Restrictions of Discrete Series to sl(2, ℝ)

Abstract

We prove that the restriction to (2, R) of the underlying HarishChandra module of a Discrete Series representation of a noncompact simple Lie group has no sl(2)-irreducible subrepresentations unless it is holomorphic. We also give a characterization of the holomorphic Discrete Series. Let G be a connected real simple Lie group. Fix a maximal compact subgroup K of G. Let g (t) denote the complex Lie algebra of G (respectively, K). Hence we have the Cartan decomposition g = t + p. Thus, p is Kinvariant under the restriction of the adjoint representation of G to K and p . Let T C K be a maximal torus. Henceforth we assume that T is a Cartan subgroup of G. Let t be the complex Lie algebra of T. Under these hypotheses Harish-Chandra in [HC2] has given a classification of the square-integrable representations of G in terms of the regular characters of T. Let (D(g, t) denote the root system of the pair (g, t) . A root of the pair (g, t) is called compact (noncompact) if its root space is contained in t (respectively, p). We now fix a noncompact root y . Then the root vectors of y span a copy by of sl(2, C) . Moreover by is invariant under the conjugation of g with respect to the Lie algebra of G. Let s,, be the real form of by associated to the conjugation of g with respect the Lie algebra of G restricted to by . Let Hy be the connected subgroup of G corresponding to sy I Let (7r, V) be the Harish-Chandra module underlying a square-integrable, irreducible representation of G. We then have Proposition 1. If the restriction of (7r, V) to by contains an irreducible bysubmodule, then G/K is a Hermitian symmetric space and (7r, V) corresponds to a holomorphic square-integrable representation. A corollary to Proposition 1 is Corollary. If G is as above, G/K is not Hermitian, and H is a semisimple subgroup of G such that its symmetric space is Hermitian, then (7r, V) restricted to [ has no [-holomorphic irreducible subrepresentations. Received by the editors April 29, 1991 and, in revised form, July 11, 1991. 1991 Mathematics Subject Classification. Primary 22E47.