A Branching Law for Subgroups Fixed by an Involution and a Concompact Analogue of the Borel-weil Theorem

We give a branching law for subgroups fixed by an involution. As an application we give a generalization of the Cartan-Helgason theorem and a noncompact analogue of the Borel-Weil theorem. 0. Introduction 0.1. Let G C be a simply-connected complex semisimple Lie group and let g = Lie G C. Let g o be a real form of g and let G be the real semisimple Lie subgroup of G C corresponding to g o. Using standard notation let G = K A N be an Iwasawa decomposition of G and let M be the centralizer of A in the maximal compact subgroup K of G. The complexified Lie algebras of K, M, A, and N are denoted respectively by k, m, a and n. Let h m be a Cartan subalgebra of m so that h = h m + a is a Cartan subalgebra of g. Let m = m − + h m + m + be a triangular decomposition of m so that b m = h m + m + is a Borel subalgebra of m and b = b m + a + n is a Borel subalgebra of g. Let Λ ⊂ h * be the set of dominant integral linear forms on h, with respect to b and, for each λ ∈ Λ, let π : g → End V λ be an irreducible representation with highest weight λ. Let 0 = v λ ∈ V λ be a highest weight vector. It follows from the Lie algebra Iwasawa decomposition g = k + a + n that U (g) = U (k)U (a + n) using the standard notation for universal enveloping algebras. However if λ ∈ Λ, then Cv λ is stable under U (a + n) and hence V λ is a cyclic U (k)-module. In fact V λ = U (k) v λ Let L λ (k) be the (left ideal) annihilator of v λ in U (k). It is then an elementary fact that if Z is any irreducible k-module, one has multiplicity of Z in V λ = dim Z L λ (k) (0.1) where Z S = {w ∈ Z | S · w = 0} for any subset S ⊂ U (k). The equation (0.1) becomes a useful branching law as soon as one can explicitly determine generators of L λ (k). It is …