Contragredient representations over local fields of positive characteristic

Abstract

It was conjectured bsy 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 parametrization of A. Genestier and V. 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 Lafforgue’s excursion operators.