Analogue of Weil representation for abelian schemes

Abstract

In this paper we construct a projective action of certain arithmetic group on the derived category of coherent sheaves on an abelian scheme $A$, which is analogous to Weil representation of the symplectic group. More precisely, the arithmetic group in question is a congruence subgroup in the group of "symplectic" automorphisms of $A\times\hat{A}$ where $\hat{A}$ is the dual abelian scheme. The "projectivity" of this action refers to shifts in the derived category and tensorings with line bundles pulled from the base. In particular, if $A$ is an abelian scheme over $S$ equipped with an ample line bundle $L$ of degree 1 then we construct an action of a central extension of $Sp_{2n}(\Bbb Z)$ by $\Bbb Z\times Pic(S)$ on the derived category of coherent sheaves on $A^n$ (the $n$-th fibered power of $A$ over $S$). We describe the corresponding central extension explicitly using the the canonical torsion line bundle on $S$ associated with $L$. As a main technical result we prove the existence of a representation of rank $d$ for a symmetric finite Heisenberg group scheme of odd order $d^2$.