On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters

TL;DR

This paper classifies the unitary dual of geometric graded Hecke algebras with unequal parameters, linking them to unipotent representations of exceptional p-adic groups, and establishes linear independence of tempered modules with real central character.

Contribution

It provides the first complete determination of the unitary dual for these Hecke algebras and connects this classification to unipotent representations of p-adic groups.

Findings

Unitary dual explicitly determined for geometric graded Hecke algebras.

Established correspondence between Hecke algebra representations and p-adic group unipotent representations.

Proved linear independence of tempered modules with real central character.

Abstract

We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's original argument) that the set of tempered modules with real central character for a geometric graded Hecke algebra is linearly independent when restricted to the Weyl group.