Character values and Hochschild homology

TL;DR

This paper proposes a conjecture relating character values of cuspidal representations of p-adic groups to weighted orbital integrals, extending previous results and introducing a new 'compactified pseudo-matrix coefficient' framework.

Contribution

It generalizes the construction of pseudo-matrix coefficients using Hochschild homology and modifies the category of smooth representations for better analysis.

Findings

Proved the conjecture for G=SL(2).

Established properties of the new space of coefficients.

Connected character values with weighted orbital integrals via 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 pseudo-matrix 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 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 general properties of that space, building on works of Waldspurger and Beuzart-Plessis.