  Let $S$ be the affine plane $\C^2$ together with an appropriate $\mathbb T =
\C^*$ action. 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 $$\Wft = \bigoplus_{m=0}^{+\infty}H^{2(m+1)}_{\mathbb
T}(S^{[m,m+1]})$$ of the middle-degree equivariant cohomology group of
$\hil{m,m+1}$. 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 $\Wft$. Our results are applied to the ring
structure of the ordinary cohomology of $\hil{m,m+1}$ and to the ring of
symmetric functions in infinitely many variables.
