Hecke algebras for p-adic groups and explicit Local Langlands Correspondence

Abstract: I will talk about several results on Hecke algebras attached to Bernstein blocks of reductive p-adic groups, where we construct a local Langlands correspondence for these Bernstein blocks. Our techniques draw inspirations from the foundational works of Deligne, Kazhdan, Langlands, Lusztig and Shahidi.

As an application, we prove the Local Langlands Conjecture for G_2, which is the first known case in literature of LLC for exceptional groups. Our correspondence satisfies an expected property on cuspidal support, which is compatible with the generalized Springer correspondence, along with a list of characterizing properties including the stabilization of character sums, formal degree property etc. In particular, we obtain (not necessarily unipotent) "mixed" L-packets containing "F-singular" supercuspidals and non-supercuspidals. I will give explicit examples of such mixed L-packets in terms of Deligne-Lusztig theory and Kazhdan-Lusztig parametrization.

If time permits, I will explain how to pin down certain choices in the construction of the correspondence using stability of L-packets; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct explicit LLC for other reductive groups, such as GSp_4, Sp_4, etc, generalizing the depth-zero L-packets of Lust-Stevens. Such explicit description of the L-packets has been useful in number-theoretic applications, e.g. modularity lifting questions.

Various parts of this talk are based on joint works with various collaborators, and attributions will be made clear throughout the talk.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.