Contragredient representations over local fields of positive characteristic

Abstract

It is conjectured by Adams-Vogan and Prasad that under the local Langlands correspondence, the L-parameter of the contragredient representation equals that of the original representation composed with the Chevalley involution of the L-group. We verify a variant of their prediction for all connected reductive groups over local fields of positive characteristic, in terms of the local Langlands parameterization of Genestier-Lafforgue. We deduce this from a global result for cuspidal automorphic representations over function fields, which is in turn based on a description of the transposes of V. Lafforgue's excursion operators. MSC (2010) 11F70; 11R58 22E55

Representations of reductive groups over local fields

“…Part (1) can be stated for L-packets on an individual group, and hence without assuming Conjecture 2.5. In this form it was formulated by Adams-Vogan in [AV16], where it was also proved for F = R. Wen-Wei Li proved that statement of (1) in [Li19] for the semisimplified basic correspondence of [GL17]. Part (1') is a variation of part (1) -it implies part (1) via linear independence of characters, provided one assumes Conjecture 2.2.…”