Symplectic birational transformations of finite order on O'Grady's sixfolds

We prove that any symplectic automorphism of finite order on a manifold of type OG6 acts trivially on the Beauville--Bogomolov--Fujiki lattice and that any birational transformation of finite order acts trivially on its discriminant group. Moreover, we classify all possible invariant and coinvariant sublattices.
Comment: Final version, to appear on Kyoto J. Math