Graded Hecke algebras for disconnected reductive groups

TL;DR

This paper introduces graded Hecke algebras for possibly disconnected complex reductive groups, develops their representation theory, and connects their irreducible representations to Langlands parameters, aiming to contribute to the local Langlands program.

Contribution

It generalizes Lusztig's graded Hecke algebras to disconnected groups and provides a canonical parametrization of their irreducible representations using geometric methods.

Findings

Complete classification of irreducible, tempered, and discrete series representations.

Parametrization of modules via perverse sheaves and equivariant homology.

Connection between irreducible representations and enhanced L-parameters.

Abstract

We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G. We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data (G,M,L) and they are closely related to Langlands parameters. Our main motivation for considering these graded Hecke algebras is that the space of irreducible H-representations is canonically in bijection with a certain set of "logarithms" of enhanced L-parameters. Therefore we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper.