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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 ${{\mathcal K}}$ 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 ${{\mathcal K}}$ to the space of invariant distributions, partly building on works of Waldspurger and Beuzart-Plessis.
author:
- 'Roman Bezrukavnikov, 