Generalizations of the Springer correspondence and cuspidal Langlands parameters
Abstract
Let H be any reductive padic group. We introduce a notion of cuspidality for enhanced Langlands parameters for H, which conjecturally puts supercuspidal Hrepresentations in bijection with such Lparameters. We also define a cuspidal support map and Bernstein components for enhanced Lparameters, in analogy with Bernstein's theory of representations of padic groups. We check that for several wellknown reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced Lparameters for H, that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible Hrepresentations) 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 Hrepresentations to the corresponding problem for supercuspidal representations of Levi subgroups of 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.05335
 Bibcode:
 2015arXiv151105335A
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Geometry;
 11S37;
 20Gxx;
 22E50
 EPrint:
 In version 2 Theorem 3.1.c and the proof of Lemma 3.3 were corrected. In version 3 Theorem 3.1.a was corrected