  Let S be a K3 surface and Aut D(S) the group of auto-equivalences of the
derived category of S. We construct a natural representation of Aut D(S) on the
cohomology of all moduli spaces of stable sheaves (with primitive Mukai
vectors) on S. The main result of this paper is the precise relation of this
action with the monodromy of the Hilbert schemes S^[n] of points on the
surface. A formula is provided for the monodromy representation, in terms of
the Chern character of the universal sheaf. Isometries of the second cohomology
of S^[n] are lifted, via this formula, to monodromy operators of the whole
cohomology ring of S^[n].
