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Epipelagic representations and invariant theory


Authors: Mark Reeder and Jiu-Kang Yu
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
Published electronically: August 5, 2013
MathSciNet review: 3164986
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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.


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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

DOI: https://doi.org/10.1090/S0894-0347-2013-00780-8
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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