## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- 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

### Local geometric Langlands correspondence: the spherical case

Edward Frenkel and Dennis Gaitsgory

#### 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

**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