Alexander Braverman, Michael Finkelberg

Duke Math. J. 152 (2), 175-206, (1 April 2010) DOI: 10.1215/00127094-2010-011
KEYWORDS: 14J60, 14D21

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 $\mathrm{Rep}({G}^{\vee})$ of finite-dimensional representations of the Langlands dual group ${G}^{\vee}$ with the tensor category ${\mathrm{Perv}}_{G(O)}({\mathrm{Gr}}_{G})$ of $G(O)$-equivariant perverse sheaves on the affine Grassmannian ${\mathrm{Gr}}_{G}=G(K)/G(O)$ of $G$. (Here $K=\mathbb{C}((t))$ and $O=\mathbb{C}[[t]]$.) As a by-product one gets a description of the irreducible $G(O)$-equivariant intersection cohomology (IC) sheaves of the closures of $G(O)$-orbits in ${\mathrm{Gr}}_{G}$ 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 ${G}_{\mathrm{aff}}$. (We refer to the (not yet constructed) affine Grassmannian of ${G}_{\mathrm{aff}}$ as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various ${G}_{\mathrm{aff}}(O)$-orbits inside the closure of another ${G}_{\mathrm{aff}}(O)$-orbit in ${\mathrm{Gr}}_{{G}_{\mathrm{aff}}}$. 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 ${G}_{\mathrm{aff}}^{\vee}$, 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