A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem
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).
KeywordsDisjoint Union Left Ideal Cartan Subalgebra Maximal Compact Subgroup Principal Series
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