ⊕ Δ+(l) g ; where Δ+(l) is the set of positive roots in l. Lemma 2.6.4 Suppose that g(1) is irreducible and that V is an irreducible con stituent of g(r + k) g(r) . Let Xh be a highest weight vector for g(1) and Y l be a lowest weight vector for V . If [ s(Xh);Ωk(Y l )]e = spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg (2.6.5) with fY 1 ; : : : ; Y n g a basis for V then, for any X 2 g(1) and Y 2 V , [ s(X);Ωk(Y )]e 2 spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg: Proof. Set E = spanC fΩk(Y 1 )e; : : : ;Ωk(Y n )eg. We rst show that for each X 2 g(1), [ s(X);Ωk(Y l )]e 2 E: (2.6.6) Observe that since (L; g(1)) is assumed to be irreducible, the Lmodule g(1) is given by g(1) = U( ul)Xh. Then, as s is linear on g(1), it suffices to show that (2.6.6) holds when X = uk Xh with uk a monomial in U( ul). This is done by induction on the order of uk. Indeed, the proof is clear once we show that (2.6.6) holds for X = Z Xh = [ Z;Xh] with Z 2 ul. By the Jocobi identity, the commutator [ s([ Z;Xh]);Ωk(Y l )] is [ s([ Z;Xh]);Ωk(Y l )] = [ s( Z); [ s(Xh);Ωk(Y l )]] [ s(Xh); [ s( Z);Ωk(Y l )]]: (2.6.7) By the lequivariance of the operator Ωk : V ! D(Ls) n, it follows that [ s( Z);Ωk(Y l )] = Ωk([ Z; Y l ]): 26 Since Z 2 ul and Y l is a lowest weight vector, we have Ωk([ Z; Y l ]) = 0, and so is the second term of the right hand side of (2.6.7). Thus we have [ s([ Z;Xh]);Ωk(Y l )]e = [ s( Z); [ s(Xh);Ωk(Y l )]]e: (2.6.8) Now, by hypotheses and the lequivariance of Ωk, it follows that [ s(Xh);Ωk(Y l )]e; [ s( Z);Ωk(Y l )]e 2 E: As Z 2 ul, by Proposition 2.3.3, we have s( Z)e = 0. Thus, by Lemma 2.6.1, we obtain [ s( Z); [ s(Xh);Ωk(Y l )]]e 2 E, and so, by (2.6.8), [ s([ Z;Xh]);Ωk(Y l )]e 2 E. Next we show that for any X 2 g(1) and Y 2 V , [ s(X);Ωk(Y )]e 2 E: (2.6.9) Once again since V is irreducible, it is given by V = U(ul)Y l . As before, it is enough to show that (2.6.9) holds for Y = Z Y l with Z 2 ul. Since Ωk(Z Y l ) = [ s(Z);Ωk(Y l )], by the Jacobi identity, the commutator [ s(X);Ωk(Z Y l )] is [ s(X);Ωk(Z Y l )] = [ s(Z); [ s(X);Ωk(Y l )]] [[ s(Z); s(X)];Ωk(Y l )]: (2.6.10) We showed above that [ s(X);Ωk(Y l )]e 2 E. Since s(Z)e = 0 and [ s(Z);Ωk(Y l )]e 2 E, by Lemma 2.6.1, the rst term of the right hand side of (2.6.10) satis es [ s(Z); [ s(X);Ωk(Y l )]]e 2 E: Moreover, as [ s(Z); s(X)] = s([Z;X]) with [Z;X] 2 g(1), by what we have shown above, the second term satis es [[ s(Z); s(X)];Ωk(Y l )]e 2 E: Hence, [ s(X);Ωk(Z Y l )]e 2 E. 27 2.7 The Ωk Systems and Generalized Verma Modules To conclude this chapter, we show that conformally invariant Ωk systems induce nonzero U(g)homomorphisms between certain generalized Verma modules. The main idea is that conformally invariant Ωk systems yield nite dimensional simple lsubmodules of generalized Verma modules, on which n acts trivially. In general, to describe the relationship between conformally invariant systems on a g0bundle V ! M and generalized Verma modules, we realize generalized Verma modules as the space of smooth distributions on M supported at the identity. How ever, in our setting that the vector bundle V is a line bundle Ls, it is not necessary to use the realization. Thus, in this section, we are going to describe the relationship without using the realization. For more general theory on the relationship between conformally invariant systems and generalized Verma modules, see Sections 3, 5, and 6 of [2]. A generalized Verma module U(g) U(q) W is a U(g)module that is induced from a nite dimensional simple lmodule W on which n acts trivially. See Section A.1 for more details on generalized Verma modules. In this section we parametrize those modules as Mq[W] = U(g) U(q) W: We rst observe that the differential operators in D(Ls) n can be described in terms of elements of Mq[Cs q ], where Cs q is the qmodule derived from the Q0 representation ( s;C). By identifying Mq[Cs q ] as U( n) Cs q , the map Mq[Cs q ] ! U( n) given by u 1 7! u is an isomorphism of vector spaces. The composition Mq[Cs q ] ! U( n) R! D(Ls) n (2.7.1) is then a vectorspace isomorphism. Let W be an irreducible constituent of g(r + k) g(k) so that the L0 intertwining operator ΩkjW : W ! D(Ls) n is not identically zero. For Y 2 W , 28 if !k(Y ) = !kjW (Y ) denotes the element in U( n) that corresponds to Ωk(Y ) = ΩkjW (Y ) in D(Ls) n via right differentiation R in (2.7.1) then the linear operator !kjW : W ! U( n) is Lequivariant. Indeed, for l 2 L and Y 2 W , we have !k(l Y ) = Ad(l)!k(Y ); where the action l Y is the standard action of L on W , which is induced from the adjoint action of L on W. De ne Mq[W]n = fv 2 Mq[W] j X v = 0 for all X 2 ng: The following result is the specialization of Theorem 19 in [2] to the present situation. Theorem 2.7.2 If D = D1; : : : ;Dm is a straight L0stable homogeneous conformally invariant system on the line bundle Ls, and if !j denotes the element in U( n) that corresponds to Dj for j = 1; : : : ;m via right differentiation R then the space F(D) = spanC f!j 1 j j = 1; : : : ;mg is an Linvariant subspace of Mq[Cs q ]n. If the ΩkjW system is ΩkjW = Ωk(Y 1 ); : : : ;Ωk(Y m); where fY 1 ; : : : ; Y m g is a basis of W , then the space F(ΩkjW ) is given by F(ΩkjW ) = spanC f!k(Y j ) 1 j j = 1; : : : ;mg Mq[Cs q ]: Corollary 2.7.3 If the ΩkjW system is conformally invariant on the line bundle Ls0 then F(ΩkjW ) is an Linvariant subspace of Mq[Cs0 q ]n. Proof. By Remark 2.5.11, if the ΩkjW system is conformally invariant then it is a straight, L0stable, and homogeneous system. Now this corollary follows from Theorem 2.7.2. 29 Now suppose that the ΩkjW system is conformally invariant over Ls0 . Then, by Corollary 2.7.3, it follows that F(ΩkjW ) is an Linvariant subspace of Mq[Cs0 q ]n. On the other hand, there exists a vector space isomorphism F(ΩkjW ) ! W Cs q ; (2.7.4) that is given by !k(Y j ) 1 7! Y j 1. It is clear that the vector space isomorphism is Lequivariant with respect to the standard action of L on the tensor products F(ΩkjW ) U( n) Cs q and W Cs q . In particular, since W is an irreducible Lmodule, if W has highest weight then F(ΩkjW ) is the irreducible Lmodule with highest weight s0 q. 1 Moreover, as F(ΩkjW ) Mq[Cs0 q ]n, the nilradical n acts on F(ΩkjW ) trivially. Therefore the inclusion map 2 HomL ( F(ΩkjW );Mq[Cs0 q ] ) induces a nonzero U(g)homomorphism φΩk 2 HomU(g);L ( Mq[F(ΩkjW )];Mq[Cs0 q ] ) of generalized Verma modules, that is given by Mq[F(ΩkjW )] φ!Ωk Mq[Cs0 q ] (2.7.5) u ( !k(Y ) 1) 7! u ( !k(Y ) 1): If F(ΩkjW ) = Cs0 q then the map in (2.7.5) is just the identity map. However, Proposition 2.7.6 below shows that it does not happen. Proposition 2.7.6 Let W be an irreducible constituent of g(r + k) g(r) with k = 1; : : : ; 2r, so that ΩkjW : W ! D(Ls) n is not identically zero. If the ΩkjW system is conformally invariant on the line bundle Ls0 then F(ΩkjW ) ̸= Cs0 q Proof. Observe that if is the highest weight for W then F(ΩkjW ) has highest weight s0 q. If F(ΩkjW ) = Cs0 q then = 0, and so the irreducible constituent W g(r+k) g(r) would have highest weight 0. It is known that if is the highest weight for g(r) then the highest weight of any irreducible constituent of g(r+k) g(r) 1See Section 3.2 for the details of what we mean by a highest weight of a nite dimensional representation of reductive group L. 30 is of the form + with some weight for g(r+k) (see for instance [21, Proposition 3.2]). Thus, the highest weight 0 for W must be of the form 0 = + ( ). However, cannot be a weight for g(r + k) for any k = 1; : : : ; 2r, since only g(r) has weight . Therefore F(ΩkjW ) ̸= Cs0 q . Corollary 2.7.7 Under the same hypotheses for Proposition 2.7.6, the generalized Verma module Mq[Cs0 q ] is reducible. Proof. If is the highest weight for W then, by the proof for Proposition 2.7.6, it follows that F(ΩkjW ) ̸= Cs0 q . Now this corollary follows from (2.7.5). 31 CHAPTER 3 Parabolic Subalgebras and Zgradings It has been observed in Section 2.5 that the Zgrading g = ⊕r j=r g(j) on g and the parabolic subalgebra q play a role to construct the Ωk systems. In this chapter we study those in detail for q a maximal twostep nilpotent parabolic subalgebra of nonHeisenberg type. The Ω1 system and the Ω2 systems of those parabolics will be studied in Chapter 4 and Chapter 7, respectively. 