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 ${\rm Rep}(G^{\vee})$ of finite-dimensional representations of the Langlands dual group $G^{\vee}$ with the tensor category ${\rm Perv}_{G({\mathcal O})}({\rm Gr}_G)$ of $G({\mathcal O})$ -equivariant perverse sheaves on the affine Grassmannian ${\rm Gr}_G=G(\mathcal{K})/G(\mathcal{O})$ of $G$ . (Here $\mathcal{K}=\mathbb{C}((t))$ and $\mathcal{O}=\mathbb{C}[[t]]$ .) As a by-product one gets a description of the irreducible $G({\mathcal O})$ -equivariant intersection cohomology (IC) sheaves of the closures of $G({\mathcal O})$ -orbits in ${\rm Gr}_G$ in terms of $q$ -analogs of the weight multiplicity for finite-dimensional representations of $G^{\vee}$ .
¶ The purpose of this article is to try to generalize the above results to the case when $G$ is replaced by the corresponding affine Kac-Moody group $G_{\rm aff}$ . (We refer to the (not yet constructed) affine Grassmannian of $G_{\rm aff}$ as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various $G_{\rm aff}(\mathcal{O})$ -orbits inside the closure of another $G_{\rm aff}(\mathcal{O})$ -orbit in ${\rm Gr}_{G_{\rm aff}}$ . 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 $G_{\rm aff}^{\vee}$ , 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