Discrete Series for Semisimple Symmetric Spaces

Abstract

We give a sufficient condition for the existence of minimal closed G-invariant subspaces of L2(G/H) for a semisimple symmetric space G/H. As a semisimple Lie group with finite center may always be considered as a symmetric space, this provides, in particular, a new and elementary proof of Harish-Chandra's result that G has a discrete series if rank (G) = rank (K), where K is a maximal compact subgroup. Let G be a connected noncompact semisimple Lie group, let z be an involution on G, and let H be the connected component of the fixed-point group Gr containing the identity. Then G/H is a semisimple symmetric space, and the group G acts by left translation on C*(G/H) and L2(G/H). In the introduction we will, for simplicity, assume that G has a finite center. By the discrete series for G/H we shall mean the set of equivalence classes of the representations of G on minimal closed invariant subspaces of L2(G/H). Let a be a Cartan involution commuting with z. The fixed-point group K for a is a maximal compact subgroup. Our main result is THEOREM 1.1. The discrete series for G/H is nonempty and infinite if