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Let M be a 2n-dimensional Kahler manifold deformation equivalent to theHilbert scheme of length n subschemes of a K3 surface S. Let Mon be the groupof automorphisms of the cohomology ring of M, which are induced by monodromyoperators. The second integral cohomology of M is endowed with theBeauville-Bogomolov bilinear form. We prove that the restriction homomorphismfrom 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, theimage of Mon in O[H^2(M)] is the subgroup generated by reflections with respectto +2 and -2 classes. As a consequence, we get counter examples to a version ofthe weight 2 Torelli question, when n-1 is not a prime power.Comment: Version 3: Latex, 54 pages. Expository change

INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE