The nonabelian Fourier transform for elliptic unipotent representations of exceptional $p$-adic groups

TL;DR

This paper introduces a new involution on elliptic unipotent Langlands parameters for certain p-adic groups, connecting it with Lusztig's nonabelian Fourier transform and extending recent theoretical developments.

Contribution

It defines an involution on elliptic unipotent parameters for split adjoint exceptional p-adic groups and links it to Lusztig's Fourier transform, advancing the understanding of unipotent representations.

Findings

Involution matches Lusztig's nonabelian Fourier transform for finite quotients.

Establishes compatibility with hyperspecial parahoric restriction.

Extends the framework of Fourier transforms to exceptional p-adic groups.

Abstract

We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric restriction map agrees with Lusztig's nonabelian Fourier transform for unipotent representations of the finite reductive quotient. This is inspired by recent works of Lusztig on the almost unipotent characters of $p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier transform of odd orthogonal groups.