Geometric construction of representations of affine algebras

Let $\mathrm{\Gamma }$ be a finite subgroup of ${\text{SL}}_2(\text{C})$. We consider $\mathrm{\Gamma }$-fixed point sets in Hilbert schemes of points on the affine plane ${\text{C}}^2$. The direct sum of homology groups of components has a structure of a representation of the affine Lie algebra $\text{ag}$ corresponding to $\mathrm{\Gamma }$. If we replace homology groups by equivariant $K$-homology groups, we get a representation of the quantum toroidal algebra $\text{Ut}$. We also discuss a higher rank generalization and character formulas in terms of intersection homology groups.