Contragredient representations and characterizing the local Langlands correspondence

TL;DR

This paper investigates the contragredient in the local Langlands correspondence, proposing a conjecture linking it to the Chevalley automorphism of the L-group, and proves this for real groups.

Contribution

It introduces a conjecture relating the contragredient to the Chevalley automorphism and proves it for real groups, advancing understanding of the local Langlands correspondence.

Findings

Conjecture that contragredient corresponds to Chevalley automorphism.

Proved the conjecture for real groups.

Explored Hermitian dual in GL(n,R).

Abstract

We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization of the local Langlands correspondence over R. We also consider the related notion of Hermitian dual, in the case of GL(n,R).