Pursuing the double affine Grassmannian I: Transversal slices via instantons on A_k-singularities
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 for simplicity we consider only untwisted and simply connected case). The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G_) of finitedimensional representations of the Langlands dual group G_ with the tensor category PervG(O)(GrG) of G(O)-equivariant perverse sheaves on the affine Grassmannian GrG = G(K)/G(O) of G (here K = C((t)) and O = C[[t]]). As a byproduct one gets a description of the irreducible G(O)-equivariant intersection cohomology sheaves of the closures of G(O)-orbits in GrG in terms of q-analogs of the weight multiplicity for finite dimensional representations of G_. The purpose of this paper is to try to generalize the above results to the case when G is replaced by the corresponding affine Kac-Moody group Gaff (we shall refer to the (not yet constructed) affine Grassmannian of Gaff as the double affine Grassmannian). More precisely, in this paper we construct certain varieties that should be thought of as transversal slices to various Gaff(O)-orbits inside the closure of another Gaff (O)-orbit in GrGaff . We present a conjecture that computes the IC sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group G_aff and we check this conjecture in a number of cases. Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian etc.) will be addressed in another publication.