Geometric construction of representations of affine algebras

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.

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• Noncommutative Instantons and Twistor Transform
• Quantum Affine Algebras and their Representations
• Stable pairs on curves and surfaces
• Vertex representations via finite groups and the McKay correspondence
• Quiver varieties and a non-commutative P^2
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• Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory
• Quiver varieties and tensor products, II
• Quot schemes for Kleinian orbifolds
• $G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products
• Geometry of canonical bases and mirror symmetry
• Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras
• Affinizations of Lorentzian Kac-Moody Algebras and Hilbert Schemes of Points on K3 Surfaces