The wavefront set over a maximal unramified field extension

TL;DR

This paper develops a framework to analyze the wavefront set of certain p-adic group representations over unramified extensions and applies it to compute geometric wavefront sets of spherical Arthur representations using perverse sheaves.

Contribution

It introduces a new method to study wavefront sets over unramified extensions and applies it to spherical Arthur representations via perverse sheaves on the dual group.

Findings

Wavefront set over unramified extension can be explicitly computed.

Method connects wavefront sets with perverse sheaves on the Langlands dual group.

Provides explicit computations for spherical Arthur representations.

Abstract

Let $(\pi,X)$ be a depth-$0$ admissible smooth complex representation of a $p$-adic reductive group that splits over an unramified extension. In this paper we develop the theory necessary to study the wavefront set of $X$ over a maximal unramified field extension of the base $p$-adic field. In the final section we then apply these methods to compute the geometric wavefront set of spherical Arthur representations of split $p$-adic reductive groups. In this case we see how the wavefront set over a maximal unramified extension can be computed using perverse sheaves on the Langlands dual group.