Geometric construction of representations of affine algebras

TL;DR

This paper constructs geometric representations of affine and quantum toroidal algebras using fixed point sets in Hilbert schemes, extending to higher ranks and providing character formulas.

Contribution

It introduces a geometric method to realize affine and quantum toroidal algebra representations via fixed point sets and homology in Hilbert schemes, including higher rank generalizations.

Findings

Homology groups form representations of affine Lie algebras.

Equivariant K-homology yields quantum toroidal algebra representations.

Provides character formulas using intersection homology.

Abstract

Let $\Gamma$ be a finite subgroup of $\SL_2(\C)$. We consider $\Gamma$-fixed point sets in Hilbert schemes of points on the affine plane $\C^2$. The direct sum of homology groups of components has a structure of a representation of the affine Lie algebra $\ag$ corresponding to $\Gamma$. If we replace homology groups by equivariant $K$-homology groups, we get a representation of the quantum toroidal algebra $\Ut$. We also discuss a higher rank generalization and character formulas in terms of intersection homology groups.