Abstract: Let G be a finite group which acts on a smooth variety M. The McKay correspondence is a principle describing the relationship between the geometry of certain resolutions of the singularities of the quotient M/G and the representation theory of G. Probably the most important example of of this principle in higher dimensions is the case where M=X^n is a power of a smooth surface with the symmetric group G=S_n permuting the factors. Then a crepant resolution of the quotient singularities is given by the Hilbert scheme of points on X. We use the derived McKay correspondence in this setting in order to compute homological invariants of naturally given vector bundles on the Hilbert scheme and to give partial geometric interpretations of two phenomena which occur on the derived category of symmetric quotient stacks; namely the categorical Heisenberg action and exceptional sequences.