In an earlier paper [14], we showed that the Hilbert scheme of points in the plane H _n =Hilb ⁿ (ℂ²) can be identified with the Hilbert scheme of regular orbits ℂ^{2 n} //S _n . Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author in [9]. In particular we prove that the dimension of the space of diagonal harmonics is equal to (n+1) ^{n -1}.