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 $\mathrm {Aut}(G)(F)/jG(F)$, where $j:G(F) \rightarrow \mathrm {Aut}(G)(F)$ 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, Vignéras, and Waldspurger for classical groups which takes any irreducible admissible representation $\pi$ of $G(F)$ for $G$ as 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 $G$ as a reductive algebraic group over a local field $F$ in terms of the (enhanced) Langlands parameter of the representation.