Unitary Highest Weight Modules Research Articles

Gelfand-Kirillov dimensions of the ℤ-graded oscillator representations of 𝔬(n,ℂ) and 𝔰𝔭(2n,ℂ)

ABSTRACTIn this paper, we give a method to compute the Gelfand–Kirillov dimensions of some polynomial–type weight modules. These modules are infinite-dimensional irreducible 𝔬(n,ℂ)-modules and 𝔰𝔭(2n,ℂ)-modules that appeared in the ℤ-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules.

BRANCHING RULES OF SINGULAR UNITARY REPRESENTATIONS WITH RESPECT TO SYMMETRIC PAIRS (A2n-1, Dn)

The irreducible decomposition of scalar holomorphic discrete series representations when restricted to semisimple symmetric pairs (G, H) is explicitly known by Schmid [Die Randwerte holomorphe funktionen auf hermetisch symmetrischen Raumen, Invent. Math.9 (1969–1970) 61–80] for H compact and by Kobayashi [Multiplicity-Free Theorems of the Restrictions of Unitary Highest Weight Modules with Respect to Reductive Symmetric Pairs, Progress in Mathematics, Vol. 255 (Birhäuser, 2007), pp. 45–109] for H non-compact. In this paper, we deal with the symmetric pair (U(n, n), SO* (2n)), and extend the Kobayashi–Schmid formula to certain non-tempered unitary representations which are realized in Dolbeault cohomology groups over open Grassmannian manifolds with indefinite metric. The resulting branching rule is multiplicity-free and discretely decomposable, which fits in the framework of the general theory of discrete decomposable restrictions by Kobayashi [Discrete decomposability of the restriction of A𝔮(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support, Ann. Math.147 (1998), 709–729].

Generalized MICZ-Kepler problems and unitary highest weight modules

For each integer n ⩾ 1, we demonstrate that a (2n + 1)-dimensional generalized MICZ-Kepler problem has a Spin(2, 2n + 2) dynamical symmetry which extends the manifest Spin(2n + 1) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight Spin(2, 2n + 2)-module with the minimal positive Gelfand–Kirillov dimension. As a byproduct, we obtain a simple geometric realization for such a unitary highest weight Spin(2, 2n + 2)-module.

Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules

Let ( G, K) be a Hermitian symmetric pair and let g and k denote the corresponding complexified Lie algebras. Let g= k⊕ p +⊕ p − be the usual decomposition of g as a k -module. There is a natural correspondence between K C -orbits in p + and a distinguished family of unitarizable highest weight modules for g called the Wallach representations. We denote by Y k the closure of the K C -orbit in p + that is associated to the kth Wallach representation. In this article we give explicit formulas for the numerator polynomials of the Hilbert series of the varieties Y k by using BGG resolutions of unitarizable highest weight modules. A preliminary result gives a new branching formula for a certain two-parameter family of finite dimensional representations of the even orthogonal groups. Our work is an extension of previous work by Enright and Willenbring.

Unitary highest weight modules

The unitary highest weight modules of a semisimple Lie group G have been classified [S, 93. Unfortunately this classification does not fit well with a more general picture of the unitary dual of G. We give such a classification for G a classical group, using derived functors and the philosophy of unipotent representations. In particular these representations are associated to coadjoint orbits of G. Let 8, = Lie(G), 0 a Cartan involution of 6 = 8, @C. A derived functor module is defined by the data q = I + u a O-stable parabolic subalgebra of 0, and 17 an irreducible unitary representation of L, the stabilizer of L in G. Considering Z7 as a representation of I, write Z7= (A, rc) where A E c* (c = center of I, * denotes dual), and rt = Z7l r,,,, . We define A(& rc) = &(,I, rc) (cf. Definition 6.1). Now fix n: and vary 1. Under certain restrictions on I, which amount to a positivity condition on the corresponding bundle over G/L, A(I, rr) is known to be irreducible and unitary. As II varies within this range the Langlands parameters of A(A, rr) vary smoothly. However, one cannot obtain all unitary highest weight modules under these restrictions. The orbit method suggests how to obtain the remaining unitary highest weight modules. The modules A(& rr) are associated to coadjoint orbits as follows. If rc is trivial we say A(]“, rr) is associated to the elliptic orbit 0 = G. II. Not all unitary highest weight modules may be obtained with rc trivial, an example of which is the oscillator representation of $(2n, R) (the metapletic group). We define 2n irreducible unitary highest weight modules rc,+ (1 (2(n-k),R)) is 113 OOOl-8708/87 $7.50

Coadjoint orbits and reductive dual pairs

Let (G, G’) be a reductive dual pair of subgroups of the metaplectic group S$(2m, R), where G is compact. Then the restriction of the oscillator representation o of Sp(2m, R) to G%’ decomposes as x,5=, T,@T: where the correspondence zj++ r( is a bijection [7]. The modules T : are unitary highest weight modules, and all such representations are obtained this way [5]. On the other hand, the unitary highest weight modules of G’ have been constructed in [3] via derived functors; in particular each such representation is associated to (is the “quantization” of) a coadjoint orbit for G’. The purpose of this paper is to establish the connection between these two realizations via the &?-orbit structure of the symplectic manifold R’“CO). These results have been known in principle for some time, the main problem being an understanding of how to quantize nilpotent and degenerate elliptic coadjoint orbits. Let 8, = Lie(G), 0; = Lie(c’). Let $ (respectively Ic/‘) be projection from Gp(2m, R)* to S,* (respectively @A*). Here ( )* = Hom,( , R). Then + and I,+’ restriced to the minimal coadjoint orbit I$,,, of Gp(2m, R)* define the graph of a correspondence between coadjoint Gand (?-orbits. Note that 0 In,” z R*” {O} via the moment map for the action of Sp(2m, R) on R”” { 0 ). Generically this is a bijection. Nilpotent orbits for C? do arise. Suppose a c-orbit 0 corresponds to a (?-orbit C’. In [ 111 it was observed that in some cases if we associate representations X(O) and X’(P’) to 0 and 0’ via geometric quantization, then X(O)@x’(0’) occurs in WJGG, (i.e., X(6) corresponds to x’(0’)). In [2] the derived functor construction was used to quantize some elliptic orbits, and this was done in greater generality. That is, for (0 = G. I. elliptic, ~(8) is cohomologically induced from a one-dimensional representation of L = stab,(i,), and similarly for G’. If ~(0) and n’(0’) have regular infinitesimal character, then n(O) corresponds to ~‘(6’). At the time [2] was written the modules n(O) with singular infinitesimal character were not understood. Furthermore, it was not known how 138 Oool-8708/87 $7.50

Hermitian symmetric spaces and their unitary highest weight modules

The purpose of this article is to determine the set of unitarizable highest weight modules corresponding to Hermitian symmetric spaces of the noncompact type. The major step is that of proving unitarity at the “last possible place.” With this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem: Bernstein-Gelfand-Gelfand, and a diagramatic representation of the set of positive noncompact roots.