Abstract:We describe the Galois action on the middle
$\ell $
-adic cohomology of smooth, projective fourfolds
$K_A(v)$
that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface A with Mukai vector v. We show this action is determined by the action on
$H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$
and on a subgroup
$G_A(v) \leqslant (A\times \hat {A})[3]$
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