3.1 kstep Nilpotent Parabolic Subalgebras We shall later construct the Ω1 system and the Ω2 systems of a maximal twostep nilpotent parabolic q. To do so, in this section we classify the kstep nilpotent parabolic subalgebras q by the subsets of simple roots. This is done in Proposition 3.1.4. Let r be any nonzero Lie algebra. Put r0 = r, r1 = [r; r], and rk = [r; rk1] for k 2 Z>0. We call rk the kth step of r for k 2 Z 0. The Lie algebra r is called nilpotent if rk = 0 for some k, and it is called kstep nilpotent if rk1 ̸= 0 and rk = 0. In particular, if [r; r] = 0 then r is called abelian, and if dim([r; r]) = 1 then r is called Heisenberg. Note that r is Heisenberg if and only if its center z(r) is one dimensional. If the nilpotent radical n of a parabolic subalgebra q = l n is kstep nilpotent (resp. abelian or Heisenberg) then we say that q is a kstep nilpotent (resp. abelian or Heisenberg) parabolic. If = Σ 2 m 2 Σ 2 Z then we say that jm j are the multiplicities of in . Proposition 3.1.4 below determines kstep nilpotent parabolic subalgebras 32 qS by the sum of the multiplicities of the simple roots of S in the highest root. Although it is a wellknown fact, we include a proof in this thesis, since we couldn't nd one in the literature. To prove the proposition it is convenient to show two technical lemmas, namely, Lemma 3.1.2 and Lemma 3.1.3. In Lemma 3.1.2 and Lemma 3.1.3, the subalgebras l and n are assumed to be the Levi factor and the nilpotent radical of qS with S = f i1 ; : : : ; ir g, respectively. Remark 3.1.1 It is easily shown by the Jacobi identity and the induction on k that we have [l; nk] nk for each k. In particular, if + 2 Δ with 2 Δ(l) and 2 Δ(nk) then + 2 Δ(nk), where Δ(l) and Δ(nk) are the subsets of roots that contribute to l and nk, respectively. Lemma 3.1.2 Suppose that is a root in Δ and let mij be the multiplicity of ij in . If Σr j=1mij = k then 2 Δ(nk1). Proof. For 2 Δ, it is well known that there exists an ordered set O = f 1; : : : ; sg of simple roots so that = Σs t=1 t having the property that each ordered partial sum is a root (see for instance [9, Corollary 10.2A]). Note that some of the roots in O belong to S and others are in (l) = Δ(l) \ . We prove this lemma by induction on the sum, Σr j=1mij , of the multiplicities of ij in S. When Σr j=1mij = 1, we have O \ S = f hg for some h 2 S Δ(n). Write = Σh t=1 t. If h = 1 then = h 2 Δ(n) = Δ(n0). If h ̸= 1 then since each partial sum is a root, we have Σh1 t=1 t 2 Δ(l). Since [l; n0] n0, it follows that = Σh t=1 t = Σh1 t=1 t + h 2 Δ(n0): Since each sum Σd t=1 t for d h is a root and all t for t > h are in Δ(l), by Remark 3.1.1, we conclude that = + h+1 + + s 2 Δ(n0): 33 Now we assume that the proposed statement holds for k1 Σr j=1mij 1. Let Σr j=1mij = k. There are two cases, s 2 S or s 2 (l). If s 2 S then the sum of the multiplicities of the simple roots in S contributing to s is equal to k 1. By induction hypothesis, we have s 2 Δ(nk2). Therefore, = ( s) + s 2 Δ([n; nk2]) = Δ(nk1). When s 2 (l), let l be the largest root in the order of O so that l 2 S. Then the sum of the multiplicities of the simple roots from S in the root Σl t=1 t is equal to k. Assuming as before, we conclude that Σl t=1 t 2 Δ(nk1). Now, once again, since each sum Σd t=1 t for d l is a root and all t for t > l are in Δ(l), by Remark 3.1.1, we conclude that = Σl t=1 t + l+1 + + s 2 Δ(nk1): Lemma 3.1.3 If 2 Δ(nk) and mij are the multiplicities of ij in then Σr j=1mij k + 1. Proof. We prove it by induction on k. Observe that if 2 Δ(n) = Δ+nΔ(l) then there exists ij 2 S so that the multiplicity of ij in is nonzero, because we would have 2 Δ(l), otherwise. Thus the case k = 0 is clear. We then assume that this holds for k = l. Let 2 Δ(nl+1). Since nl+1 = [nl; n], the root may be written as = ′ + ′′ with ′ 2 Δ(nl) and ′′ 2 Δ(n). Denoting by mij ( ) the multiplicities of ij in , we have Σr j=1 mij ( ) = Σr j=1 mij ( ′ + ′′) = Σr j=1 mij ( ′) + Σr j=1 mij ( ′′) (l + 1) + 1 = l + 2: By induction the lemma follows. We remark that if the highest root is = Σ 2 m then for any root = Σ 2 n , it follows that n m for all 2 . 34 Proposition 3.1.4 Let g be a complex simple Lie algebra with highest root , and qS = l n be the parabolic subalgebra of g that is parametrized by S with S = f i1 ; : : : ; ir g . Then n is kstep nilpotent if and only if k = mi1 +mi2 + +mir , where mij are the multiplicities of ij in . Proof. First we show that if k = Σr j=1mij then n is kstep nilpotent. If k = Σr j=1mij then, by Lemma 3.1.2, we have 2 Δ(nk1); in particular, nk1 ̸= 0. If nk ̸= 0 then there would exist 2 Δ(nk). If nij are the multiplicities of ij in then, by Lemma 3.1.3, it follows that Σr j=1 nij k + 1 > k: This contradicts the remark above. Therefore nk = 0, and so n is kstep nilpotent. Conversely, suppose that n is kstep nilpotent. If Σr j=1mij = l then, as we showed above, n is lstep nilpotent. Hence, l = k. To nish this section we introduce subdiagrams of Dynkin diagrams that associate to parabolics qS and classi cation types of them. First, Theorem 2.2.3 shows that there exists a bijection between the standard parabolics qS and the subsets S of simple roots. This allows us to associate qS to subdiagrams of Dynkin diagrams. The subdiagrams that associates to qS are obtained by deleting the nodes of the Dynkin diagram of g that correspond to the simple roots in S, and the edges in incident on them. We call such subdiagrams deleted Dynkin diagrams. With the multiplicities of simple roots in the highest root of g in hand, by Proposition 3.1.4, we can also see the number of steps of nilradical n of qS from the deleted Dynkin diagram. Example 3.1.5 below describes the deleted Dynkin diagram of a given parabolic qS and how we read the diagram. For simplicity, we depict deleted Dynkin diagrams by crossing out the deleted nodes. 35 Example 3.1.5 Take g = sl(6;C). The set of simple roots is = f 1; 2; 3; 4; 5g with Dynkin diagram ◦ 1 ◦ 2 ◦ 3 ◦ 4 ◦ 5 : Choose S = f 2; 4g. Then the deleted Dynkin diagram of parabolic subalgebra qS corresponding to the subset S is ◦ 1 2 ◦ 3 4 ◦ 5 : Moreover, Figure B.2 in Appendix B shows that the multiplicity of each simple root in the highest root of g of type An is 1, so this parabolic qS is a twostep nilpotent parabolic. In later sections we often refer to parabolic subalgebras qS by their corresponding subset S of simple roots. To this end, we are going to de ne classi cation types of parabolics qS. In De nition 3.1.6 below, we mean by classi cation type T of g type An, Bn, Cn, Dn, E6, E7, E8, F4, or G2. De nition 3.1.6 If g is a complex simple Lie algebra of classi cation type T and S is a subset of of simple roots then we say that a parabolic subalgebra qS of g is of type T (S), or type T (i1; : : : ; ik) if S = f i1 ; : : : ; ik g. For example, the parabolic subalgebra qS in Example 3.1.5 is of type A5(2; 4). Any maximal parabolic subalgebra is of type T (i) for some i 2 . In this thesis we use the Bourbaki conventions [4] for the labels of the simple roots (see Figure B.1 in Appendix B for the labels). 3.2 Maximal TwoStep Nilpotent Parabolic q of NonHeisenberg type The aim of this section is to study the 2grading g = ⊕2 j=2 g(j) on g, that is induced from a maximal twostep nilpotent parabolic subalgebra q of nonHeisenberg type. 36 Assume that g has rank greater than one and that q is a simple root, so that the parabolic subalgebra q = qf qg = l n parameterized by q is a maximal twostep nilpotent parabolic with dim([n; n]) > 1. Let ⟨ ; ⟩ be the inner product induced on h corresponding to the Killing form . Write jj jj2 = ⟨ ; ⟩ for 2 Δ. The coroot of is _ = 2 =⟨ ; ⟩. Recall from Section 2.2 that q denotes the fundamental weight for q. As Δ(l) = f 2 Δ j 2 span( nf qg)g and Δ(n) = Δ+nΔ(l), we have ⟨ q; ⟩ 8>>< >>: = 0 if 2 Δ(l) > 0 if 2 Δ(n) : Observe that if H q 2 h is de ned by (H;H q) = q(H) for all H 2 h and if Hq = 2 jj qjj2H q (3.2.1) then (Hq) is the multiplicity of q in . In particular, it follows from Proposition 3.1.4 that for 2 Δ+, (Hq) only can assume the values of 0, 1, or 2. Therefore, if g(j) denotes the jeigenspace of ad(Hq) then the action of ad(Hq) on g induces a 2grading g = g(2) g(1) g(0) g(1) g(2) with parabolic subalgebra q = g(0) g(1) g(2): Here we have l = g(0) and n = g(1) g(2). The subalgebra n, the opposite of n, is given by n = g(1) g(2): Observe that L acts on each of the subspaces g(j) via the adjoint representation. The goal of this section is to show that g(j) are irreducible Lmodules for j ̸= 0. 37 Via the Killing form, g(1) and g(2) are dual to g(1) and g(2), respectively. Thus, we will show that g(1) and g(2) are Lirreducible; hence, so are true for g(1) and g(2). The following proposition is well known. However, since the argument used in the proof will be referred in the proof for Corollary 3.2.3 below, we give a proof. Proposition 3.2.2 Assume that g is a graded complex semisimple Lie algebra with g = ⊕ j g(j), and let q = g(0) ⊕ j>0 g(j) with g(1) ̸= 0. Then g(1) is g(0)irreducible if and only if q is a maximal parabolic. Proof. We rst show that if q is not maximal then g(1) is not g(0)irreducible. Under this assumption there are at least two distinct simple roots in nΔ(g(0)), say 1 and 2. Let X 1 and X 2 be root vectors for 1 and 2, respectively. If U(g(0)) denotes the universal enveloping algebra of g(0) then U(g(0))X 1 and U(g(0))X 2 are two g(0)submodules of g(1). Since 1 and 2 are simple, U(g(0))X 1 ̸= U(g(0))X 2 . Hence g(1) is reducible. To prove the converse, as g(0) = z(g(0)) g(0)ss and the center z(g(0)) acts by scalars on g(1), it suffices to show that g(1) is an irreducible g(0)ssmodule. As in [9, Corollary 10.2A] we write 2 Δ+ as = i1 + + in with ij 2 (not necessarily distinct) in such a way that each partial sum i1+ + ij is a root. If q is maximal then there exists unique simple root 2 nΔ(g(0)). Each root 2 Δ(g(1)) is of the form = i1 + + ik + + im + + in; where the sum i1 + + ik q is a root with ij 2 Δ(g(0)). Let X q and X be root vectors for q and , respectively. If Xj is a root vector for ij then 0 ̸= ad(Xn)ad(Xn1) ad(Xm+1)ad(Xm)ad(X q)X 38 is a nonzero element in (U(g(0)ss)X ) \ g . Since 2 Δ(g(1)) is arbitrary, it is followed that g(1) = U(g(0)ss)X . We quote the Theorem of the Highest Weight to conclude that g(1) is g(0)ssirreducible with lowest weight . Let l = z(l) lss be the decomposition of l, that