Spherical varieties and Langlands duality

Details Spherical varieties and Langlands duality Spherical varieties and Langlands duality

2006-11-10

arxiv.org/abs/math/0611323v2

Mathematics

Representation Theory

Scientific paper

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 an algebraic model for the loop space of X. In this paper, we associate to X a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The construction of $\check H$ 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 $\check H$. Combinatorial shadows of the group $\check H$ govern many aspects of the geometry of X such as its compactifications and invariant differential operators. When X is a symmetric variety, the group $\check H$ coincides with that associated to the corresponding real form of G via the (real) geometric Satake correspondence.

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