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Pursuing the double affine Grassmannian I: transversal slices via instantons on A_k-singularities

Research paper by Alexander Braverman, Michael Finkelberg

Indexed on: 13 Nov '09Published on: 13 Nov '09Published in: Mathematics - Algebraic Geometry



Abstract

This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck space of instantons on the quotient of the affine plane by a finite cyclic subgroup of SL(2).