corresponds to L = Z(L)◦Lss with Z(L)◦ the identity component of the center of L and Lss the semisimple part of L. We say that a weight 2 h is a highest weight of a nite dimensional Lmodule V if jhss is a highest weight of V as an Lssmodule, where hss = h \ lss. A lowest weight of a nite dimensional Lmodule is similarly de ned. Corollary 3.2.3 If q = g(0) g(1) g(2) is the maximal twostep nilpotent parabolic of nonHeisenberg type determined by q then g(1) is the irreducible Lmodule with lowest weight q. Proof. Observe that since a root vector for q is an element of g(1), we have g(1) ̸= ∅. As Ad(L) preserves g(1), Proposition 3.2.2 implies that g(1) is Lirreducible. Next we show that g(2) is the irreducible Lmodule with highest weight . Since the argument of the proof works for general rgrading g = ⊕r j=r g(j), we give the proof in the general setting. Proposition 3.2.4 Assume that g = ⊕r j=r g(j) is a graded complex simple Lie algebra with n = ⊕r j=1 g(j). If the positive system Δ+ is chosen so that Δ+ = Δ+(g(0)) [ Δ(n) and is the highest root of g with respect to Δ+ then g(r) is the irreducible g(0)module with highest weight . Proof. As g is simple and is the highest root with respect to Δ+, g = U(g)X = U( n)(U(g(0))X ): Observe that since X 2 g(r) and g(r) is g(0)stable, we have U(g(0))X g(r). On 39 the other hand, as n = ⊕1 j=r g(j), it follows that U( n)g(r) ⊕r1 j=r g(j): As g = ⊕r j=r g(j), this shows that U(g(0))X g(r). Corollary 3.2.5 If q = g(0) g(1) g(2) is the maximal twostep nilpotent parabolic of nonHeisenberg type determined by q then g(2) is the irreducible Lmodule with highest weight . Proof. Observe that is the highest root of g for Δ+ = Δ+(l)[Δ(n). Now, as Ad(L) preserves g(2), Proposition 3.2.4 implies that g(2) is Lirreducible. To conclude this section we show that z(n) = g(2) and z( n) = g(2), where z(n) and z( n) are the centers of n and n, respectively. Because of the identi cation of g(j) with g(j) via the Killing form, it suffices to show that z(n) = g(2). The following technical lemma will simplify the expositions. Lemma 3.2.6 If q = g(0) g(1) g(2) is a maximal twostep nilpotent parabolic of nonHeisenberg type with n = g(1) g(2) then z(n) \ g(1) = f0g. Proof. One can easily check that z(n) is an lmodule by using the Jacobi identity and the fact that n is an lmodule. Therefore the intersection z(n) \ g(1) is an l submodule of g(1). The irreducibility of g(1) from Corollary 3.2.3 then forces that z(n) \ g(1) = f0g or g(1). However, the second is impossible; otherwise, we would have [n; n] = [g(1); g(1)] = 0; contrary to [n; n] ̸= 0. Therefore, z(n) \ g(1) = f0g. 40 Lemma 3.2.7 If q = g(0) g(1) g(2) is a maximal twostep nilpotent parabolic of nonHeisenberg type with n = g(1) g(2) then z(n) = g(2). Proof. Since g(2) z(n), it suffices to show the other inclusion. Take X 2 z(n). Since n = g(1) g(2), there exist Xj 2 g(j) for j = 1; 2 so that X = X1 + X2. Since X;X2 2 z(n), we have for any Y 2 n, [Y;X1] = [Y;X1] + [Y;X2] = [Y;X] = 0: Thus X1 2 z(n) \ g(1). Lemma 3.2.6 then concludes that X1 = 0, and so we have X = X2 2 g(2). Since X 2 z(n) is arbitrary, this yields that z(n) g(2). Now, since l = g(0), g(2) = z(n) and g(2) = z( n), we write the 2grading g = ⊕2 j=2 g(j) as g = z( n) g(1) l g(1) z(n) (3.2.8) with parabolic subalgebra q = l g(1) z(n): (3.2.9) 3.3 The Simple Subalgebras l and ln The purpose of this section is to study the structure of the Levi subalgebra l = z(l) lss. The material of this section will play a role in Chapter 5 and Chapter 6 when we decompose l z(n) into Lirreducible subspaces. The center z(l) is of the form z(l) = ∩ 2 (l) ker( ). Since g has rank greater than one and (l) = nf qg, z(l) is nonzero and onedimensional. It is clear from (3.2.1) that Hq is an element of z(l). Therefore we have z(l) = CHq. Next we consider the structure of lss. Observe that the Dynkin diagram of g can be extended by attaching the lowest root to the diagram. If g is not of type An then 41 there is exactly one simple root, that is connected to in the extended diagram (see Figure B.3 in Appendix B). Let denote such a unique simple root. It is easy to see that qf g is the Heisenberg parabolic of g; that is, the twostep nilpotent parabolic with dim([n; n]) = 1. Hence, if qf qg is a maximal twostep nilpotent parabolic with dim([n; n]) > 1 then 2 (l) = nf qg. If we delete the node corresponding to q then we obtain one, two, or three subgraphs with one subgraph containing . This implies that the subalgebra lss is either simple or the direct sum of two or three simple subalgebras with only one simple subalgebra containing the root space g for . The three subgraphs occur only when q is of type Dn(n 2). So, if q is not of type Dn(n 2) then there are at most two subgraphs. In this case we denote by l (resp. ln ) the simple subalgebra of l whose subgraph in the deleted Dynkin diagram contains (resp. does not contain) the node for . Thus the Levi subalgebra l may decompose into l = CHq l ln : (3.3.1) Then, for the rest of this chapter, we assume that q is not of type Dn(n2), so that the Levi subalgebra l can be expressed as (3.3.1). Recall from De nition 3.1.6 that if g is of type T then we say that the parabolic subalgebra q determined by i 2 is of type T (i). Then the parabolic subalgebras q under consideration are given as follows: Bn(i) (3 i n); Cn(i) (2 i n 1); Dn(i) (3 i n 3); (3.3.2) and E6(3); E6(5); E7(2); E7(6); E8(1); F4(4): (3.3.3) Note that in type An the nilradical n of any maximal parabolic subalgebra is abelian. Write (l ) = f 2 j 2 Δ(l )g and (ln ) = f 2 j 2 Δ(ln )g. Example 3.3.4 below exhibits the subgraphs for l and ln of q of type B5(3) with (l ) and 42 (ln ). One can nd those data in Appendix C for each maximal parabolic subalgebra in (3.3.2) or (3.3.3). Example 3.3.4 Let q be the parabolic subalgebra of type B5(3) with deleted Dynkin diagram ◦ 1 ◦ 2 3 ◦ 4 +3 ◦ 5 : Figure B.3 in Appendix B shows that = 2. Therefore, the subgraph for l is ◦ 1 ◦ 2 and that for ln is ◦ 4 +3 ◦ 5 with (l ) = f 1; 2g and (ln ) = f 4; 5g. Remark 3.3.5 It is clear from the extended Dynkin diagrams that ⟨ ; ⟩ > 0 and ⟨ ; ⟩ = 0 for any other simple roots . In particular, ⟨ ; ⟩ = 0 for all 2 (ln ). 3.4 Technical Facts on the Highest Weights for l , ln , g(1), and z(n) In this section we summarize technical lemmas on the Lhighest weights for l , ln , g(1), and z(n). These technical facts will be used in later computations. Proposition 3.2.4 shows that z(n) has highest weight , which is the highest root of g. We denote by , n , and the highest weights for l , ln , and g(1), respectively. In Appendix C we give the explicit values for these highest weights for each of the parabolic subalgebras under consideration. We remark that all these highest weights are indeed roots in Δ+. Observe that the highest weights and n of l and ln , respectively, are also the highest roots of l and ln as simple algebras; in particular, the multiplicities of 2 (l ) (resp. 2 (ln )) in (resp. n ) are all strictly positive. 43 Lemma 3.4.1 If q is the simple root that determines q = l g(1) z(n) then + q and n + q are roots. Proof. We only prove that + q 2 Δ; the other assertion that n + q 2 Δ can be proven similarly. It suffices to show that ⟨ ; q⟩ < 0, since both and q are roots. For 2 we observe that ⟨ ; q⟩ < 0 if is adjacent to q in the Dynkin diagram and ⟨ ; q⟩ = 0 otherwise. An observation on the deleted Dynkin diagrams shows that there exists a unique simple root k in (l ) that is adjacent to q. Since is the highest root for l as a simple algebra, the multiplicity of k in is strictly positive. Thus ⟨ ; q⟩ < 0. Lemma 3.4.2 If , n , , and are the highest weights of l , ln , g(1), and z(n), respectively, then the following hold: (1) 2 Δ, but n =2 Δ. (2) 2 Δ. (3) ; n 2 Δ. Proof. To prove n =2 Δ, we recall a wellknown fact that if n and m are the largest nonnegative integers so that n n 2 Δ and + m n 2 Δ, respectively, then ⟨ ; _ n ⟩ is given by ⟨ ; _ n ⟩ = n m (see for instance [9, Section 9.4]). Observe that the roots in Δ(ln ) are orthogonal to ; in particular, ⟨ ; _ n ⟩ = 0. Thus, we have n = m. As n 2 Δ+ and is the highest root, + n =2 Δ. Therefore, n = m = 0, which concludes that n is not a root. To prove 2 Δ, it suffices to show that ⟨ ; ⟩ > 0, since both and are roots. Write in terms of simple roots in (l ). Observe that each 2 (l ) has positive multiplicity m in . As is orthogonal to for any 2 (l )nf g, we have ⟨ ; ⟩ = m ⟨ ; ⟩ > 0. To prove the assertion (2), we show that ⟨ ; ⟩ > 0. Since, for simple and ̸= , we have ⟨ ; ⟩ = 0 and ⟨ ; ⟩ > 0, it suffices to show that the multiplicity 44 n of in is n > 0. Observe that the root = Σ 2 belongs to Δ(g(1)). The multiplicity of in is one. As g(1) is an irreducible Lmodule with highest weight , the root is of the form = Σ 2 (l) c with c nonnegative integers. Therefore = + Σ 2 (l) c , and so n = 1 + c > 0. Next we show that n 2 Δ. The other assertion in (3) is proven in a similar manner. It suffices to show that ⟨ ; n ⟩ > 0. We write as = Σ 2 (l ) m ϖ + Σ 2 (ln ) n eϖ with m ; n 2 Z 0; (3.4.3) where ϖ and eϖ are the fundamental weights of 2 (l ) and 2 (ln ), re spectively. The root n is an integer combination of simple roots in (ln ) of the form n = Σ 2 (ln ) m with m 2 Z>0: Then ⟨ϖ ; n ⟩ = 0 for all 2 (l ), and ⟨ eϖ ; n ⟩ > 0 for all 2 (ln ). It follows from Lemma 