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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
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.