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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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Epipelagic representations and invariant theory
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by Mark Reeder and Jiu-Kang Yu PDF
J. Amer. Math. Soc. 27 (2014), 437-477 Request permission

Abstract:

We introduce a new approach to the representation theory of reductive $p$-adic groups $G$, based on the geometric invariant theory (GIT) of Moy-Prasad quotients. Stable functionals on these quotients are used to give a new construction of supercuspidal representations of $G$ having small positive depth, called epipelagic. With some restrictions on $p$, we classify the stable and semistable functionals on Moy-Prasad quotients. The latter classification determines the nondegenerate $K$-types for $G$ as well as the depths of irreducible representations of $G$. The main step is an equivalence between Moy-Prasad theory and the theory of graded Lie algebras, whose GIT was analyzed by Vinberg and Levy. Our classification shows that stable functionals arise from $\mathbb {Z}$-regular elliptic automorphisms of the absolute root system of $G$. These automorphisms also appear in the Langlands parameters of epipelagic representations, in accordance with the conjectural local Langlands correspondence.
References
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Additional Information
  • Mark Reeder
  • Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
  • Email: reederma@bc.edu
  • Jiu-Kang Yu
  • Affiliation: The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
  • Email: jkyu@ims.cuhk.edu.hk
  • Received by editor(s): August 13, 2012
  • Received by editor(s) in revised form: June 20, 2013
  • Published electronically: August 5, 2013
  • Additional Notes: The first author was supported by NSF grants DMS-0801177 and DMS-0854909
    The second author was supported by NSF grant DMS-0854909
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 27 (2014), 437-477
  • MSC (2010): Primary 22E50, 11S15, 11S37
  • DOI: https://doi.org/10.1090/S0894-0347-2013-00780-8
  • MathSciNet review: 3164986