3.4.1 that ln acts on g(1) nontrivially. Thus, there exists ′ 2 (ln ) so that n ′ ̸= 0 in (3.4.3), and so we obtain ⟨ ; n ⟩ n ′m ′ > 0. When g is not simply laced then there are two root lengths in Δ. A root is called long or short accordingly. The following technical lemma will simplify arguments concerning the long roots later. We regard any root as a long root, when g is simply laced. Lemma 3.4.4 Suppose that 2 Δ is a long root. For any 2 Δ, the following hold. (1) If 2 Δ then ⟨ ; _⟩ = 1. (2) If + 2 Δ then ⟨ ; _⟩ = 1. (3) If 2 Δ then ∓ =2 Δ. (4) 2 =2 Δ. 45 Proof. Assume that 2 Δ. Since is a long root, we have 1 jj jj2=jj jj2 > 0. Thus, 1 jj jj2 jj jj2 ⟨ ; _⟩ + 1 > 0; which implies that 0 < jj jj2 jj jj2 ⟨ ; _⟩ < 1 + jj jj2 jj jj2 2: Therefore ⟨ ; _⟩ = 1. Part (2) may be shown similarly, and (3) and (4) follow from (1) and (2) with the fact that ⟨ ; _⟩ = p ; q ; , where p ; = maxfj 2 Z 0 j j 2 Δg and q ; = maxfj 2 Z 0 j + j 2 Δg. Lemma 3.4.5 If , n , , and are the highest weights of l , ln , g(1), and z(n), respectively, then the following hold: (1) + n 2 Δ. (2) n =2 Δ. (3) If is a long root then =2 Δ. Proof. Lemma 3.4.2 shows that 2 Δ. Then in order to prove (1), it is enough to show that ⟨ n ; ⟩ < 0. It follows from Remark 3.3.5 that ⟨ n ; ⟩ = 0. On the other hand, we have ⟨ n ; ⟩ > 0 by the proof for (3) of Lemma 3.4.2. Therefore, ⟨ n ; ⟩ = ⟨ n ; ⟩ ⟨ n ; ⟩ < 0: When n is a long root of g, the assertion (2) follows from (1) and Lemma 3.4.4. The data in Appendix C shows that n is a long root unless q is of type Bn(n 1). If q is of type Bn(n 1) then we have = "1 + "2, = "1 + "n, and n = "n. Thus n =2 Δ. To show (3), observe that, by Lemma 3.4.2, we have ; 2 Δ. Since is assumed to be a long root, it follows from Lemma 3.4.4 that ⟨ ; _ ⟩ = ⟨ ; _ ⟩ = 1. 46 Therefore ⟨ ; _ ⟩ = 0, which forces that jj jj2 = jj jj2 + jj jj2: (3.4.6) Since is a root, we have jj jj ̸= 0. As is assumed to be a long root, (3.4.6) implies that ( ) =2 Δ. Remark 3.4.7 Direct observation shows that is a long root, unless q is of type Cn(i). If q is of type Cn(i) then the data in Appendix C shows = 2"1, = "1+"i+1, and = "1 "i. Thus + =2 Δ, but 2 Δ. 47 CHAPTER 4 The Ω1 System The aim of this chapter is to determine the complex parameter s1 2 C for the line bundle Ls so that the Ω1 system of a maximal twostep nilpotent parabolic q of nonHeisenberg type is conformally invariant on Ls1 . The special value is given in Theorem 4.2.5. 4.1 Normalizations The purpose of this section is to x normalizations for root vectors. In the next section we are going to construct the Ω1 system and determine its special value of s. To do so, it is essential to set up convenient normalizations. If ; 2 Δ then de ne p ; = maxfj 2 Z 0 j j 2 Δg and q ; = maxfj 2 Z 0 j + j 2 Δg: (4.1.1) In particular, we have ⟨ ; _⟩ = p ; q ; : (4.1.2) It is known that we can choose X 2 g and H 2 h for each 2 Δ in such a way that the following conditions hold (see for instance [7, Sections III.4 and III.5]). The reader may want to notice that our normalizations are different from those used in [1]. (H1) For each 2 Δ+, fX ;X ;H g is an sl(2;C) triple; in particular, [X ;X ] = H : 48 (H2) For each ; 2 Δ+, [H ;X ] = (H )X . (H3) For 2 Δ we have (X ;X ) = 1. (H4) For ; 2 Δ we have (H ) = ⟨ ; ⟩. (H5) For ; 2 Δ with + ̸= 0, there is a constant N ; so that [X ;X ] = N ; X + if + 2 Δ, N ; = 0 if + =2 Δ: (H6) If 1; 2; 3 2 Δ+ with 1 + 2 + 3 = 0 then N 1; 2 = N 2; 3 = N 3; 1 : (H7) If ; 2 Δ and + 2 Δ then N ; N ; = q ; (1 + p ; ) 2 (H ): In particular, N ; is nonzero if + 2 Δ. We call the constants N ; structure constants. 4.2 The Ω1 System In this section we shall build the Ω1 system and determine its special value. As we have observed in Section 2.5, we use the covariant map 1 and the associated L intertwining operators ~ 1jV , where V are irreducible constituents of g(1) g(2) . By De nition 2.5.1, the covariant map 1 is given by 1 : g(1) ! g(1) z(n) X 7! ad(X)!0 49 with !0 = Σ j2Δ(z(n)) X j X j . It is clear that 1 is not identically zero. Indeed, if X = X with the highest weight for g(1) then 1(X ) = ad(X )!0 = Σ Δ (z(n)) N ; jX j X j with Δ (z(n)) = f j 2 Δ(z(n)) j j 2 Δg. By Lemma 3.4.2, we have 2 Δ with the highest weight for z(n), so Δ (z(n)) ̸= ∅. Since the vectors X j X j for j 2 Δ (z(n)) are linearly independent, we have 1(X ) ̸= 0. For each irreducible constituent V of g(1) z(n) , there exists an associated Lintertwining operator ~ 1jV 2 HomL(V ;P1(g(1))) so that, for all Y 2 V , ~ 1jV (Y )(X) = Y ( 1(X)): Observe that the duality for V is de ned with respect to the Killing form . More over, via the Killing form , we have g(1) z(n) = g(1) z( n). Thus, if Y = X X t with 2 Δ(g(1)) and t 2 Δ(z(n)) then Y ( 1(X)) is given by Y ( 1(X)) = Σ j2Δ(z(n)) (X ; ad(X)X j ) (X t ;X j ); (4.2.1) as 1(X) = Σ j2Δ(z(n)) ad(X)X j X j . Now we wish to determine all the irreducible constituents V of g(1) z( n), so that ~ 1jV are not identically zero. Observe that P1(g(1)) = Sym1(g(1)) = g(1) and that g(1) is an irreducible Lmodule, as q is a maximal parabolic. Thus, if ~ 1jV is not identically zero then V = g(1). Proposition 4.2.2 below shows that the converse also holds. Proposition 4.2.2 Let V be an irreducible constituent of g(1) z( n). Then ~ 1jV is not identically zero if and only if V = g(1). 50 Proof. First observe that g(1) is an irreducible constituent of g(1) z( n). Indeed, since 1 is linear, we have 1(g(1)) = g(1) as an Lmodule; in particular, g(1) is an irreducible constituent of g(1) z(n). Therefore g(1) = g(1) is an irreducible constituent of g(1) z( n) = (g(1) z(n)) . To prove ~ 1jg(1) is a nonzero map, it suffices to show that ~ 1jg(1)(Y ) ̸= 0 for some Y 2 g(1) g(1) z( n). To do so, consider a map 1 : g(1) ! g(1) z( n) X 7! ad( X ) !0 with !0 = Σ t2Δ(z(n)) X t X t . This is a nonzero Lintertwining operator. Thus 1(g(1)) = g(1) as an Lmodule, and 1(X ) is a weight vector with weight for all 2 Δ(g(1)). As g(1) has highest weight , the lowest weight for g(1) is . Now we set c = Σ t2Δ (z(n)) N ; tN ; t with Δ (z(n)) = f t 2 Δ(z(n)) j t 2 Δg. By Lemma 3.4.2, it follows that 2 Δ; in particular, Δ (z(n)) ̸= ∅. The normalization (H7) in Section 4.1 shows that N ; tN ; t < 0 for all t 2 Δ (z(n)). Therefore c ̸= 0. Then de ne Y l 2 g(1) by means of Y l = 1 c 1(X ) = 1 c Σ t2Δ (z(n)) N ; tX t X t : We claim that ~ 1jg(1)(Y l )(X) ̸= 0. By (4.2.1), the polynomial ~ 1jg(1)(Y l )(X) is ~ 1jg(1)(Y l )(X) = Y l ( 1(X)) = 1 c Σ t2Δ (z(n)) j2Δ(z(n)) N ; t (X t ; ad(X)X j ) (X t ;X j ) = 1 c Σ t2Δ (z(n)) N ; t (X t ; ad(X)X t): 51 Write X = Σ 2Δ(g(1)) X , where 2 n is the coordinate dual to X with respect to the Killing form . Then, ~ 1jg(1)(Y l )(X) = 1 c Σ t2Δ (z(n)) N ; t (X t ; ad(X)X t) = 1 c Σ 2Δ(g(1)) t2Δ (z(n)) N ; t (X t ; ad(X )X t) = 1 c Σ t2Δ (z(n)) N ; tN ; t = = (X;X ): (4.2.3) Hence ~ 1jg(1)(Y l )(X) ̸= 0. Since only g(1) contributes to the construction of the Ω1 systems, we simply refer to the Ω1 system as the Ω1jg(1) system. As we observed in Section 2.5, the operator Ω1jg(1) : g(1) ! D(Ls) n is obtained via the composition of maps g(1) ~ 1j !g(1) P1(g(1)) ! g(1) ,! U( n) R! D(Ls) n: By (4.2.3), we have ~ 1jg(1)(Y l )(X) = (X;X ). Therefore, Ω1(Y l ) = R(X ): Now, for all 2 Δ(g(1)), set Y = 1(X ): Then, as Y l = (1=c ) 1(X ), we have Ω1(Y ) = c R(X ): 52 Since both Ω1jg(1) and 1 are L0intertwining operators and g(1) = U(l)X , for any 2 Δ(g(1)), we obtain Ω1(Y ) = c R(X ) (4.2.4) with some constant c . Then, for Δ(g(1)) = f 1; : : : ; mg, the Ω1 system is given by R(X 1); : : : ;R(X m): The following theorem shows that the Ω1 system is conformally invariant on L0. Theorem 4.2.5 Let g be a complex simple Lie algebra, and let q be a maximal two step nilpotent parabolic subalgebra of nonHeisenberg type. Then the Ω1 system is conformally invariant on Ls if and only if s = 0. Proof. By Remark 2.5.11, we only need to show that the condition (S2) in De nition 2.1.4 holds if and only if s = 0. By Theorem 2.4.1, ( [ s(Y );R(X j )] f ) ( n) = ( R([(Ad( n1)Y )q;X j ] n) f ) ( n) + s q ( [Ad( n1)Y;X j ]q ) f( n) for any Y 2 g and any f 2 C1(N 0;C s ). Hence, the condition (S2) holds if and only if s = 0. 53 CHAPTER 5 Irreducible Decomposition of l z(n) Our next goal is to construct the Ω2 systems and to nd their special values. To do so, we need to detect the irreducible constituents V of l z(n) so that ~ 2jV is not identically zero. (see Section 2.5 for the general construction of the Ωk systems). In this chapter and the next one, we shall show preliminary results to nd such irreducible constituents. 