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Representation Models for Classical Groups and Their Higher Symmetries Astérisque, Tome S131 (1985), P Astérisque I. M. GELFAND A. V. ZELEVINSKY Representation models for classical groups and their higher symmetries Astérisque, tome S131 (1985), p. 117-128 <http://www.numdam.org/item?id=AST_1985__S131__117_0> © Société mathématique de France, 1985, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Société Mathématique de France Astérisque, hors série, 1985, p. 117-128 REPRESENTATION MODELS FOR CLASSICAL GROUPS AND THEIR HIGHER SYMMETRIES BY I.M. GELFAND and A.V. ZELEVINSKY The main results presented in this talk are published with complete proofs in [1]. We give also some new results obtained by the authors jointly with V.V. SERGANOVA. The conversations with V.V. SERGANOVA enable us also to clarify the formulations of [1] related to supermanifolds. We are very grateful to her. Let G be a reductive algebraic group over C. A representation of G which decomposes into the direct sum of all its (finite dimensional) irreducible al­ gebraic representations each occurring exactly once is called a representation model for G. The H. WeyPs unitary trick shows that the construction of such a model is equivalent to the construction of a representation model for the compact form of G ; the language of complex groups is more convenient for us here. The classical example of a representation model for SO3 is the natural representation in the space of functions on the two dimensional sphere. We study two approaches to representation models which are in some sense dual to each other. The first one is to realize a representation model of G as induced representation Ind^r. When r = 1, we say that this is a geometric realization of a model; in this case a model is realized in the space of regular functions on the homogeneous space G/M. The second approach is to realize a model as the restriction to G of a certain special representation of an overgroup of G. More precisely, we shall construct an overgroup L D G and a representation of the Lie algebra 1 of L which is an extension of the action of the Lie algebra g of G on the representation model of G. The action of 1 on a representation model for G will be called higher symmetry of the model (in [1] we used the term "hidden 118 LM. GELFAND and A.V. ZELEVINSKY symmetry" but the present terminology looks more natural). An important example of higher symmetry was recently discovered by L.C. BIEDENHARN and D. FLATH [2] : they construct an action of the Lie algebra so$ on the representation model for SL3. We shall construct geometric realizations and higher symmetries of repre­ sentation models for all classical groups G. Thus, G will be one of the groups GL(n,C) (n > 2), 0(n,C) [n > 3) or Sp(2n, C) [n > 1) (the letter C will be omitted for brevity). All our constructions exhibit the remarkable duality between the orthogonal and symplectic groups, and between symmetric and exterior algebras. As a consequence supergroups and supermanifolds will naturally appear in a "purely even" situation. The systematic study of representation models was initiated in [3]. For classical groups the result of [3] may be formulated as follows : THEOREM 1 [3]. — For any classical group G define the subgroup M and the representation r of M according to the following table [the embedding of M into G is standard, and A(n) denotes the natural representation of a subgroup of GL[n) in the exterior algebra A*(^n))- G GL(n) 0(2n + 1) 0(2n) Sp{2n) M 0(n) 0(n+ 1) x O(n) O(n) x 0[n) GL{n) T A(n) A(n + 1) ® 1 A(n) ® 1 A(n) Then in each case the induced representation IndM r is a representation model for G. In fact, in [3] the model Ind^ r was constructed for any connected reduc­ tive group G. The homogeneous space G/M is the complexification of the symmetric space of maximal rank corresponding to G. The first main result of [1] is the construction of a new series of models which are in some sense dual to models from THEOREM 1. THEOREM 2 [1]. — For any classical group G different from GL(2n-\-l) andO[2n+l) define the