The characters of some representations of Harish-Chandra

Abstract

Let G be a connected semisimpte real matrix group and K a maximal compact subgroup. Assume that G/K is Hermitian symmetric. For such groups G, HarishChandra has constructed in [2] a class {Ta} of Frechet representations of G parameterized by a discrete cone. If 2 satisfies a certain set of inequalities. Th is infinitesimally equivalent to a holomorphic discrete series representation. In this special case Martens has obtained a character formula 0h for Th (cf. [8]). This formula, as it is defined, makes sense as a function for any 2, and one can show that it determines an invariant, though not necessarily tempered, eigendistribution. It is natural therefore to ask whether Oh is the character of 2r~ in general. (The method used in [8] depends strongly on the special condition imposed upon 2.) The main result of this paper is a positive answer to this question. Recently Schmid has obtained semi-explicit formulas for characters of discrete series for G as above. He has also shown that Blattner's conjecture, which predicts exact multiplicities of irreducible K-modules in discrete series, holds, provided that the formal multiplicities of K-modules in the distribution 0 h given by 0 h, are the same as their multiplicities in Th (el. [9], Chapter 7). Hence in view of Schmid's results and those in this paper, Blattner's conjecture holds for all groups G such that G/K is Hermitian symmetric. Unlike all previous partial solutions this proof of Blattner's conjecture works for all discrete series representations of G.