Genericity and contragredience in the local Langlands correspondence

Abstract

We prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. The proof holds for tempered representations of quasi-split real K-groups and quasi-split p-adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.

Distinguished regular supercuspidal representations

“…In this subsection, we give a proof of this conjecture for regular supercuspidal parameters, following the arguments of [Kal13] closely. From now on, we fix a regular supercuspidal L-parameter ϕ for G.…”

Contragredient representations and characterizing the local Langlands correspondence

“…For discrete series L-packets of real groups, Conjecture 1.1 follows from [18,Lemma 7.4.1], and it is known for some representations of some quasisplit p-adic groups by [13]. It would be reasonable to impose Conjecture 1.1 as a condition on the local Langlands correspondence in cases where it is not known.…”

Contragredient representations over local fields of positive characteristic

“…• When F is Archimedean, the local Langlands correspondence is Langlands' paraphrase of Harish-Chandra's works. The "first layer" of the Adams-Vogan-Prasad conjecture is established by Adams-Vogan [1], and Kaletha [18,Theorem 5.9] obtained the necessary refinement for the "second layer".…”