Henryk Hecht

The characters of some representations of Harish-Chandra

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 Blattners 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 Schmids results and those in this paper, Blattners conjecture holds for all groups G such that G/K is Hermitian symmetric. Unlike all previous partial solutions this proof of Blattners conjecture works for all discrete series representations of G.

A Remark on Casselman’s Comparison Theorem

This paper is an outgrowth of our attempt to understand a comparison theorem of Casselman which asserts that Lie algebra homology groups, with respect to certain nilpotent algebras of a Harish-Chandra module and its C ∞ completion, coincide. Let us start with a precise statement of this theorem.