Spherical varieties and langlands duality

Abstract

Let G be a connected reductive complex algebraic group. This paper is devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as a finite-dimensional algebraic model for the loop space of X. The theory we develop associates to X a connected reductive complex algebraic subgroup H of the dual group Ǧ. The construction of Ȟ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of Ȟ. The group Ȟ encodes many aspects of the geometry of X. © 2010 Independent University of Moscow.