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March 29, 2026Open Access

Irreducibility of Wave-Front Sets for Depth Zero Cuspidal Representations

Key Points

The research aims to verify the irreducibility of wave-front sets for depth zero cuspidal representations in specific mathematical contexts.
Utilized past results from Barbasch and Moy, DeBacker, and Okaka to frame the problem.
Reduced the problem to an analogous statement already proven for finite groups of Lie type.
Applied techniques from representation theory and algebraic geometry.
Confirmed that for large residue characteristics, Zariski closure yields an irreducible variety.
Demonstrated that depth zero irreducible cuspidal representations maintain this quality across reductive groups.

Abstract

We show that recent results 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 results of Barbasch and Moy, DeBacker, and Okaka to reduce the statement to an analogous statement for finite groups of Lie type, which was proven by Lusztig, Achar and Aubert, and Taylor.

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