Irreducibility of Wave-Front Sets for Depth Zero Cuspidal Representations

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.