Equivariant cohomology of incidence Hilbert schemes and infinite dimensional Lie algebras

Abstract

Let S be the affine plane \({\mathbb C^2}\) together with an appropriate \({\mathbb T = \mathbb C^*}\) action. Let S[m,m+1] be the incidence Hilbert scheme. Parallel to Li and Qin (2007, Incidence Hilbert schemes and infinite dimensional Lie algebras, Hangzhou), we construct an infinite dimensional Lie algebra that acts on the direct sum $$\widetilde {\mathbb H}_{\mathbb T} = \bigoplus_{m=0}^{+\infty}H^{2(m+1)}_{\mathbb T}(S^{[m,m+1]})$$ of the middle-degree equivariant cohomology group of S[m,m+1]. The algebra is related to an infinite dimensional Heisenberg algebra. In addition, we study the transformations among three different linear bases of \({\widetilde {\mathbb H}_{\mathbb T}}\) . Our results are applied to the ring structure of the ordinary cohomology of S[m,m+1] and to the ring of symmetric functions in infinitely many variables.