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 $j:G(F)\to \mathrm{Aut}(G)(F)$ 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-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.

arXiv admin note: substantial text overlap with arXiv:1512.04347

References in corpus (2)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• A `relative' local Langlands correspondence