David Kazhdan'
title: Character values and Hochschild homology
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Introduction
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Let $G$ be a reductive group over a local nonArchimedean field $F$.

The goal of the article is to present an algebraic expression for a character of an admissible representation of $G$ on a compact element.

The statement is presented as a conjecture (see Conjecture \[char\_form\_conj\]) for a general reductive group, it is proved in the paper for $G=SL(2)$. We also describe a modification of the category $Sm=Sm(G)$ of finitely generated smooth representations, the so called [*compactified category*]{} of smooth $G$-modules, which plays a key role in our algebraic description of character values and may have an independent interest.

To describe the context for these constructions recall a conjecture of [@cusp] proved in [@SS] and [@thes].

Let ${{\mathcal H}}=\cup_K {{\mathcal H}}_K$ be the Hecke algebra of locally constant compactly supported ${{\mathbb C}}$-valued measures. Thus $Sm(G)$ is identified with the category of finitely generated nondegenerate ${{\mathcal H}}$ modules [@BZ].

Let $C({{\mathcal H}})={{\mathcal H}}/[{{\mathcal H}},{{\mathcal H}}]={{\mathcal H}}_G=HH_0({{\mathcal H}})=HH_0(Sm)$ be the cocenter of ${{\mathcal H}}$. Here ${{\mathcal H}}_G$ denotes coinvariants with respect to the conjugation action, while $HH_*$ stands for Hochschild homology, and its second appearance refers to the notion of Hochschild 