Localization of \(\mathfrak g\)-modules on the affine Grassmannian. (English) Zbl 1202.17009

Frenkel, Edward; Gaitsgory, Dennis

Ann. Math. (2) 170, No. 3, 1339-1381 (2009).

This work is in the context of the local geometric Langlands correspondence. A general framework and conjectures for this correspondence were proposed by the authors in [“Local geometric Langlands correspondence and affine Kac-Moody algebras”, Algebraic geometry and number theory, Prog. Math. 253, 69–260 (2006; Zbl 1184.17011)].
Let \(G\) be a simple complex algebraic group and \(\mathfrak{g}\) the Lie algebra of \(G\). Let \(\hat{\mathfrak{g}}\) be the affine Kac-Moody algebra, the universal central extension of the formal loop algebra \(\mathfrak{g}((t))\). Representations of \(\hat{\mathfrak{g}}\) have a parameter, an invariant bilinear form on \(\mathfrak{g}\), which is called the level. Representations corresponding to the bilinear form which is equal to minus one half of the Killing form are called representations of critical level. The authors consider the category of modules over the affine Kac-Moody algebra \(\hat{\mathfrak{g}}\) of critical level with regular central character. In (loc.cit.) they conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian \(G((t))/G[[t]]\). In this paper the authors prove this conjecture for the corresponding \(I^0\) equivariant categories, where \(I^0\) is the radical of the Iwahori subgroup of \(G((t))\).

Reviewer: Dubravka Ban (Carbondale)

MSC:

17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
11G45	Geometric class field theory
22E67	Loop groups and related constructions, group-theoretic treatment

Keywords:

geometric Langlands correspondence; affine Kac-Moody algebras; affine Grassmannian

Citations:

Zbl 1184.17011

References:

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