Correspondance th\^eta locale $\ell$-modulaire I : groupe m\'etaplectique, repr\'esentation de Weil et $\Theta$-lift

TL;DR

This paper extends the theory of the Weil representation and theta correspondence to fields of characteristic different from 2, introducing modular tools for studying dual pairs, scalar extension, and reduction modulo .

Contribution

It develops the modular Weil representation for symplectic groups over various fields and explores its implications for theta lifts and dual pairs in modular settings.

Findings

Construction of the modular Weil representation over fields of characteristic .

Analysis of the splitting behavior of dual pairs in the modular setting.

Framework for studying scalar extension and reduction modulo  in theta correspondence.

Abstract

Let $F$ be a field which is, either local non archimedean, or finite, of residual charcateristic $p$ but of characteristic different from $2$. Let $W$ be a symplectic space of finite dimension over $F$. Suppose $R$ is a field of characteristic $\ell \neq p$ so that there exists a non trivial smooth additive character $\psi : F \to R^\times$. Then the Stone-von Neumann theorem of the Heisenberg group $H(W)$ is still valid for representations with coefficients in $R$. It leads to a projective representation of the group $\text{Sp}(W)$ which lifts to a genuine smooth representation of a central extension of $\text{Sp}(W)$ by $R^\times$: this is the modular Weil representation of the metaplectic group. For any dual pair $(H_1,H_2)$, their lifts to the metaplectic group may splitor not according to the different cases at stake. Eventually, computing the biggest isotypic quotient of the modular Weil representation allows to define the $\Theta$-lift. Some new lines of investigation are thus available with these new tools such as studying scalar extension and reduction modulo $\ell$.