Parabolic sheaves on surfaces and affine Lie algebra $\hat{gl}_n$

Abstract

We give an example of geometric construction (via Hecke correspondences) of certain representations of the affine Lie algebra $\hat{gl}_n$. The construction is similar to the one of [FK] for the Lie algebra $sl_n$. Given a surface with a smooth embedded curve $C$ we consider the moduli spaces $K_\alpha$ of rank $n$ parabolic sheaves satisfying certain conditions. The top dimensional irreducible components of $K_\alpha$ are numbered by the isomorphism classes of $\alpha$-dimensional nilpotent representations of the cyclic quiver $\tilde{A}_{n-1}$. Summing up over all $\alpha\in{\Bbb N}[{\Bbb Z}/n{\Bbb Z}]$ we obtain a vector space $M$ with a basis of fundamental classes of top dimensional components of $K_\alpha$. The natural correspondences give rise to the action of Chevalley generators $e_i,f_i\in\hat{sl}_n$ on $M$. We compute explicitly the matrix coefficients of $e_i,f_i$ in the above basis. The central charge of $M$ depends on the genus of the curve $C$ and the degree of its normal bundle.