  We study moduli spaces of O'Grady's ten-dimensional irreducible symplectic
manifolds. These moduli spaces are covers of modular varieties of dimension 21,
namely quotients of hermitian symmetric domains by a suitable arithmetic group.
The interesting and new aspect of this case is that the group in question is
strictly bigger than the stable orthogonal group. This makes it different from
both the K3 and the K3^[n] case, which are of dimension 19 and 20 respectively.