5.1 Irreducible Decomposition of l z(n) We continue with q = l g(1) z(n) a maximal twostep nilpotent parabolic subalgebra of nonHeisenberg type listed in (3.3.2) or (3.3.3), and Q = LN = NG(q). The Levi subgroup L acts on l z(n) g g via the standard action on the tensor product induced by the adjoint representation on l and z(n). As L is complex reductive, this action is completely reducible. Since l = z(l) l ln with z(l) = CHq, we have l z(n) = ( CHq z(n) ) ( l z(n) ) ( ln z(n) ) : (5.1.1) It is clear that CHq z(n) = z(n) = g(2) as an Lmodule. Thus, by Corollary 3.2.5, CHq z(n) is Lirreducible. It is also easy to show that ln z(n) is Lirreducible. Let L (resp. Ln ) be the analytic subgroup of L with Lie algebra l (resp. ln ). As in Section 3.2, we call a weight for a nite dimensional Lmodule V a highest weight for V if the restriction jhss onto hss is a highest weight for V as an Lssmodule. Proposition 5.1.2 Suppose that ln ̸= 0. If n and are the highest weights of ln and z(n), respectively, then ln z(n) is the irreducible Lmodule with highest weight 54 n + . Proof. First we observe that Ln acts trivially on z(n). By Corollary 3.2.5, we have z(n) = g(2) = U(lss)X . By the observation made in Remark 3.3.5, it follows that ? for all 2 Δ(ln ). Thus z(n) = U(l )X . Hence Ln acts trivially; in particular, the irreducible Lmodule z(n) is L irreducible. On the other hand, it is clear that L acts on ln trivially. Therefore the representation (L; Ad Ad; ln z(n)) is equivalent to (L Ln ; Ad^ Ad; ln z(n)), where ^ denotes the outer tensor product. Since ln and z(n) have highest weight n and , respectively, the lemma follows. Now we focus on the decomposition of l z(n) into irreducible Lsubmodules. As noted in the proof for Lemma 5.1.2, the subgroup Ln acts trivially on l z(n). Hence we study l z(n) as an L module. For 2 h with ⟨ ; _⟩ 2 Z 0 for all 2 (l ), we will denote by V ( ) the irreducible constituent with highest weight jh , where h = h\l . For classical algebra g, we use the standard realization of the roots "i, the dual basis of the standard orthonormal basis for Rn. Theorem 5.1.3 The Lmodule l z(n) is reducible. If V ( ) denotes the irreducible representation of L with highest weight jh then the irreducible decomposition of l z(n) is given as follows. 1. Bn(i); 3 i n : l z(n) = 8>>< >>: V ( + ) V ( ) V ( + ("1 + "3)) if i = 3 V ( + ) V ( ) V ( + ("1 + "i)) V ( + ("2 + "3)) if 4 i n 2. Cn(i); 2 i n 1 : l z(n) 55 = 8>>< >>: V ( + ) V ( ) V ( + 2"2) if i = 2 V ( + ) V ( ) V ( + ("2 + "i)) V ( + ("1 + "2)) if 3 i n 1 3. Dn(i); 3 i n 3 : l z(n) = 8>>< >>: V ( + ) V ( ) V ( + ("1 + "3)) if i = 3 V ( + ) V ( ) V ( + ("1 + "i)) V ( + ("2 + "3)) if 4 i n 3 4. All exceptional cases (E6(3), E6(5), E7(2), E7(6), E8(1), F4(4)): l z(n) = V ( + ) V ( ) V ( + 0); where 0 is the following root contributing to z(n): E6(3) : 0 = 1 + 2 + 2 3 + 3 4 + 2 5 + 6 E6(5) : 0 = 1 + 2 + 2 3 + 3 4 + 2 5 + 6 E7(2) : 0 = 1 + 2 2 + 3 3 + 4 4 + 3 5 + 2 6 + 7 E7(6) : 0 = 1 + 2 2 + 2 3 + 4 4 + 3 5 + 2 6 + 7 E8(1) : 0 = 2 1 + 3 2 + 4 3 + 6 4 + 5 5 + 4 6 + 2 7 + 8 F4(4) : 0 = 1 + 2 2 + 4 3 + 2 4. 5.2 Technical Results on l z(n) In general, the study of tensor product decomposition of irreducible nite dimen sional representations is complicated. Techniques from representation theory and algebraic geometry have been used to study the problem (See for instance [21]). In our setting l = V ( ) and z(n) = V ( ), the standard techniques suffice to decompose 56 V ( ) V ( ) under L action. We have already observed that this action is com pletely reducible. The goal is to nd all the constituents and their multiplicities. To this end, it is enough to study V ( ) V ( ) as an l module. Our main technique is to analyze the character formula for l z(n) = V ( ) V ( ) as an l module. We will freely use the standard notions of dominant weights and regular weights. When we say that is Δ(l )dominant (resp. Δ(l )regular), we mean that ⟨ ; ⟩ 0 (resp. ⟨ ; ⟩ ̸= 0) for all 2 Δ+(l ). For V ( ), the nite dimensional l module with highest weight jh , and a weight 2 h , we denote by m ( ) the multiplicity jh in V ( ); that is, the dimension of the weight space V ( ) jh in V ( ). A weight is either Δ(l )regular or not. If is Δ(l )regular then no nontrivial element w in theWeyl group W(l ) of l xes . Otherwise, there is w ̸= 1 in W(l ) so that w = . Hence, if is a Δ(l )regular weight then there is a unique w 2 W(l ) so that w is Δ(l )dominant. We will write d( ) = w . De ne sgn( ) = 8>>< >>: 0 if some w ̸= 1 in W(l ) xes (1)l(w ) otherwise, where w 2 W(l ) so that w = d( ); where l(w ) is the length of w . We denote by (l ) half the sum of positive roots in Δ+(l ). Then if (resp. ′) is the character for V ( ) (resp. V ( ′)) then the character formula for the character ′ for the l module V ( ) V ( ′) is ′ = Σ ′′2Δ(V ( )) m ( ′′)sgn( ′′ + ′ + (l )) d( ′′+ ′+ (l )) (l ); (5.2.1) where Δ(V ( )) is the set of the weights for V ( ). This character formula is due to Klimyk [14, Corollary]. Among the standard facts, we use the following to analyze (5.2.1): (I) The constituent V ( + ′) occurs exactly once in V ( ) V ( ′). Moreover, if v 57 and v ′ are highest weight vectors of V ( ) and V ( ′), respectively, then v v ′ is a highest weight vector of V ( ) V ( ′). (II) If ′′ is the highest weight of some irreducible constituent of V ( ) V ( ′) then ′′ is of the form ′′ = + for some weight of V ( ′). (III) If all weights of V ( ) have multiplicity one then each irreducible constituent of V ( ) V ( ′) has multiplicity one. The unique irreducible constituent V ( + ′) is called the Cartan component of V ( ) V ( ′) (see for instance [21, page 1230]). In our setting l z(n) = V ( ) V ( ), the weights and are roots. By Fact (I) the highest weights of the irreducible constituents of l z(n) are of the form + j with j 2 Δ(z(n)). The character formula (5.2.1) is particularly simple when (l ) consists solely of long roots. We obtain a couple of results under this assumption. Lemma 5.2.2 Suppose that (l ) consists solely of long roots of g. If + j is not Δ(l )dominant then sgn( + j + (l )) = 0. Proof. We show that there exists 2 (l ) so that s xes + j + (l ). Since ⟨ (l ); _⟩ = 1 for all 2 (l ), it suffices to show that ⟨ + j ; _⟩ = 1 for some 2 (l ). Under our hypothesis + j is not Δ(l )dominant. Hence there exists 2 (l ) so that ⟨ + j ; _⟩ < 0. On the other hand, since is the highest weight of l , it follows that ⟨ ; _⟩ 0. We have ⟨ j ; _⟩ < ⟨ ; _⟩ 0; (5.2.3) and j + 2 Δ. Since (l ) contains only long roots, Lemma 3.4.4 shows that ⟨ j ; _⟩ = 1. Then (5.2.3) forces ⟨ ; _⟩ = 0, since ⟨ ; _⟩ is an integer. Therefore ⟨ + j ; _⟩ = 1. 58 Remark 5.2.4 If + j is Δ(l )dominant then + j + (l ) is Δ(l )dominant and Δ(l )regular. Hence, we have sgn( + j + (l )) = 1. Proposition 5.2.5 Suppose that (l ) consists solely of long roots of g. Then V ( + j) is an irreducible constituent of l z(n) if and only if + j is Δ(l )dominant. Proof. One of the directions is obvious. We then show that V ( + j) is an irreducible constituent if + j is Δ(l )dominant. By Klimyk's character formula, the character is of the form = Σ j2Δ(z(n)) m ( j)sgn( + j + (l )) d( + j+ (l )) (l ): (5.2.6) Since the weights of z(n) are roots of g, they have multiplicity one. Thus m ( j) = 1 for all j 2 Δ(z(n)). Moreover, Lemma 5.2.2 and Remark 5.2.4 show that sgn( + j + (l )) = 8>>< >>: 1 if + j is Δ(l )dominant 0 otherwise: Thus (5.2.6) is reduced to = Σ + j ; (5.2.7) where the sum runs over all j 2 Δ(z(n)) so that + j is Δ(l )dominant. Now the proposed assertion follows. Corollary 5.2.8 If (l ) consists solely of long roots of g then V ( ) occurs in the decomposition of l z(n) into irreducibles. Proof. By Lemma 3.4.2, we have 2 Δ(z(n)). Thus there exists j 2 Δ(z(n)) so that + j = . Since is Δ(l )dominant, the corollary follows from Proposition 5.2.5. Remark 5.2.9 Theorem 5.1.3 shows that V ( ) in fact occurs in l z(n) in every case. 59 5.3 Proof of Theorem 5.1.3 In the previous section we have shown that the character formula (5.2.1) is simple, when (l ) consists solely of long roots. Then in order to prove Theorem 5.1.3, we consider two cases, namely, Case 1: (l ) consists solely of long roots. Case 2: (l ) contains at least one short root. When g is simply laced, we regard any roots as long roots. Direct observation shows that the parabolic subalgebras q in (3.3.2) and (3.3.3) are then classi ed as follows: Case 1: Bn(i), Dn(i), E6(3), E6(5), E7(2), E7(6), E8(1) Case 2: Cn(i), F4(4) We start by proving Theorem 5.1.3 for parabolic subalgebras q in Case 1. Proof. [Proof for Theorem 5.1.3 for Case 1] Let be the set of all roots j 2 Δ(z(n)) so that + j is Δ(l )dominant. It follows from Fact (III) and Proposition 5.2.5 that the character is of the form = Σ j2 + j : (5.3.1) Moreover, Fact (I) and Corollary 5.2.8 show that V ( + ) and V ( ) occur in the decomposition. Therefore (5.3.1) might be expressed as = + + + Σ j2nf ; g + j : It remains to identify the roots in nf ; g. This is done in a case by case fashion. We include the computation for type E6(3). Other cases may be handled similarly. 