Local Geometric Langlands Correspondence: the Spherical Case

Details Local Geometric Langlands Correspondence: the Spherical Case Local Geometric Langlands Correspondence: the Spherical Case

2007-11-07

arxiv.org/abs/0711.1132v1

Mathematics

Quantum Algebra

14 pages

Scientific paper

A module over an affine Kac--Moody algebra g^ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical 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 arXiv:math/0508382.

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