Contragredient representations over local fields of positive characteristic

TL;DR

This paper verifies a conjecture relating contragredient representations and L-parameters over local fields of positive characteristic, using the Genestier-Lafforgue parameterization and global automorphic results.

Contribution

It proves a variant of the Adams-Vogan and Prasad conjecture for all connected reductive groups over positive characteristic local fields, extending the understanding of local Langlands correspondence.

Findings

Confirmed the conjecture for all connected reductive groups over positive characteristic fields.

Connected the local conjecture to global automorphic representations over function fields.

Provided a description of transposes of Lafforgue's excursion operators.

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.