Discrete series characters and the Lefschetz formula for Hecke operators

Abstract

This paper consists of three independent but related parts. In the rst part (xx1{ 6) we give a combinatorial formula for the constants appearing in the \numerators of characters of stable discrete series representations of real groups (see x3) as well as an analogous formula for individual discrete series representations (see x6). Moreover we give an explicit formula (Theorem 5.1) for certain stable virtual characters on real groups; by Theorem 5.2 these include the stable discrete series characters, and thus we recover the results of x3 in a more natural way. In the second part (x7) we use the character formula given in Theorem 5.1 to rewrite the Lefschetz formula of [GM] (for the local contribution at a single xed point component to the trace of a Hecke operator on weighted cohomology) in the same spirit as that of Arthur's Lefschetz formula [A]: in terms of stable virtual characters on real groups (see Theorem 7.14.B). We then sum the contributions of the various xed point components and show that, in the case of middle weighted cohomology, the resulting global Lefschetz xed point formula agrees with Arthur's Lefschetz formula. This gives a topological proof of Arthur's formula.