The wavefront sets of unipotent supercuspidal representations

Ciubotaru, Dan; Mason-Brown, Lucas; Okada, Emile

doi:10.2140/ant.2024.18.1863

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The wavefront sets of unipotent supercuspidal representations

Dan Ciubotaru, Lucas Mason-Brown and Emile Okada

Vol. 18 (2024), No. 10, 1863–1889

DOI: 10.2140/ant.2024.18.1863

Abstract

We prove that the double (or canonical unramified) wavefront set of an irreducible depth-0 supercuspidal representation of a reductive $p$ -adic group is a singleton provided $p > 3 (h - 1)$ , where $h$ is the Coxeter number. We deduce that the geometric wavefront set is also a singleton in this case, proving a conjecture of Mœglin and Waldspurger. When the group is inner to split and the representation belongs to Lusztig’s category of unipotent representations, we give an explicit formula for the double and geometric wavefront sets. As a consequence, we show that the nilpotent part of the Deligne–Langlands–Lusztig parameter of a unipotent supercuspidal representation is precisely the image of its geometric wavefront set under Spaltenstein’s duality map.

Keywords

wavefront set, supercuspidal representations, Langlands correspondence, unipotent representations, local character expansion

Mathematical Subject Classification

Primary: 20C33, 22E50

Milestones

Received: 11 July 2022

Revised: 27 July 2023

Accepted: 24 October 2023

Published: 7 October 2024

Authors

Dan Ciubotaru
Mathematical Institute University of Oxford Oxford United Kingdom

Lucas Mason-Brown
Mathematical Institute University of Oxford Oxford United Kingdom

Emile Okada
Department of Mathematics National University of Singapore Singapore

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Vol. 18, No. 10, 2024