Advanced Studies in Pure Mathematics

Local geometric Langlands correspondence: the spherical case

Edward Frenkel and Dennis Gaitsgory

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Abstract

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].

Article information

Source
Algebraic Analysis and Around: In honor of Professor Masaki Kashiwara's 60th birthday, T. Miwa, A. Matsuo, T. Nakashima and Y. Saito, eds. (Tokyo: Mathematical Society of Japan, 2009), 167-186

Dates
Received: 2 December 2007
First available in Project Euclid: 28 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543447759

Digital Object Identifier
doi:10.2969/aspm/05410167

Mathematical Reviews number (MathSciNet)
MR2499556

Zentralblatt MATH identifier
1192.17012

Citation

Frenkel, Edward; Gaitsgory, Dennis. Local geometric Langlands correspondence: the spherical case. Algebraic Analysis and Around: In honor of Professor Masaki Kashiwara's 60th birthday, 167--186, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05410167. https://projecteuclid.org/euclid.aspm/1543447759


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