Parabolic sheaves on surfaces and affine Lie algebra gl_n

1999/03/30 by Michael Finkelberg, Alexander Kuznetsov, Finkelberg, Michael +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.math/9903181

openalex publication_date 1999/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an example of geometric construction (via Hecke correspondences) of certain representations of the affine Lie algebra 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_α of rank n parabolic sheaves satisfying certain conditions. The top dimensional irreducible components of K_α are numbered by the isomorphism classes of α-dimensional nilpotent representations of the cyclic quiver A_n-1. Summing up over all α∈\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_α. The natural correspondences give rise to the action of Chevalley generators e_i,f_i∈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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