subgroup M and the representation r of M according to the following table [the embedding of M into G is standard, and o~[n) denotes the natural representation of a subgroup of GL[n) in the symmetric algebra S*[Cn)). G GL(2n) 0(2n) Sp{4n + 2) Sp(4n) M Sp{2n) GL{n) Sp{2n + 2) x Sp(2n) Sp{2n) x Sp{2n) T <r(2n) a(n) a(2n + 2) <g> 1 a(2n) <g> 1 REPRESENTATION MODELS 119 Then in each case the induced representation Ind^ r is a representation model for G. The spaces G/M in THEOREM 2 also are complexifications of symmetric spaces but not of maximal rank. There is a specific correspondence between models from THEOREMS 1 and 2. Namely, each of the models from THEOREM 1 transforms into a model from THEOREM 2 if all groups and representations are changed as follows : GL{n) is every where replaced by GL(2n), 0(n) by Sp(2n), Sp(2n) by 0(4n), and A(n) by <r(2n). The meaning of this procedure is not clear. For the groups GL{2n + 1) and 0(2n + 1) not covered by THEOREM 2 there are also important representation models constructed in [4]. THEOREM 3 [4]. — Let G = GL(2n + 1) or 0{2n + 1). Define the subgroup M0 in G as follows. For G — GL(2n + 1) put M0 = Sp(2n), the embedding of MQ into G being the composition of standard embeddings Sp{2n) C GL{2n) c GL{2n + 1). For G = 0{2n + 1) put M0 = GL(n), the embedding of M0 into G being the composition of standard embeddings GL{n) C 0(2n) C 0(2n + 1). Then in each case Ind^o 1 is a representation model for G. It turns out that each of the models from THEOREM 2 also may be realized as Ind^o 1; for this we choose a subgroup M0 in M such that r = Indj}fo 1. To describe Mo we need some terminology. Let GL(n —1/2) denote the affine subgroup in GL(n), and Sp(2n — 1) the affine subgroup in Sp(2n), i.e., in each case the stabilizer of a nonzero linear functional on the space of the standard representation. Therefore, in the series of groups GL{n) the index n now may be half-integer, and in the series of groups Sp{n) n may be any positive integer. This terminology enables us to unify THEOREMS 2 and 3 : THEOREM 4 [1]. — For any classical group G define the subgroup M0 in G according to the following table (for G = GL(2n + 1) or 0(2n + 1) the embedding of Mo into G is described in THEOREM 3, and in other cases MQ is embedded in a standard way into the subgroup M from THEOREM 2). G GL{n) 0(n) Sp{2n) Mo Sp(n - 1) GL((n-l)/2) Sp{n - 1) X Sp{n) T 1 1 1 Then in each case IndMo 1 is a representation model for G. THEOREM 4 gives geometric realizations of representation models for all classical groups. In particular, each model from THEOREM 2 is naturally 120 I.M. GELFAND and A.V. ZELEVINSKY interpreted as a geometric realization. The symmetry between THEOREMS 1 and 2 suggests that such an interpretation must exist for models from THEOREM 1. We shall show that each representation model from THEOREM 1 is naturally realized as the space of functions on a certain supermanifold (THEOREMS 6 and 7). Our next goal is to show that each of the representation models from THEOREM 4 admits higher symmetry, i.e., the natural action of the Lie algebra 1 of an overgroup L D G. The overgroup L will be always classical or the product of two classical groups (instead of 0(n) it will be sometimes more convenient to take its connected component of identity SO(n)). For any classical group L we define the homogeneous space T(L) as follows : r(GL(2n)) is the Grassmannian of n-dimensional subspaces in C2n, r(GL(2n + l)) is the Grassmannian of (n + l)-dimensional subspaces in C2n+1, r(0(2n + l)) is the manifold of all isotropic n-dimensional subspaces in C2n, r(0(2n + l)) is the manifold of all isotropic n-dimensional subspaces in C271"*"1, r(5p(2n)) is the manifold of all Lagrangian (n-dimensional) subspaces in C2n. T(L) is connected when L / 0(2n), and T(0(2n)) has two connected components (two isotropic subspaces X and Y belong to the same component of T(0(2n)) if dimensions of X D Z and Y fl Z have the same parity for some (and hence for any) Z G T(0(2n)) ). Finally, for L\ and L2 classical we put T(Li x L2) — T(Li) X T(L2).
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