  In this paper, we consider moduli spaces of stable sheaves on abelian
surfaces. Our main assumption is the primitivity of the associated Mukai
vector. We construct many isomorphisms of muduli spaces induced by
Fourier-Mukai functor. As an application, we show that deformation type of
these spaces are determined by their dimension. We next show that the fiber of
the albanese map is irreducible symplectic manifolds In particular, we describe
the period by using Mukai lattice.
  We also discuss deformation type of moduli spaces of stable sheaves on K3
surfaces.