60 The parabolic subalgebra q of type E6(3) corresponds to the deleted Dynkin dia gram ◦2 ◦ 1 3 ◦ 4 ◦ 5 ◦ 6: The subgraph corresponding to l is ◦ 2 ◦ 4 ◦ 5 ◦ 6: So the simple subalgebra l is isomorphic to sl(5;C). Write the fundamental weights of sl(5;C) corresponding to 2, 4, 5, 6 as ϖ1, ϖ2, ϖ3, ϖ4, respectively. The l module z(n) has highest weight . As ⟨ ; i⟩ = i;2 with i;2 the Kronecker delta for all i = 2; 4; 5; 6, we have z(n) = V (ϖ1). Thus, the adjoint representation l on z(n) is equivalent to the standard representation of sl(5;C) on C5. We then identify the weights of the adjoint action of l on z(n) with those of the standard action of sl(5;C) on C5; that is, Δ(z(n)) = f"1; "2; "3; "4; "5g: In terms of the fundamental weights we have "1 = ϖ1; "2 = ϖ1 + ϖ2; "3 = ϖ2 + ϖ3; "4 = ϖ3 + ϖ4; "5 = ϖ4: The highest weight of l is = ϖ1 + ϖ4. Therefore, the weights j 2 Δ(z(n)) that make + j Δ(l )dominant are j = ϖ1, ϖ4, or ϖ1 + ϖ2. Here, we have + ϖ1 = + , + (ϖ4) = ϖ1 = , and + (ϖ1 + ϖ2) = + 0 with 0 the root in Δ(z(n)) listed in Theorem 5.1.3. We next show Theorem 5.1.3 for parabolic subalgebras q in Case 2, namely, Cn(i) for 2 i n 1, and F4(4). Proof. [Proof for Theorem 5.1.3 for Case 2] The character formula of the tensor 61 product (5.2.6) is of the form = Σ j2Δ(z(n)) sgn( + j + (l )) d( + j+ (l )) (l ): (5.3.2) Here, we use the fact that m ( j) = 1 for j roots in z(n). Our strategy is to rst nd all j 2 Δ(z(n)) so that + j is Δ(l )dominant. We then consider the contributions from roots j with + j not Δ(l )dominant. The case Cn(i) for 2 i n 1 is demonstrated rst. Later, we handle the F4(4) case. Let q be of type Cn(i) for 2 i n 1. The deleted Dynkin diagram is ◦ 1 : : : ◦ i1 i ◦ i+1 ◦ n1 ks ◦ n and the subgraph corresponding to l is ◦ 1 ◦ 2 ◦ 3 : : : ◦ i1: (5.3.3) The data in Appendix C shows that Δ+(l ) = f"j "k j 1 j < k ig and Δ(z(n)) = f"j + "k j 1 j < k ig [ f2"j j 1 j ig: We have = "1 "i and = 2"1. If is the set of all j 2 Δ(z(n)) so that + j is Δ(l )dominant then, by Remark 5.2.4, the character may be written as = Σ j2 + j + Σ j2Δ(z(n))n sgn( + j + (l )) d( + j+ (l )) (l ): (5.3.4) One can see by direct computation that = 8>>>>>>< >>>>>>: f ; "1 + "2; 2"2g if i = 2 f ; "1 + "2; "1 + "3; "2 + "3g if i = 3 f ; "1 + "2; "1 + "i; "2 + "3; "2 + "ig if 4 i n 1 : 62 When i = 2, we have = Δ(z(n)), and so, is = Σ j2 + j = + + +("1+"2) + +(2"2): Since = "1 "2, we have + ("1 + "2) = 2"1 = . When i = 3, it follows that Δ(z(n))n = f2"2; 2"3g. Since we have s"1"2( + 2"2 + (l )) = + 2"2 + (l ) and s"2"3( + 2"3 + (l )) = + 2"3 + (l ), both weights are not Δ(l )regular and do not contribute to the character. Therefore, when i = 3, = Σ j2 + j = + + +("1+"2) + +("1+"3) + +("2+"3): Since = "1 "3, we have + ("1 + "3) = 2"1 = . If 4 i n 1 then j 2 Δ(z(n))n is "1 + "k for 3 k i 1; "2 + "k for 4 k i 1; "r + "k for 3 r < k i, or 2"r for 2 r i: An observation shows that, for each j 2 Δ(z(n))n with j ̸= 2"3, there exists w 2 W(l ) with w ̸= 1 so that w xes + j + (l ). Indeed, it is clear from (5.3.3) that l is of type Ai1. Thus (l ) is given by (l ) = Σi s=1 (i (2s 1) 2 ) "s: (5.3.5) If w = 8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>: s"k1"k when j = "1 + "k; "2 + "k s"r1"r when j = "r + "k s"1"2 when j = 2"2 s"r2"r when j = 2"r for 4 r i 1 s"i1"i when j = 2"i 63 then w( + j + (l )) = + j + (l ). Therefore sgn( + j + (l )) = 0 for such j . Now suppose that j = 2"3. We rst show that + 2"3 + (l ) is Δ(l )regular. By (5.3.5), we have + 2"3 + (l ) = (i + 1 2 ) "1 + (i 3 2 ) "2 + (i 1 2 ) "3 + Σi1 s=4 (i (2s 1) 2 ) "s + ( i + 1 2 ) "i: (5.3.6) The coefficients of "s and "t with s ̸= t in (5.3.6) are different. Since roots in Δ+(l ) are of the form "s "t with s < t, this shows that the weight + 2"3 + (l ) is Δ(l )regular. The re ection s"2"3 conjugates +2"3 + (l ) to the Δ(l )dominant weight s"2"3( + 2"3 + (l )) = + ("2 + "3) + (l ): Thus sgn( + j + (l )) = 1 and d( + j + (l )) = +("2 +"3)+ (l ); we have sgn( + j + (l )) d( + j+ (l )) (l ) = +("2+"3): Hence, = Σ j2 + j + Σ j2Δ(z(n))n sgn( + j + (l )) d( + j+ (l )) (l ) = Σ j2 + j +("2+"3) (5.3.7) with = f ; "1 + "2; "1 + "i; "2 + "3; "2 + "ig for 4 i n 1. Then we obtain = Σ j2 + j +("2+"3) = + + +("1+"2) + +("1+"i) + +("2+"3) + +("2+"i) +("2+"3) = + + +("1+"2) + +("1+"i) + +("2+"i): Since = "1 "i, we have + ("1 + "i) = 2"1 = . 64 Next we consider the case that q is of type F4(4). The deleted Dynkin diagram is ◦ 1 ◦ 2 +3 ◦ 3 4 and the subgraph corresponding to l is ◦ 1 ◦ 2 +3 ◦ 3: The simple subalgebra l is isomorphic to so(7;C). If we write the fundamental weights of l = so(7;C) corresponding to 1, 2, 3 as ϖ1, ϖ2, ϖ3, respectively, then the highest weights for l and for z(n) are written in terms of the fundamental weights as = ϖ2 and = ϖ1; we have l = V (ϖ2) and z(n) = V (ϖ1). Therefore the adjoint action of l on itself (resp. on z(n)) is equivalent to the standard action of so(7;C) on ^2C7 (resp. on C7). We then identify the l module l z(n) as the so(7;C)module (^2C7) (C7), and consider the irreducible decomposition of (^2C7) (C7). Let Δ+ be the standard choice of a positive system of so(7;C) and be half the sum of the positive roots; that is, Δ+ = f"1 "2; "2 "3; "1 "3g [ f"1; "2; "3g and = 5 2 "1 + 3 2 "2 + 1 2 "3: If = f 2 Δ(C7) j ϖ2 + is dominantg with Δ(C7) the set of weights for C7 then the character ϖ2 ϖ1 for (^2C7) (C7) = 65 V (ϖ2) V (ϖ1) is ϖ2 ϖ1 = Σ 2Δ(C7) mϖ1( )sgn(ϖ2 + + ) d(ϖ2+ + ) = Σ 2Δ(C7) sgn(ϖ2 + + ) d(ϖ2+ + ) = Σ 2 ϖ2+ + Σ 2Δ(C7)n sgn(ϖ2 + + ) d(ϖ2+ + ) : We need determine the contributions from 2 Δ(C7)n. The weights for C7 under the standard action of so(7;C) are Δ(C7) = f "1; "2; "3; 0g: In terms of the fundamental weights ϖ1, ϖ2, and ϖ3, we have "1 = ϖ1; "2 = ϖ1 + ϖ2; "3 = ϖ2 + 2ϖ3: Therefore, the weights for C7 may be written in terms of the fundamental weights as Δ(C7) = f ϖ1; (ϖ1 + ϖ2); (ϖ2 + 2ϖ3); 0g: If is a weight for C7 so that ϖ2 + is Δ(l )dominant then must be = ϖ1;ϖ2 ϖ2;ϖ2 + 2ϖ3; or 0: (5.3.8) Thus, Δ(C7)n = fϖ1;ϖ1 + ϖ2;ϖ2 2ϖ3g = f"1; "2;"3g: Observe that when = "1 or "2, there exists a Weyl group element w 2 W of so(7;C) that xes ϖ2+ + . Indeed, for either case = "1 or "2, the root re ection s"1"2 xes ϖ2 + + , as ϖ2 = "1 + "2. Thus sgn(ϖ2 + + ) = 0 when = "1 or "2. On the other hand, when = "3, we have ϖ2 "3 + = 7 2 "1 + 5 2 "2 1 2 "3: (5.3.9) 66 The coefficients of "s and "t with s ̸= t in (5.3.9) are different. Since roots in Δ+ are of the form "s "t with s < t or "s, this shows that the weight ϖ2 "3 + is Δ(l )regular. The re ection s"3 conjugates ϖ2"3+ to the Δ(l )dominant weight s"3(ϖ2 "3 + ) = 7 2 "1 + 5 2 "2 + 1 2 "3: Thus sgn(ϖ2 "3 + ) = 1 and d(ϖ2 "3 + ) = "1 + "2 = ϖ2; we have sgn(ϖ2 "3 + ) d(ϖ2"3+ ) = ϖ2 : Hence, ϖ2 ϖ1 = Σ 2 ϖ2+ + Σ 2Δ(C7)n sgn(ϖ2 + + ) d(ϖ2+ + ) = Σ 2 ϖ2+ ϖ2 : By (5.3.8), we have = fϖ1;ϖ2 ϖ2;ϖ2 + 2ϖ3; 0g. Therefore, ϖ2 ϖ1 = Σ 2 ϖ2+ ϖ2 = ϖ2+ϖ1 + ϖ2+(ϖ1ϖ2) + ϖ2+(ϖ2+2ϖ3): We have ϖ2+ϖ1 = + , ϖ2+(ϖ1ϖ2) = ϖ1 = , and ϖ2+(ϖ2+2ϖ3) = + 0 with 0 the root in Δ(z(n)) in Theorem 5.1.3. This completes the proof. 67 CHAPTER 6 Special Constituents of l z(n) In this chapter, by using the decomposition results in Chapter 5, we shall determine the candidates of the irreducible constituents of l z(n) that will contribute to the Ω2 systems; that is, the irreducible constituents V ( ) so that ~ 2jV ( ) are not identically zero. 6.1 Special Constituents Given V ( ), an irreducible constituent in l z(n), we build an Lintertwining map ~ 2jV ( ) 2 HomL(V ( ) ;P2(g(1))) with V ( ) the dual of V ( ) with respect to the Killing form . From ~ 2jV ( ) , we construct operator Ω2jV ( ) : V ( ) ! D(Ls) n. To do so, it is necessary to determine which irreducible constituents V ( ) have property that ~ 2jV ( ) ̸= 0. We start by observing the vector space isomorphism P2(g(1)) = Sym2(g(1)) . With the natural Laction on P2(g(1)) and Sym2(g(1)) , this vector space isomor phism is Lequivariant. Thus, if ~ 2 V ( ) is a nonzero map then V ( ) is an irreducible constituent of Sym2(g(1)) g(1) g(1); in particular, by Fact (II) in Section 5.2, is of the form = + ϵ for some ϵ 2 Δ(g(1)), where is the highest weight of g(1). One can see from the decompositions in Theorem 5.1.3 that V ( ) is an irreducible constituent of l z(n) for any q under consideration. By Lemma 3.4.2, we have = + ϵ for some ϵ 2 Δ(g(1)). Now we claim that ~ 2jV ( ) is identically zero. It is 68 wellknown that g(1) g(1) = Sym2(g(1)) ^2(g(1)) (6.1.1) as an Lmodule. Since each weight for g(1) is a root of g, by Fact (III) in Section 5.2, the Lmodule decomposition (6.1.1) is multiplicity free. Proposition 6.1.2 The Lmodule V ( ) is an irreducible constituent of ^2(g(1)). Proof. De ne a linear map φ : z(n) ! ^2(g(1)) by means of φ(W) = Σ 2Δ(g(1)) ad(W)X ^ X : By using an argument similar to that for Lemma 2.5.4, one can show that φ is L equivariant. Then, since z(n) = V ( ) as an irreducible Lmodule, it suffices to show that φ is a nonzero map. Write Δ (g(1)) = f 2 Δ(g(1)) j 2 Δg. By Lemma 3.4.2, we have 2 Δ. Hence Δ (g(1)) ̸= ∅. By writing ′ = for 2 Δ (g(1)), φ(X ) is given by φ(X ) = Σ 2Δ(g(1)) ad(X )X ^ X = Σ 2Δ (g(1)) N ; X ′ ^ X : Observe that for each 2 Δ (g(1)), we have 2 Δ (g(1)). Moreover, by Property (H6) of our normalizations in Section 4.1, it follows that N ; ′ = N ; . Therefore, N ; X ′ ^ X + N ; ′X ^ X ′ = 2N ; X ′ ^ X : (6.1.3) Since N ; ̸= 0 for 2 Δ (g(1)), equation (6.1.3) is nonzero. On the other hand, if 2 Δ (g(1)) and 2 Δ (g(1)) is so that ̸= ; ′ then X ′ ^ X and X ^ X are linearly independent. Hence, φ(X ) ̸= 0. De nition 6.1.4 An irreducible constituent V ( ) of l z(n) is called special if ̸= and there exists ϵ 2 Δ(g(1)) so that = +ϵ, where and are the highest weights for g(1) and z(n), respectively. 69 Proposition 6.1.5 Let V ( ) be an irreducible constituent of l z(n). Then ~ 2 V ( ) is not identically zero only if V ( ) is a special constituent of l z(n). Proof. At the beginning of this section we observed that if ~ 2jV ( ) ̸= 0 then must be of the form = + ϵ for some ϵ 2 Δ(g(1)). Then V ( ) is either a special constituent or V ( ) (by Lemma 3.4.2, satis es the form). However, by Proposition 6.1.2, it follows that ~ 2jV ( ) is identically zero. Therefore, V ( ) must be a special constituent. We will show in Chapter 7 that the converse of Proposition 6.1.5 also holds for certain special constituents (see Proposition 7.1.6). 