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VOL. 54 | 2009 Local geometric Langlands correspondence: the spherical case
Edward Frenkel, Dennis Gaitsgory

Editor(s) Tetsuji Miwa, Atsushi Matsuo, Toshiki Nakashima, Yoshihisa Saito


A module over an affine Kac–Moody algebra $\widehat{\mathfrak{g}}$ is called spherical if the action of the Lie subalgebra $\mathfrak{g} [[t]]$ on it integrates to an algebraic action of the corresponding group $G [[t]]$. Consider the category of spherical $\widehat{\mathfrak{g}}$-modules of critical level. In this paper we prove that this category is equivalent to the category of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a categorical version of the well-known description of spherical vectors in representations of groups over local non-archimedian fields. It may be viewed as a special case of the local geometric Langlands correspondence proposed in [FG2].


Published: 1 January 2009
First available in Project Euclid: 28 November 2018

zbMATH: 1192.17012
MathSciNet: MR2499556

Digital Object Identifier: 10.2969/aspm/05410167

Rights: Copyright © 2009 Mathematical Society of Japan


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