Unipotent Representations of Complex Groups and Extended Sommers Duality

TL;DR

This paper describes a new duality map linking unipotent representations of complex reductive groups to data involving nilpotent orbits and conjugacy classes in the Langlands dual group, advancing understanding of the unitary dual.

Contribution

It introduces a duality map connecting unipotent representations with pairs of nilpotent orbits and conjugacy classes in the dual group, generalizing previous notions and providing a new structural perspective.

Findings

Construction of a duality map D from pairs ( ext{O}^{\u0010}, ar{C}) to covers of nilpotent orbits.

Description of unipotent representations in terms of the Langlands dual group.

Framework for understanding the building blocks of the unitary dual of complex reductive groups.

Abstract

Let $G$ be a complex reductive algebraic group. In arXiv:2108.03453, we have defined a finite set of irreducible admissible representations of $G$ called `unipotent representations', generalizing the special unipotent representations of Arthur and Barbasch-Vogan. These representations are defined in terms of filtered quantizations of symplectic singularities and are expected to form the building blocks of the unitary dual of $G$. In this paper, we provide a description of these representations in terms of the Langlands dual group $G^{\vee}$. To this end, we construct a duality map $D$ from the set of pairs $(\mathbb{O}^{\vee},\bar{C})$ consisting of a nilpotent orbit $\mathbb{O}^{\vee} \subset \mathfrak{g}^{\vee}$ and a conjugacy class $\bar{C}$ in Lusztig's canonical quotient $\bar{A}(\mathbb{O}^{\vee})$ to the set of finite covers of nilpotent orbits in $\mathfrak{g}^*$.