  Earlier we showed that the Hilbert scheme of $n$ points in the plane can be
identified with the Hilbert scheme of regular $S_n$ orbits on $C^{2n}$. Using
this result, together with a recent theorem of Bridgeland, King and Reid 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 particular we prove that the dimension
of the space of diagonal harmonics is equal to $(n+1)^{n-1}$.
