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

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