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Hall algebras of curves, commuting varieties and Langlands duality

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Abstract

We construct an isomorphism between the (universal) spherical Hall algebra of a smooth projective curve of genus g and a convolution algebra in the (equivariant) K-theory of the genus g commuting varieties \({C_{\mathfrak{gl}_r}=\big\{ (x_i, y_i) \in \mathfrak{gl}_r^{2g}\;;\; \sum_{i=1}^g [x_i,y_i]=0\big\}}\) . We can view this isomorphism as a version of the geometric Langlands duality in the formal neighborhood of the trivial local system, for the group GL r . We extend this to all reductive groups and we compute the image, under our correspondence, of the skyscraper sheaf supported on the trivial local system.

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences d’Orsay, Bât. 425, Université de Paris-Sud 11, 91405, Orsay-Cedex, France

    O. Schiffmann

  2. Département de Mathématiques, Université de Paris 7, 175 ruedu Chevaleret, 75013, Paris, France

    E. Vasserot

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  1. O. Schiffmann
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  2. E. Vasserot
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Schiffmann, O., Vasserot, E. Hall algebras of curves, commuting varieties and Langlands duality. Math. Ann. 353, 1399–1451 (2012). https://doi.org/10.1007/s00208-011-0720-x

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  • DOI: https://doi.org/10.1007/s00208-011-0720-x

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