Character values and Hochschild homology

Abstract

We present a conjecture (and a proof for G = SL(2)) generalizing a result of J. Arthur which expresses a character value of a cuspidal representation of a p-adic group as a weighted orbital integral of its matrix coefficient. It also generalizes a conjecture by the second author proved by Schneider-Stuhler and (independently) the first author. The latter statement expresses an elliptic character value as an orbital integral of a pseudo-matrix coefficient defined via the Chern character map taking value in zeroth Hochschild homology of the Hecke algebra. The present conjecture generalizes the construction of pseudomatrix coefficient using compactly supported Hochschild homology, as well as a modification of the category of smooth representations, the so called compactified category of smooth G-modules. This newly defined ”compactified pseudo-matrix coefficient” lies in a certain space K on which the weighted orbital integral is a conjugation invariant linear functional, our conjecture states that evaluating a weighted orbital integral on the compactified pseudo-matrix coefficient one recovers the corresponding character value of the representation. We also discuss the properties of the averaging map from K to the space of invariant distributions, partly building on works of Waldspurger and BeuzartPlessis.