PURSUING THE DOUBLE AFFINE GRASSMANNIAN, I: TRANSVERSAL SLICES VIA INSTANTONS ON Ak-SINGULARITIES

This article is the first in a series that describes a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. (For simplicity, we only consider the untwisted and simply connected case here.) The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G(v)) of finite-dimensional representations of the Langlands dual group G(v) with the tensor category Perv(G)(0)(Gr(G)) of G(0)-equivariant perverse sheaves on the affine Grassmannian GrG = G(X)/G(0) of G. (Here K = C((t)) and 0 = C[[t].) As a by-product one gets a description of the irreducible G(0)-equivariant intersection cohomology (IC) sheaves of the closures of G(0)-orbits in Gr(G) in terms of q-analogs of the weight multiplicity for finite-dimensional representations of G(v). The purpose of this article is to try to generalize the above results to the case when G is replaced by the corresponding qffine Kac-Moody group G(aff). (We refer to the (not yet constructed) affine Grassmannian of G(aff) as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various G(aff)(0)-orbits inside the closure of another G(aff)(9)orbit in Gr(Gaff). We present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group G(aff)(V), 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, and so forth) will be addressed in another publication.