Hecke algebras for p-adic groups, the explicit Local Langlands Correspondence and stability
De Yujie Xu
Apparaît dans la collection : Automorphic forms, endoscopy and trace formulas / Formes automorphes, endoscopie et formule des traces
I will talk about my joint work with Aubert where we prove the Local Langlands Conjecture for $G_2$ (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of (arbitrary) reductive $p$-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties (including stability). In particular, we obtain 'mixed' L-packets containing F-singular supercuspidals and nonsupercuspidals. Our methods are inspired by the Langlands-Shahidi method, Deligne-Lusztig and Lusztig theories etc. If time permits, I will explain how to characterize our correspondence using stability of L-packets, by computing character formulae in terms of (generalized) Green functions ; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct LLC for other reductive groups, such as $G S p(4), S p(4)$, etc. The latter parts are based on recent joint work with Suzuki.
