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

Abstract

We give an example of geometric construction (via Hecke correspondences) of certain representations of the affine Lie algebra ${\widehat{gl}}_n$. The construction is similar to the one of [FK] for the Lie algebra $s{l}_n$. 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 $\alpha \in \mathbb{N}[\mathbb{Z}/n\mathbb{Z}]$ 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 $e_i,f_i\in {\widehat{sl}}_n$ 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.

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