Journal of Lie Theory 34 (2024), No. 3, 503--510
Copyright Heldermann Verlag 2024
Irreducibility of Wave-Front Sets for Depth Zero Cuspidal Representations
Avraham Aizenbud
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
aizenr@gmail.com
Dmitry Gourevitch
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
dimagur@weizmann.ac.il
Eitan Sayag
Dept. of Mathematics, Ben Gurion University of the Negev, Be'er Sheva, Israel
eitan.sayag@gmail.com
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.
Keywords: Representation, reductive group, algebraic group, nilpotent orbit, wave-front set, character, non-commutative harmonic analysis, generalized Gelfand-Graev models.
MSC: 20G05, 20G25, 22E35, 22E46, 20C33.
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