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

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 $\ell$.