The stable wave front set of theta representations

TL;DR

This paper calculates the stable wave front set of theta representations for specific p-adic group covers, combining techniques from finite group theory, Lusztig's formula, and Hecke algebra analysis.

Contribution

It introduces a method to compute wave front sets of theta representations in Brylinski-Deligne covers using adapted theorems and algebraic analysis.

Findings

Computed stable wave front sets for certain covers

Connected wave front sets to Kawanaka wave front sets of finite groups

Utilized Lusztig's formula and Hecke algebra actions

Abstract

We compute the stable wave front set of theta representations for certain tame Brylinski-Deligne covers of a connected reductive $p$-adic group. The computation involves two main inputs. First we use a theorem of Okada, adapted to covering groups, to reduce the computation of the wave front set to computing the Kawanaka wave front set of certain representations of finite groups of Lie type. Second, to compute the Kawanaka wave front sets we use Lusztig's formula. This requires a careful analysis of the action of the pro-$p$ Iwahori-Hecke algebra on the theta representation, using the structural results about Hecke algebras developed by Gao-Gurevich-Karasiewicz and Wang.