Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

We describe the Galois action on the middle l-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-et(2)(A((k) over bar), Q(l)(1)) and on a subgroup G(A)(v) = (A x (A) over cap) [3], which depends on v. This generalizes the analysis carried out by Hassett and Tschinkel over C [21]. As a consequence, over number fields, we give a condition under which K-2(A) and K-2(<(A)over cap) are not derived equivalent. The points of G(A)(v) correspond to involutions of K-A(v). Over C, they are known to be symplectic and contained in the kernel of the map Aut(K-A(v)) -> O(H-2 (K-A(v),Z)). We describe this kernel for all varieties K-A(v) of dimension at least 4. When K-A(v) is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds K-A (0, 1, s) over C where A is (1, 3)-polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of K-A(0, 1, s).