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FORTHCOMING Character values and Hochschild homology
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1803.09442.pdf
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Accepted version
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375.31 KB
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Author(s)
Bezrukavnikov, Roman
Kazhdan, David
Publisher
American Mathematical Society
Citation
Bezrukavnikov, Roman and David Kazhdan. "Character values and Hochschild homology."
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://arxiv.org/abs/1803.09442
