Towards a theta correspondence in families for type II dual pairs

Abstract

Let $R$ be a commutative $\mathbb{Z}[1/p]$-algebra, let $m\le n$ be positive integers, and let $G_n={\text{GL}}_n(F)$ and $G_m={\text{GL}}_m(F)$ where $F$ is a $p$-adic field. The Weil representation is the smooth $R[G_n\times G_m]$-module $C_c^{\infty }({\text{Mat}}_{n\times m}(F),R)$ with the action induced by matrix multiplication. When $R=\mathbb{C}$ or is any algebraically closed field of banal characteristic compared to $G_n$ and $G_m$, the local theta correspondence holds by the work of Howe and Mínguez. At the level of supercuspidal support, we interpret the theta correspondence as a morphism of varieties ${\theta }_R$, which we describe as an explicit closed immersion. For arbitrary $R$, we construct a canonical ring homomorphism ${\theta }_R^{\#}:{\mathfrak{Z}}_R(G_n)\to {\mathfrak{Z}}_R(G_m)$ that controls the action of the center ${\mathfrak{Z}}_R(G_n)$ of the category of smooth $R[G_n]$-modules on the Weil representation. We use the rank filtration of the Weil representation to first obtain ${\theta }_{\mathbb{Z}[1/p]}^{\#}$, then obtain ${\theta }_R^{\#}$ for arbitrary $R$ by proving ${\mathfrak{Z}}_R(G_n)$ is compatible with scalar extension. In particular, the map $\text{Spec}({\mathfrak{Z}}_R(G_m))\to \text{Spec}({\mathfrak{Z}}_R(G_n))$ induced by ${\theta }_R^{\#}$ recovers ${\theta }_R$ in the $R=\mathbb{C}$ case and in the banal case. We use gamma factors to prove ${\theta }_R^{\#}$ is surjective for any $R$. Finally, we describe ${\theta }_R^{\#}$ in terms of the moduli space of Langlands parameters and use this description to give an alternative proof of surjectivity in the tamely ramified case.

42 pages