Epipelagic representations and invariant theory

By Mark Reeder, J. Yu. 2013

125 citations recorded by the source corpus.

LetG be a reductive p-adic group. We give a new construction of small-depth “epipelagic” supercuspidal representations ofG(k), using stable orbits in Geometric Invariant Theory (GIT). In contrast to previously known methods, this construction works without any restrictions on the residue characteristic p. We construct appropriate Langlands parameters for these representations, with some restrictions on p. The GIT arises from graded Moy-Prasad filtrations, which we show are isomorphic to the graded Lie algebras whose GIT was analyzed by Vinberg and Levy. This leads to a classification of stable orbits in Moy-Prasad filtrations, as well as epipelagic supercuspidal representations, in terms of Z-regular elliptic automorphisms σ of the absolute root system of G. Transferred to the L-group of G, σ generates the image of tame inertia under the corresponding (wild) Langlands parameter. For unramified groups and sufficiently large p we also classify the Moy-Prasad filtrations which have semi-stable orbits, which solves the long-outstanding problem of classifying non-degenerate K-types for G(k).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29d88eb95362bf710f8ad852b2ce3d6d18b69354

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Jessica Fintzen
2. Tasho Kaletha
3. Mark Reeder
4. J. Yu

In Mathematical Physics

1. Jessica Fintzen
2. Tasho Kaletha
3. Mark Reeder
4. J. Yu

Citations in the corpus

• Epipelagic $$L$$L-packets and rectifying characters (2012, methodology): “The purpose of this section is to recall some material from [17,48,58], which is relevant to our situation.”
• Regular supercuspidal representations (2016, background): “tive p-adic groups, and whose approach formed the basis of Yu’s more general construction. The class of toral supercuspidal representations is general enough to include the epipelagic representations [RY14] when pdoes not divide the order of the inertial action, and the representations considered by”
• Rigid inner forms of real and p-adic groups (2013, background): “racter identities hold. Furthermore, the same is true [KalEpi] for a different class of supercuspidal Langlands parameters, for which the corresponding L-packets consist of epipelagic representations [RY14]. The cohomology set B(G) bas has a map to H1(Γ,G/Z(G)) which is surjective when the center of”
• Types for tame p-adic groups (2018, methodology): “ls). We expect that we can use these tori to produce supercuspidal representations that were not constructed by Yu. Examples of such representations are provided by the construction of Reeder and Yu ([RY14]), whose ingredients exist also when p | |W| (whenever they exist for some large prime p), see”
• Geometric representations of graded and rational Cherednik algebras (2014, unknown): “Out treatment here is strongly influenced by the work of Gross, Levy, Reeder and Yu (see [22] and [33]).”