Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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



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
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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 22E50, 11S15, 11S37

Retrieve articles in all journals with MSC (2010): 22E50, 11S15, 11S37

Additional Information

Mark Reeder
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467

Jiu-Kang Yu
Affiliation: The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

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.