Some Properties of Square-Integrable Representations of Semisimple Lie Groups

Abstract

In the theory of irreducible representations of a compact Lie group, the formula for the multiplicity of a weight and the so-called theorem of the highest weight are among the important results. At least conjecturally, both of these statements have analogues for the discrete series of representations of a semisimple Lie group. Let G be a connected, semisimple Lie group, K c G a maximal compact subgroup, and suppose that rk K = rk G. Exactly in this situation, G has a non-empty discrete series [8]. Blattner's conjecture predicts how a given discrete series representation should break up under the action of K; precise statements can be found in [10], [15], [16]. Formally, the conjectured multiplicity formula looks just like the formula for the multiplicity of a weight. Partial results toward the conjecture have been proved in [10], [15]. More recently, the full conjecture was established for those linear groups G, whose quotient G/K admits a Hermitian symmetric structure [16]. As this paper was being completed, H. Hecht and I succeeded in proving Blattner's conjecture for all linear groups, by extending the arguments of [16]. According to Blattner's conjecture, any particular discrete series representation 7t contains a distinguished irreducible K-module V, with multiplicity one; moreover, w contains no irreducible K-module with a highest weight which is lower, in the appropriate sense, than that of V,:. For most discrete series representations, it was known that these two properties characterize at, up to infinitesimal equivalence, among all irreducible representations of G [10], [15]. In this paper, I shall give an infinitesimal characterization, by lowest K-type, for all discrete series representations. The result, which is stated as Theorem (1.3) below, closely resembles the theorem of the highest weight. I shall also draw a number of conclusions from it. The methods of this paper have some further, less immediate consequences, which will be taken up elsewhere. For the remainder of the introduction, I assume that G is a linear group.