Uhlenbeck spaces for A² and affine Lie algebra sl_n

2002/02/20 by Michael Finkelberg, Finkelberg, Michael, Dennis Gaitsgory +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT

paper · pdf · doi:10.48550/arxiv.math/0202208

35pp., LaTeX

arxiv created 2002/02/20 · openalex publication_date 2002/02/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce an Uhlenbeck closure of the space of based maps from projective line to the Kashiwara flag scheme of an untwisted affine Lie algebra. For the algebra sl_n this space of based maps is isomorphic to the moduli space of locally free parabolic sheaves on P¹× P¹ trivialized at infinity. The Uhlenbeck closure admits a resolution of singularities: the moduli space of torsion free parabolic sheaves on P¹× P¹ trivialized at infinity. We compute the Intersection Cohomology sheaf of the Uhlenbeck space using this resolution of singularities. The moduli spaces of parabolic sheaves of various degrees are connected by certain Hecke correspondences. We prove that these correspondences define an action of sl_n in the cohomology of the above moduli spaces.

Citations

• Stable pairs on curves and surfaces

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