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Cocenters and representations of pro-$p$ Hecke algebras

Authors: Xuhua He and Sian Nie
Journal: Represent. Theory 21 (2017), 82-105
MSC (2010): Primary 20C08, 20C20, 22E50
Published electronically: June 23, 2017
MathSciNet review: 3665615
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Abstract: In this paper, we study the relation between the cocenter $\overline {{\tilde {\mathcal H}}}$ and the representations of an affine pro-$p$ Hecke algebra ${\tilde {\mathcal H}}={\tilde {\mathcal H}}(0, -)$. As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of ${\tilde {\mathcal H}}$ is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter $\overline {\tilde {\mathcal H}}$.

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Additional Information

Xuhua He
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
MR Author ID: 733194

Sian Nie
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, People’s Republic of China

Keywords: Affine Coxeter groups, Hecke algebras, $p$-adic groups
Received by editor(s): May 11, 2016
Received by editor(s) in revised form: October 10, 2016, December 1, 2016, February 26, 2017, and May 10, 2017
Published electronically: June 23, 2017
Additional Notes: The first author was partially supported by NSF DMS-1463852. The second author was partially supported by NSFC (No. 11501547 and No. 11621061.) and by the Key Research Program of Frontier Sciences, CAS, Grant No. QYZDB-SSW-SYS007.
Article copyright: © Copyright 2017 American Mathematical Society