Epipelagic representations and invariant theory

Reeder, Mark; Yu, Jiu-Kang

doi:10.1090/S0894-0347-2013-00780-8

Epipelagic representations and invariant theory
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by Mark Reeder and Jiu-Kang Yu;

J. Amer. Math. Soc. 27 (2014), 437-477

DOI: https://doi.org/10.1090/S0894-0347-2013-00780-8

Published electronically: August 5, 2013

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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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