Unipotent Representations of Complex Groups and Extended Sommers Duality

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 ${\mathfrak{g}}^{\ast }$.

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