Genericity and contragredience in the local Langlands correspondence

TL;DR

This paper proves conjectures about how the local Langlands correspondence behaves under taking contragredients and changing Whittaker data for certain groups, advancing understanding of representation dualities.

Contribution

It establishes the behavior of the local Langlands correspondence under contragredients and Whittaker data changes for quasi-split real and p-adic classical groups, confirming recent conjectures.

Findings

Proved conjectures of Adams-Vogan and D. Prasad.

Derived formulas for changes in Whittaker data.

Validated behavior for tempered representations of specific groups.

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.