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Jordan decompositions of cocenters of reductive $p$-adic groups


Authors: Xuhua He and Ju-Lee Kim
Journal: Represent. Theory 23 (2019), 294-324
MSC (2010): Primary 22E50; Secondary 11F70
DOI: https://doi.org/10.1090/ert/528
Published electronically: September 16, 2019
MathSciNet review: 4007169
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Abstract: Cocenters of Hecke algebras $\mathcal {H}$ play an important role in studying mod $\ell$ or $\mathbb C$ harmonic analysis on connected $p$-adic reductive groups. On the other hand, the depth $r$ Hecke algebra $\mathcal {H}_{r^+}$ is well suited to study depth $r$ smooth representations. In this paper, we study depth $r$ rigid cocenters $\overline {\mathcal {H}}^\mathrm {rig}_{r^+}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell \neq p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an explicit basis of $\overline {\mathcal {H}}^\mathrm {rig}_{r^+}$.


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Additional Information

Xuhua He
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong.
MR Author ID: 733194
Email: xuhuahe@math.cuhk.edu.hk

Ju-Lee Kim
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge Massachusetts 02139
MR Author ID: 653104
Email: juleekim@mit.edu

Received by editor(s): October 17, 2017
Received by editor(s) in revised form: October 30, 2018
Published electronically: September 16, 2019
Additional Notes: The first author was partially supported by NSF DMS-1463852 and DMS-1128155 (from IAS)
Article copyright: © Copyright 2019 American Mathematical Society