Equivariant cohomology of incidence Hilbert schemes and loop algebras

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