Irreducibility of wave-front sets for depth zero cuspidal representations

TL;DR

This paper proves that for large residue characteristic, the wave-front sets of depth zero cuspidal representations are irreducible varieties, confirming a conjecture by Moeglin-Waldspurger using existing results and reduction techniques.

Contribution

It establishes the irreducibility of wave-front sets for depth zero cuspidal representations, extending known results to a broader class of groups over local fields.

Findings

Wave-front sets are irreducible varieties for large residue characteristic.

Reduction to finite groups of Lie type confirms the irreducibility.

Supports Moeglin-Waldspurger's conjecture in the depth zero case.

Abstract

We show that the results of [BM97, DeB02b, Oka, Lus85, AA07, Tay16] imply a positive answer to the question of Moeglin-Waldspurger on wave-front sets in the case of depth zero cuspidal representations. Namely, we deduce that for large enough residue characteristic, the Zariski closure of the wave-front set of any depth zero irreducible cuspidal representation of any reductive group over a non-Archimedean local field is an irreducible variety. In more details, we use [BM97, DeB02b, Oka] to reduce the statement to an analogous statement for finite groups of Lie type, which is proven in [Lus85, AA07, Tay16].