# Generalizing the MVW involution, and the contragredient

## TL;DR

This paper constructs a new automorphism for certain quasi-split reductive groups over fields, generalizing a classical involution for classical groups, and proposes a conjecture relating contragredients of representations to Langlands parameters.

## Contribution

It introduces a novel automorphism for quasi-split reductive groups over fields, extending the MVW involution to a broader class of groups, and formulates a conjecture linking contragredients to Langlands parameters.

## Key findings

- Constructed an automorphism i_G for quasi-split reductive groups over fields.
- Generalized the MVW involution for classical groups to a wider class of groups.
- Proposed a conjecture relating contragredients of representations to Langlands parameters.

## Abstract

For certain quasi-split reductive groups $G$ over a general field $F$, we construct an automorphism $\iota_G$ of $G$ over $F$, well-defined as an element of ${\rm Aut}(G)(F)/jG(F)$ where $j:G(F) \rightarrow {\rm Aut}(G)(F)$ is the inner-conjugation action of $G(F)$ on $G$. The automorphism $\iota_G$ generalizes (although only for quasi-split groups) an involution due to Moeglin-Vigneras-Waldspurger in [MVW] for classical groups which takes any irreducible admissible representation $\pi$ of $G(F)$ for $G$ a classical group and $F$ a local field, to its contragredient $\pi^\vee$. The paper also formulates a conjecture on the contragredient of an irreducible admissible representation of $G(F)$ for $G$ a reductive algebraic group over a local field $F$ in terms of the (enhanced) Langlands parameter of the representation.

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Source: https://tomesphere.com/paper/1705.03262