Journal of Lie Theory

no. 3
Irreducibility of Wave-Front Sets for Depth Zero Cuspidal Representations
Avraham Aizenbud1; Dmitry Gourevitch1; Eitan Sayag2
1Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
2Dept. of Mathematics, Ben Gurion University of the Negev, Be'er Sheva, Israel
Journal of Lie Theory, Volume 34 (2024) no. 3, pp. 503-510
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.
DOI: 10.5802/jolt.1346
Classification: 20G05, 20G25, 22E35, 22E46, 20C33
Keywords: Representation, reductive group, algebraic group, nilpotent orbit, wave-front set, character, non-commutative harmonic analysis, generalized Gelfand-Graev models

Avraham Aizenbud  1 ; Dmitry Gourevitch  1 ; Eitan Sayag  2

1 Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
2 Dept. of Mathematics, Ben Gurion University of the Negev, Be'er Sheva, Israel 
Avraham Aizenbud; Dmitry Gourevitch; Eitan Sayag. Irreducibility of Wave-Front Sets for Depth Zero Cuspidal Representations. Journal of Lie Theory, Volume 34 (2024) no. 3, pp. 503-510. doi: 10.5802/jolt.1346
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     title = {Irreducibility of {Wave-Front} {Sets} for {Depth} {Zero} {Cuspidal} {Representations}},
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