Groups acting on moduli spaces of hyper-Kähler manifolds

The period morphism of polarized hyper-Kähler manifolds of K3${}^{[m]}$-type gives an embedding of each connected component of the moduli space of polarized hyper-Kähler manifolds of K3${}^{[m]}$-type into their period space, which is the quotient of a Hermitian symmetric domain by an arithmetic group. Following work of Stellari and Gritsenko-Hulek-Sankaran, we study the ramification of covering maps between these period spaces that arise from the action of some groups of isometries.