Induced birational transformations on O'Grady's sixfolds

We introduce the notion of induced birational transformations of irreducible holomorphic symplectic sixfolds of the sporadic deformation type discovered by O'Grady. We give a criterion to determine when a manifold of OG6 type is birational to a moduli space of sheaves on an abelian surface. Then we determine when a birational transformation of the moduli space is induced by an automorphism of the abelian surface. Referring to the Mongardi-Rapagnetta-Sacca birational model of manifolds of OG6 type, we give a result to determine when a birational transformation is induced at the quotient. We give an application of these criteria in the non-symplectic case.