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×
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
S
p
2n
(Z)
by
Z×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
.