Generalizations of the Springer correspondence and cuspidal Langlands parameters

Abstract

Let $${\mathcal H}$$ be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for $${\mathcal H}$$ , which conjecturally puts supercuspidal $${\mathcal H}$$ -representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein’s theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for $${\mathcal H}$$ , that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible $${\mathcal H}$$ -representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for $${\mathcal H}$$ -representations to the corresponding problem for supercuspidal representations of Levi subgroups of $${\mathcal H}$$ . The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig’s generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.