  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 $\mathfrak{g}^*$.