6.2 Types of Special Constituents The aim of this section is to determine and classify all the special constituents of l z(n). Such a classi cation will play a role in the explicit construction of the Ω2 systems. We use the decomposition results in Chapter 5 for the rest of this chapter. The parabolic subalgebra q under consideration is assumed to be one in (3.3.2) or (3.3.3). Since l z(n) = (CHq z(n)) (lss z(n)) and CHq z(n) = V ( ), it suffices to consider lss z(n) = (l z(n)) (ln z(n)). We start by observing that, by Proposition 5.1.2, ln z(n) = V ( n + ). Proposition 6.2.1 Suppose that ln ̸= 0. Then the irreducible constituent V ( n + ) is special. Proof. We need to show that n + = + for some 2 Δ(g(1)). This is precisely the statement (1) of Lemma 3.4.5. We next investigate the Cartan component V ( + ) of l z(n) = V ( ) V ( ). 70 Lemma 6.2.2 The Cartan component V ( + ) of l z(n) is not special. Proof. Lemma 3.4.5 and Remark 3.4.7 show that + =2 Δ(g(1)), which implies that + ̸= + for all 2 Δ(g(1)). We determine all the special constituents of l z(n) in two steps. First we assume that g is a classical algebra, and then consider the case that g is an exceptional algebra. For classical cases the parabolic subalgebras q under consideration are of type Bn(i) (3 i n), Cn(i) (2 i n 1), or Dn(i) (3 i n 3). It will be convenient to write 2 Δ(g(1)) in terms of the fundamental weights of l and ln . It is clear from the deleted Dynkin diagrams that, for each of the cases, (l ) and (ln ) are given by (l ) = f r j 1 r i 1g and (ln ) = f i+s j 1 s n ig; where j are the simple roots with the standard numbering. By using the standard realizations of roots, we have r = "r "r+1 for 1 r i 1, i+s = "i+s "i+s+1 for 1 s n i 1, and n = 8>>>>>>< >>>>>>: "n if g is of type Bn 2"n if g is of type Cn "n1 + "n if g is of type Dn. The data in Appendix C shows that Δ(g(1)) is Δ(g(1)) = 8>>< >>: f"j "k j 1 j i and i + 1 k ng [ f"j j 1 j ig if q is of type Bn(i) f"j "k j 1 j i and i + 1 k ng if q is of type Cn(i) or Dn(i). 71 Since we have two simple algebras l and ln , we use the notation ϖr for the funda mental weights of r 2 (l ) and ~ϖs for those of i+s 2 (ln ). Direct computation then shows that each 2 Δ(g(1)) is exactly one of the following form: = 8>>>>>>< >>>>>>: ϖ1 + Σni s=1 ~ms ~ϖs; (ϖr + ϖr+1) + Σni s=1 ~ms ~ϖs with 1 r i 2, or ϖi1 + Σni s=1 ~ms ~ϖs (6.2.3) for some ~ms 2 Z. Proposition 6.2.4 Let V ( ) be an irreducible constituent of l z(n). 1. If q is of type Bn(i) (3 i n) or Dn(i) (3 i n3) then V ( ) is a special constituent if and only if = 2"1. 2. If q is of type Cn(i) (2 i n 1) then V ( ) is a special constituent if and only if = "1 + "2. Proof. Suppose that q is of type Bn(i), Cn(i), or Dn(i). By De nition 6.1.4, we need to nd all of the form = + for some 2 Δ(g(1)). Here , the highest weight for g(1), is = 8>>< >>: "1 + "i+1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) "1 if q is of type Bn(n): We write in terms of the fundamental weights of l and ln ; that is, = 8>>< >>: ϖ1 + ~ϖ1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) ϖ1 if q is of type Bn(n); (6.2.5) where ϖ1 and ~ϖ1 are the fundamental weights of 1 = "1 "2 and i+1 = "i+1 "i+2, respectively. As ln acts trivially on both l and z(n), the highest weight for a 72 constituent V ( ) l z(n) is of the form = Σi1 j=1 njϖj for nj 2 Z 0. (6.2.6) If there exists 2 Δ(g(1)) so that = + then (6.2.5) and (6.2.6) imply that = is of the form = 8>>< >>: (n1 1)ϖ1 + Σi1 j=2 njϖj ~ϖ1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) (n1 1)ϖ1 + Σi1 j=2 njϖj if q is of type Bn(n) (6.2.7) for nj 2 Z 0. On the other hand, we observed that the root must be one of the forms in (6.2.3). Then observation shows that if satis es both (6.2.3) and (6.2.7) then must be = 8>>< >>: ϖ1 ~ϖ1 or (ϖ1 + ϖ2) ~ϖ1 if q is of type Bn(i) with i ̸= n, Cn(i), or Dn(i) ϖ1 or (ϖ1 + ϖ2) if q is of type Bn(n): Therefore = + is = 2ϖ1 or ϖ2, which shows that = 2"1 or "1 + "2. As = "1 "i for q of type Bn(i), Cn(i), or Dn(i), Theorem 5.1.3 shows that both V (2"1) and V ("1+"2) occur in l z(n). Now the assertions follow from the fact that the highest root of g is = "1 + "2 if g is of type Bn or Dn, and = 2"1 if g is of type Cn. If g is an exceptional algebra then the parabolic subalgebras q under consideration are E6(3);E6(5);E7(2);E7(6);E8(1); and F4(4): (6.2.8) Lemma 6.2.9 If q is of exceptional type as in (6.2.8) then V ( + 0) in Theorem 5.1.3 is a special constituent. Proof. This is done by a direct computation. The roots ϵ in Δ(g(1)) so that + 0 = + ϵ are given in Table 6.4 below. 73 Proposition 6.2.10 There exists a unique special constituent in l z(n). Proof. If q is of classical type then this proposition follows from Proposition 6.2.4. For q of exceptional type, by Theorem 5.1.3, the tensor product l z(n) decomposes into l z(n) = V ( + ) V ( ) V ( + 0) with 0 2 Δ(n) as in Theorem 5.1.3. Then Lemma 6.2.2 and Lemma 6.2.9 show that V ( + 0) is the unique special constituent. Since the weight ϵ 2 Δ(g(1)) so that + ϵ is the highest weight of a special constituent will play a role later, we introduce the notation related to ϵ. De nition 6.2.11 We denote by ϵ the root contributing to g(1) so that V ( + ϵ ) is the special constituent of l z(n). Similarly, we denote by ϵn the root for g(1) so that V ( + ϵn ) = ln z(n). In Table 6.1, Table 6.2, Table 6.3, and Table 6.4 we summarize the results of this section. Table 6.1 and Table 6.2 contain the highest weight of each special constituent occurring in l z(n) for each parabolic q of classical algebras and exceptional algebras. Table 6.3 and Table 6.4 list the roots , ϵ , and ϵn for each q. A dash indicates that no special constituent exists for the case. Table 6.1: Highest Weights for Special Constituents (Classical Cases) Type V ( + ϵ ) V ( + ϵn ) Bn(i); 3 i n 2 2"1 "1 + "2 + "i+1 + "i+2 Bn(n 1) 2"1 "1 + "2 + "n Bn(n) 2"1 Cn(i); 2 i n 1 "1 + "2 2"1 + 2"i+1 Dn(i); 3 i n 3 2"1 "1 + "2 + "i+1 + "i+2 74 Table 6.2: Highest Weights for Special Constituents (Exceptional Cases) Type V ( + ϵ ) V ( + ϵn ) E6(3) 1 + 2 2 + 2 3 + 4 4 + 3 5 + 2 6 2 1 + 2 2 + 2 3 + 3 4 + 2 5 + 6 E6(5) 2 1 + 2 2 + 3 3 + 4 4 + 2 5 + 6 1 + 2 2 + 2 3 + 3 4 + 2 5 + 2 6 E7(2) 2 1 + 2 2 + 4 3 + 5 4 + 4 5 + 3 6 + 2 7 E7(6) 2 1 + 3 2 + 4 3 + 6 4 + 4 5 + 2 6 + 7 2 1 + 2 2 + 3 3 + 4 4 + 3 5 + 2 6 + 2 7 E8(1) 2 1 + 4 2 + 5 3 + 8 4 + 7 5 + 6 6 + 4 7 + 2 8 F4(4) 2 1 + 4 2 + 6 3 + 2 4 Table 6.3: The Roots , ϵ , and ϵn (Classical Cases) Type ϵ ϵn Bn(i); 3 i n 2 "1 + "i+1 "1 "i+1 "2 + "i+2 Bn(n 1) "1 + "n "1 "n "2 Bn(n) "1 "1 Cn(i); 2 i n 1 "1 + "i+1 "2 "i+1 "1 + "i+1 Dn(i); 3 i n 3 "1 + "i+1 "1 "i+1 "2 + "i+2 75 Table 6.4: The Roots , ϵ , and ϵn (Exceptional Cases) Type ϵ ϵn E6(3) 2 + 3 + 2 4 + 5 + 6 1 + 2 + 3 + 4 E6(5) 1 + 2 + 3 + 2 4 + 5 2 + 4 + 5 + 6 E7(2) 1 + 2 + 2 3 + 2 4 + 5 + 6 + 7 E7(6) 1 + 2 + 2 3 + 3 4 + 2 5 + 6 1 + 3 + 4 + 5 + 6 + 7 E8(1) 1 + 2 + 2 3 + 3 4 + 3 5 + 3 6 + 2 7 + 8 F4(4) 1 + 2 2 + 3 3 + 4 with E6(3) : = 1 + 2 + 3 + 2 4 + 2 5 + 6 E6(5) : = 1 + 2 + 2 3 + 2 4 + 5 + 6 E7(2) : = 1 + 2 + 2 3 + 3 4 + 3 5 + 2 6 + 7 E7(6) : = 1 + 2 2 + 2 3 + 3 4 + 2 5 + 6 + 7 E8(1) : = 1 + 3 2 + 3 3 + 5 4 + 4 5 + 3 6 + 2 7 + 8 F4(4) : = 1 + 2 2 + 3 3 + 4 76 By Proposition 6.1.5, only special constituents could contribute to the construc tion of the Ω2 systems. Next we want to show that ~ 2jV ̸= 0 when V is a special constituent. An observation on the highest weights for the special constituents will simplify the argument. We classify them by their highest weights and call them type 1a, type 1b, type 2, and type 3. De nition 6.2.12 We say that a special constituent V ( ) of l z(n) is of 1. type 1a if = + ϵ is not a root with ϵ ̸= and both and ϵ are long roots, 2. type 1b if = + ϵ is not a root with ϵ ̸= and either or ϵ is a short root, 3. type 2 if = + ϵ = 2 is not a root, or 4. type 3 if = + ϵ is a root, where is the highest weight for g(1) and ϵ = ϵ or ϵn is the root in Δ(g(1)) de ned in De nition 6.2.11. Example 6.2.13 The following are examples of each type of special constituents: 1. type 1a: V ( + ϵ ) for type Bn(n 1) ( + ϵ = ("1 + "n) + ("1 "n) ) 2. type 1b: V ( + ϵn ) for type Bn(n 1) ( + ϵn = ("1 + "n) + ("2) ) 3. type 2: V ( + ϵn ) for type Cn(i) ( + ϵn = 2("1 + "i+1) = 2 ) 4. type 3: V ( + ϵ ) for type Cn(i) ( + ϵ = "1 + "2 ) Table 6.5 summarizes the types of special constituents for each parabolic subagle bra q. One may want to observe that almost all the special constituents are of type 1a. We regard any roots as long roots, if g is simply laced. A dash indicates that no special constituent exists in the case. 77 Table 6.5: Types of Special Constituents Type V ( + ϵ ) V ( + ϵn ) Bn(i); 3 i n 2 Type 1a Type 1a Bn(n 1) Type 1a Type 1b Bn(n) Type 2 Cn(i); 2 i n 1 Type 3 Type 2 Dn(i); 3 i n 3 Type 1a Type 1a E6(3) Type 1a Type 1a E6(5) Type 1a Type 1a E7(2) Type 1a E7(6) Type 1a Type 1a E8(1) Type 1a F4(4) Type 2 Remark 6.2.14 It is observed from Table 6.3 and Table 6.4 that we have ϵ =2 Δ, unless V ( + ϵ) is of type 3. Remark 6.2.15 Table 6.5 shows that when V ( + ϵ) is a special constituent of type 1a, the parabolic subalgebra q is of type Bn(i) (3 i n 1), Dn(i), E6(3), E6(5), E7(2), E7(6), or E8(1). The data in Appendix C shows that when q is of type Bn(i) for 3 i n 1, the simple root q = "i "i+1 that parametrizes q is a long root and Δ(z(n)) contains solely long roots. Since we regard any roots as long roots for g simply laced, it follows that when V ( + ϵ) is of type 1a, the simple root q and any root j 2 Δ(z(n)) are all long roots. 