Incidence Hilbert schemes and infinite dimensional Lie algebras

Let S be a smooth projective surface. Using correspondences, we construct an infinite dimensional Lie algebra that acts on the direct sum He S = M+∞ m=0 H∗(S[m,m+1]) of the cohomology groups of the incidence Hilbert schemes S[m,m+1]. The algebra is related to an extension of an infinite dimensional Heisenberg algebra. The space He S is a highest weight representation of this algebra. Our result provides a representation-theoretic interpretation of Cheah’s generating function of Betti numbers of the incidence Hilbert schemes. As a consequence, an additive basis of H∗(S[m,m+1]) is obtained.