Epipelagic representations and invariant theory

Reeder, Mark ¹   ; Yu, Jiu-Kang ²

¹ Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
² The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Journal of the American Mathematical Society, Tome 27 (2014) no. 2, pp. 437-477

Résumé

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.

Détail

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

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