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

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- by Xuhua He and Ju-Lee Kim
- Represent. Theory
**23**(2019), 294-324 - DOI: https://doi.org/10.1090/ert/528
- Published electronically: September 16, 2019
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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 $\overline {\mathcal {H}}^\mathrm {rig}_{r^+}$.## References

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## Bibliographic 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)
- © Copyright 2019 American Mathematical Society
- Journal: Represent. Theory
**23**(2019), 294-324 - MSC (2010): Primary 22E50; Secondary 11F70
- DOI: https://doi.org/10.1090/ert/528
- MathSciNet review: 4007169