  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.
