The MVW involution of the metaplectic group
Shuichiro Takeda, Justin Trias
TL;DR
This work proves the MVW involution for the metaplectic group Mp(W) over a non-archimedean field F with char(F) \ eq 2 and residue characteristic p, for representations with coefficient field of char \\ell \ eq p. The authors reduce the existence of the involution to a key conjugacy statement: for every \\tilde{g} \ ightarrow Mp(W) and \\delta \ ightarrow GSp(W) with similitude -1, the elements \\tilde{g} and its \\delta-conjugate satisfy \\[^{\\delta}\\tilde{g} \\sim \\tilde{g}^{-1}. They establish this via a lifting of the Sp(W) conjugacy using Jordan decomposition in Mp(W), handling semisimple, Jordan-decomposable, and general cases (the latter using purely inseparable extensions). Consequently, for any algebraically closed coefficient field R with char \\ell \ eq p and any irreducible genuine representation \\pi of Mp(W), one has \\pi^{\\delta} \\simeq \\\\pi^{\\vee}, verified through the trace-character method within the invariant-distributions framework and its Matryoshka stratification. This yields a broad MVW duality result for metaplectic groups, extending complex-analytic results and enabling modular applications in positive characteristic.
Abstract
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 $p$.