Title: Generalizing the MVW involution, and the contragredient,
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.