  Let M be a 2n-dimensional Kahler manifold deformation equivalent to the
Hilbert scheme of length n subschemes of a K3 surface S. Let Mon be the group
of automorphisms of the cohomology ring of M, which are induced by monodromy
operators. The second integral cohomology of M is endowed with the
Beauville-Bogomolov bilinear form. We prove that the restriction homomorphism
from Mon to the isometry group O[H^2(M)] is injective, for infinitely many n,
and its kernel has order at most 2, in the remaining cases. For all n, the
image of Mon in O[H^2(M)] is the subgroup generated by reflections with respect
to +2 and -2 classes. As a consequence, we get counter examples to a version of
the weight 2 Torelli question, when n-1 is not a prime power.
