Equivariant cohomology of incidence Hilbert schemes and loop algebras

TL;DR

This paper constructs an infinite-dimensional Lie algebra acting on the equivariant cohomology of incidence Hilbert schemes of points on the affine plane, revealing connections to loop algebras and symmetric functions.

Contribution

It introduces a new Lie algebra action on the equivariant cohomology of incidence Hilbert schemes, linking geometric structures to loop algebras and symmetric functions.

Findings

Constructed an infinite-dimensional Lie algebra acting on cohomology

Connected the algebra to the loop algebra of an infinite Heisenberg algebra

Analyzed transformations among different bases of the cohomology space

Abstract

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.