78 6.3 Technical Results In this section we collect technical results on the special constituents, so that certain arguments will go smoothly in Chapter 7. The weight vectors X and the structure constants N ; are normalized as in Section 4.1. Lemma 6.3.1 Let V ( +ϵ) be a special constituent l z(n) of type 1a, and 2 Δ+(l). If ϵ + 2 Δ then 2 Δ. Proof. We show that ⟨ ; ⟩ > 0. Since + ϵ is the highest weight of an irreducible lmodule, it is Δ(l)dominant. Thus, ⟨ + ϵ; ⟩ = ⟨ ; ⟩ + ⟨ϵ; ⟩ 0: (6.3.2) Observe that, as + ϵ is of type 1a, ϵ is a long root of g. Since + ϵ is assumed to be a root, Lemma 3.4.4 implies that ⟨ ; ϵ_⟩ = 1; in particular, ⟨ϵ; ⟩ < 0. Now, by (6.3.2), we have ⟨ ; ⟩ ⟨ϵ; ⟩ > 0: Lemma 6.3.3 Let V ( + ϵ) be a special constituent of l z(n) of type 1a. Then, for 2 Δ+(l) with + ϵ 2 Δ, we have ad(X )ad(X +ϵ)X j = 0 for all j 2 Δ(z(n)). Proof. If ( + ϵ) j =2 Δ then there is nothing to prove. So we assume that ( + ϵ) j 2 Δ and + ( + ϵ) j 2 Δ. Since + ϵ is assumed to be of type 1a, the root is long. Lemma 3.4.4 then implies that ⟨( + ϵ) j ; _⟩ = 1: (6.3.4) 79 By Remark 6.2.14, we have ⟨ϵ; _⟩ = 0. Thus (6.3.4) becomes ⟨ ; _⟩ ⟨ j ; _⟩ = 1: (6.3.5) Since is the highest weight for g(1), j 2 Δ(z(n)), and 2 Δ+(l), neither + nor j + is a root. Then, as is a long root, (6.3.5) holds if and only if ⟨ ; _⟩ = 0 and ⟨ j ; _⟩ = 1. On the other hand, since + ϵ is a root by hypothesis and by Lemma 6.3.1, is a root. In particular, by Lemma 3.4.4, ⟨ ; _⟩ = 1. Now we have ⟨ ; _⟩ = 1 and ⟨ ; _⟩ = 0, which is a contradiction. For any ad(h)invariant subspace W g and any weight 2 h , we write Δ (W) = f 2 Δ(W) j 2 Δg: In Chapter 7, we will construct the Ω2jV ( +ϵ) systems and nd their special values, when V ( +ϵ) is of either type 1a or type 2. When we do so, the roots 2 Δ +ϵ(g(1)) and j 2 Δ +ϵ(z(n)) will play a role. Therefore, for the rest of this section, we shall show several technical results about those roots, so that certain argument will become simple. First of all, we need check that Δ +ϵ(g(1)) and Δ +ϵ(z(n)) are not empty. It is clear that Δ +ϵ(g(1)) ̸= ∅, since , ϵ 2 Δ +ϵ(g(1)). Moreover, Lemma 6.3.6 below shows that when V ( + ϵ) is of type 2, we have Δ +ϵ(g(1)) = f g. Lemma 6.3.6 If V ( +ϵ) is a special constituent of l z(n) of type 2 then Δ +ϵ(g(1)) = f g. Proof. First we claim that has the maximum height among the roots 2 Δ(g(1)). As g(1) is the irreducible Lmodule with highest weight , any root 2 Δ(g(1)) is of the form = Σ 2 (l) n with n 2 Z 0. Then if ht( ) and ht( ) denote the 80 heights of and , respectively, then ht( ) = ht( ) + Σ 2 (l) n ht( ): Now as V ( + ϵ) is of type 2, by de nition, we have + ϵ = 2 . If 2 Δ2 (g(1)) then 2 2 Δ(g(1)). In particular, the height ht(2 ) satis es ht( ) ht(2 ). If = Σ 2 (l) n with n 2 Z 0 then ht( ) ht(2 ) = 2ht( ) ht( ) = 2ht( ) ht( ) + Σ 2 (l) n = ht( ) + Σ 2 (l) n : This forces that Σ 2 (l) n = 0. Therefore = . Lemma 6.3.7 If V ( + ϵ) is a special constituent of l z(n) then Δ +ϵ(z(n)) ̸= ∅. Proof. By Fact (II) in Section 5.2, the highest weight + ϵ of V ( + ϵ) l z(n) is of the form + ϵ = 8>>< >>: + ′ if V ( + ϵ) l z(n) n + ′′ if V ( + ϵ) = ln z(n) for some ′; ′′ 2 Δ(z(n)), where and n are the highest weights for l and ln , respectively. Then we have ′; ′′ 2 Δ +ϵ(z(n)). The following simple technical lemma will simplify an argument in later proofs. Lemma 6.3.8 Let ; ; 2 Δ with , ̸= . If + =2 Δ and + 2 Δ then the following hold: (1) ; 2 Δ, and (2) N ; N ; = N ; N ; . Proof. For the rst assertion, we show that 2 Δ. Suppose that =2 Δ, so ⟨ ; ⟩ 0. By hypothesis, we have ⟨ ; ⟩ 0. Thus it follows that ⟨ + ; ⟩ = ⟨ ; ⟩ + ⟨ ; ⟩ ⟨ ; ⟩ > 0: 81 Therefore, = ( + ) is a root. Now let X , X , and X be the root vectors of , , and , respectively, normalized as in Section 4.1. Since 2 Δ, we have N ; ̸= 0 (see Property (H7) in Section 4.1). Moreover, the conditions that ; + 2 Δ imply that N ; ̸= 0. On the other hand, we have [X ;X ] = 0 by assumption, and [X ;X ] = 0 by hypothesis. So it follows from the Jacobi identity that 0 = [X ; [X ;X ]] = [X ; [X ;X ]] = N ; N ; X + ̸= 0; which is absurd. Therefore 2 Δ. Since it may be shown similarly that 2 Δ, we omit the proof. Observe that the condition + =2 Δ implies that ad(X )ad(X ) = ad(X )ad(X ) by the Jacobi identity. Therefore, ad(X )ad(X )X = ad(X )ad(X )X , which implies that N ; N ; = N ; N ; : Lemma 6.3.9 Let W be any ad(h)invariant subspace of g with Δ +ϵ(W)nf ; ϵg ̸= ∅. If V ( + ϵ) is a special constituent of l z(n) of type 1a, type 1b, or type 2 then, for any 2 Δ +ϵ(W)nf ; ϵg, we have , ϵ 2 Δ. Proof. If V ( + ϵ) is of type 1a, type 1b, or type 2 then, by de nition, + ϵ is not a root. Then this lemma simply follows from Lemma 6.3.8 Remark 6.3.10 A direct observation shows that if V ( + ϵ) is a special constituent of type 1a then Δ +ϵ(g(1))nf ; ϵg ̸= ∅. Lemma 6.3.11 If V ( + ϵ) is a special constituent of l z(n) of type 1a then, for any 2 Δ +ϵ(g(1)) and any j 2 Δ +ϵ(z(n)), we have j 2 Δ. Proof. By Lemma 6.3.8, we have j ; j ϵ 2 Δ. So, let ̸= ; ϵ. We show that ⟨ j ; ⟩ > 0. Observe that since 2 Δ(g(1)) and j 2 Δ(z(n)), we have j + =2 Δ. 82 Thus ⟨ j ; ⟩ 0. Since 2 Δ +ϵ(g(1))nf ; ϵg and j 2 Δ +ϵ(z(n)), by Lemma 6.3.9, we have , ϵ j 2 Δ. Then we rst claim that if ⟨ j ; ⟩ = 0 then ( ) + (ϵ j) 2 Δ. Since V ( + ϵ) is assumed to be of type 1a, both and ϵ are long roots. Thus, by Lemma 3.4.4, ⟨ j ; _⟩ = ⟨ ; ϵ_⟩ = 1; in particular, ⟨ j ; ⟩, ⟨ ; ϵ⟩ > 0. By Remark 6.2.14, we have ⟨ ; ϵ⟩ = 0. Then, ⟨ ; ϵ j⟩ = ⟨ ; j⟩ ⟨ ; ϵ⟩ < 0: Therefore, as ; ϵ j 2 Δ, it follows that ( ) + (ϵ j) 2 Δ. On the other hand, since ⟨ ; ϵ⟩ = 0 and ⟨ j ; ⟩ is assumed to be 0, we have jj( ) + (ϵ j)jj2 = jj jj2 + jj jj2 + jjϵjj2 + jj j jj2 2⟨ ; ⟩ 2⟨ ; ϵ⟩ 2⟨ j ; ⟩ 2⟨ j ; ϵ⟩: For = ; j and = ; ϵ, by Lemma 3.4.4, we have ⟨ ; _⟩ = 2⟨ ; ⟩=jj jj2 = 1, as and ϵ are long roots. Therefore, 2⟨ ; ⟩ = jj jj2, and so, jj( ) + (ϵ j)jj2 = jj jj2 + jj j jj2 jj jj2 jjϵjj2: Since and ϵ are assumed to be long roots, this shows that jj( )+(ϵ j)jj2 0, which contradicts that ( ) + (ϵ j) is a root. Hence, ⟨ j ; ⟩ > 0. Lemma 6.3.12 If V ( + ϵ) is a special constituent of l z(n) of type 1a or type 2 then, for any j 2 Δ +ϵ(z(n)), Δ +ϵ(g(1)) Δ j (g(1)): In particular, Δ j (g(1)) ̸= ∅ for any j 2 Δ +ϵ(z(n)). Proof. If V ( + ϵ) is of type 1a then the assertion follows from Lemma 6.3.11. If V ( + ϵ) is of type 2 then Lemma 6.3.6 implies that Δ +ϵ(g(1)) = f g. Now this lemma follows from Lemma 6.3.9 by taking = j 2 Δ +ϵ(z(n)). 83 If V ( + ϵ) is a special constituent of l z(n) then, for 2 Δ, we write ( ) = ( + ϵ) : Lemma 6.3.13 If V ( + ϵ) is a special constituent of l z(n) of type 1a or type 2 then, for any j 2 Δ +ϵ(z(n)), Δ ( j )(g(1)) ̸= ∅: Proof. Since j 2 Δ +ϵ(z(n)), we have ( + ϵ) j 2 Δ. As V ( + ϵ) is assumed to be of type 1a or type 2, by de nition, it follows that + ϵ =2 Δ. Thus, by Lemma 6.3.8, we have j 2 Δ and ϵ j 2 Δ. Then, ( j) = ( + ϵ) j = ϵ j 2 Δ; that is, 2 Δ ( j )(g(1)). Lemma 6.3.14 If V ( + ϵ) is a special constituent of type 1a or type 2 then Σ j2Δ +ϵ(z(n)) N ;ϵ jN ; jϵNϵ; jNϵ; j > 0; where N ; are the structure constants for ; 2 Δ, de ned in Section 4.1. Proof. It follows from Property (H7) of our normalizations in Section 4.1 that N ;ϵ jN ; jϵ = q ;ϵ j (1 + p ;ϵ j ) 2 jj jj2 and Nϵ; jNϵ; j = qϵ; j (1 + pϵ; j ) 2 jjϵjj2: In particular, by (4.1.1) in Section 4.1, we have N ;ϵ jN ; jϵ 0 and Nϵ; jNϵ; j 0. By Lemma 6.3.7 and Lemma 6.3.9, Δ +ϵ(z(n)) ̸= ∅ and j ϵ 2 Δ for any j 2 Δ +ϵ(z(n)). Therefore, for all j 2 Δ +ϵ(z(n)), we have N ;ϵ jN ; jϵNϵ; jNϵ; j > 0: 84 Lemma 6.3.15 If V ( + ϵ) is a special constituent of type 1a then, for any 2 Δ +ϵ(g(1))nf ; ϵg and any j 2 Δ +ϵ(z(n)), [X j ;X ] = [X ( j );X ] = 0: Proof. We show that j + and ( j)+ are neither zero nor roots. First of all, if j + = 0 then j = 2 Δ(l), which contradicts that j 2 Δ(z(n)). Next, if ( j) + = 0 then since ( j) + = ϵ + j , we would have + ϵ = j 2 Δ. On the other hand, as V ( + ϵ) is assumed to be of type 1a, ϵ is a long root. As 2 Δ +ϵ(g(1))nf ; ϵg, by Lemma 6.3.9, we have ϵ 2 Δ. Then, by Lemma 3.4.4, it follows that + ϵ ̸2 Δ, which is a contradiction. To show j + is not a root, observe that, by Lemma 3.4.4, we have ⟨ j + ; _⟩ = 1 + 1 2 = 2: Thus, if j + 2 Δ then ( j + ) + 2 would be a root. However, since is a long root, it is impossible. The fact that ( j) + =2 Δ can be shown in a similar manner. Lemma 6.3.16 If V ( + ϵ) is a special constituent of type 1a then, for any 2 Δ +ϵ(g(1))nf ; ϵg and any j 2 Δ +ϵ(z(n)), p j; = 0 and q j; = 1; where p ; and q ; are the constants de ned in (4.1.1) in Section 4.1. In particular, we have N j; N( j );( ) = jj j jj2 2 : (6.3.17) Proof. Observe that, by Lemma 6.3.11, ( ) + ( j) = j is a root. As V ( +ϵ) is assumed to be of type 1a, is a long root. By Remark 6.2.15, the root j is also a long root. Therefore j is a long root. Now the rst part of the lemma follows immediately from Lemma 3.4.4, and the second follows from Property (H7) in our normalizations in Section 4.1. 85 Lemma 6.3.18 If V ( +ϵ) is a special constituent of type 1a or type 2 then, for any 2 Δ +ϵ(g(1)) and any j 2 Δ +ϵ(z(n)), N ; jN ( j ); ( ) = N ( ); jN ( j ); : Proof. Observe that, by Property (H3) in Section 4.1, we have (X ; 



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