A nonabelian Fourier transform for tempered unipotent representations

TL;DR

This paper introduces a new involution on the space of tempered unipotent representations of split simple p-adic groups, exploring its properties and conjecturing a connection with Lusztig's nonabelian Fourier transform, supported by proofs for specific groups.

Contribution

It defines a novel involution on unipotent representations and formulates a conjecture linking it to Lusztig's Fourier transform, with evidence provided for certain groups.

Findings

Defined an involution on unipotent representations.

Formulated a conjecture relating the involution to Lusztig's Fourier transform.

Provided proofs for $ ext{SL}_n$ and $ ext{PGL}_n$.

Abstract

We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for $\mathsf{SL}_n$ and $\mathsf{PGL}_n$.