A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem

Abstract

Let g be a complex semisimple Lie algebra and let t be the subalgebra of fixed elements in g under the action of an involutory automorphism of g. Any such involution is the complexification of the Cartan involution of a real form of g. If V _λ is an irreducible finite-dimensional representation of g, the Iwasawa decomposition implies that V _λ is a cyclic U(t)module where the cyclic vector is a suitable highest weight vector v _λ. In this paper we explicitly determine generators of the left ideal annihilator L _λ (t) of v _λ in U(t). One of the applications of this result is a branching law which determines how V _λ decomposes as a module for t Other applications include (1) a new structure theorem for the subgroup M (conventional terminology) and its unitary dual, and (2) a generalization of the Cartan-Helgason theorem where, in the generalization, the trivial representation of M (using conventional terminology) is replaced by an arbitrary irreducible representation τ of M. For the generalization we establish the existence of a unique minimal representation of g associated to τ.

Another application (3) yields a noncompact analogue of the Borel-Weil theorem. For a suitable semisimple Lie group G and maximal compact subgroup K, where g = (Lie G)_ℂ and t = (Lie K) _ℂ the representation V _λ embeds uniquely (see [W], §8.5) as a finite dimensional (g, K) submodule V _λ of the Harish-Chandra module H(δ, ξ) of a principal series representation of G. The functions in H(δ, ξ) are uniquely determined by their restriction to K. As a noncompact analogue of the Borel-Wei! theorem the functions in V _λ are given as solutions of differential equations arising from the generators of L _λ(t) (rather than Cauchy-Riemann equations as in the Borel-Weil theorem).

Research supported in part by the NSF Contract DMS-0209473 and the KG&G Foundation.