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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Regular supercuspidal representations
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by Tasho Kaletha HTML | PDF
J. Amer. Math. Soc. 32 (2019), 1071-1170 Request permission

Abstract:

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive $p$-adic group $G$ arise from pairs $(S,\theta )$, where $S$ is a tame elliptic maximal torus of $G$, and $\theta$ is a character of $S$ satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive $p$-adic groups.
References
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Additional Information
  • Tasho Kaletha
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • MR Author ID: 939928
  • Received by editor(s): March 19, 2017
  • Received by editor(s) in revised form: October 16, 2018, January 28, 2019, and April 17, 2019
  • Published electronically: July 18, 2019
  • Additional Notes: This research was supported in part by NSF grants DMS-1161489, DMS-1801687 and a Sloan Fellowship
  • © Copyright 2019 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 32 (2019), 1071-1170
  • MSC (2010): Primary 22E50, 11S37, 11F70
  • DOI: https://doi.org/10.1090/jams/925
  • MathSciNet review: 4013740