Analogue of Weil representation for abelian schemes

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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×A where 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 Sp₂ₙ(Z) by Z× Pic(S) on the derived category of coherent sheaves on Aⁿ (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 d².