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.