Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups I

TL;DR

This paper provides a precise formula for the wavefront set of certain unipotent representations of reductive p-adic groups, confirming conjectures and offering explicit descriptions for specific cases.

Contribution

It establishes a formula for the wavefront set of Iwahori-spherical representations with real infinitesimal character and confirms the singleton nature of the algebraic wavefront set in this context.

Findings

Wavefront set formula for Iwahori-spherical representations

Confirmation of the singleton algebraic wavefront set conjecture

Explicit description of wavefront sets for spherical representations

Abstract

The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an explicit description of the wavefront set of an irreducible spherical representation with real Satake parameter.