The MVW involution of the metaplectic group

The MVW involution – named after Colette Moeglin, Marie-France Vignéras, and Jean-Loup Waldspurger – is a fundamental dualizing involution in the representation theory of $p$-adic classical groups. It extends the well-known transpose-inverse automorphism for general linear groups. In this work, we establish the existence of the MVW involution for the metaplectic group over a non-archimedean local field $F$ of characteristic different from $2$ and with residue characteristic $p$. Our construction applies to representations over any coefficient field of characteristic distinct from $p$.