In the representation theory of reductive p-adic groups, a lot of general structure was developed by Bernstein: supercuspidal representations, cuspidal support, Bernstein components and so on.

Langlands predicted a beautiful correspondence between irreducible representations of a reductive p-adic group G and L-parameters defined in terms of the Galois group and the complex dual group of G. A modern formulation of this local Langlands correspondence makes it into a (conjectural) bijection, by enhancing L-parameters. Thus, in principle every property of the set of irreducible G-representations should have some counterpart for enhanced L-parameters.

In this talk we will discuss the development of the Bernstein theory for enhanced L-parameters, in particular the property of cuspidality and the cuspidal support map. These can be defined in a natural way, and they put many earlier results and observations in a new framework.

Remarkably, it can all be formulated and proven entirely in terms of complex algebraic groups, no knowledge of p-adic groups is needed for that. An important ingredient is a generalization of the Lusztig-Springer correspondence (about representations of Weyl groups) to disconnected complex reductive groups.

This is joint work with Anne-Marie Aubert and Ahmed Moussaoui.

Hopefully, it will be useful to reduce a proof of (cases of) the local Langlands correspondence to the level of supercuspidal representations and cuspidal Langlands parameters.

## Date:

Mon, 19/08/2019 - 14:00 to 15:00

## Location:

Ross 70