Let $G$ be a connected reductive algebraic group over (say) $\mathbb{C}$. The set $\hat{G}_u$ of isomorphism classes of unipotent irreducible character sheaves has some complicated classification in terms of two-sided cells in $W$ and so on... I can't understand from the literature:
Should it be correct that $\hat{G}_u$ is in bijection with the set of isomorphism classes of irreducible $G^{\vee}$-equivariant perverse sheaves on the nilpotent cone of the Langlands-dual group? Does it follow "synthetically" from the complicated classification? Should there be some equivalence of categories explaining this bijection?
Thanks