## Duke Mathematical Journal

### Pursuing the double affine Grassmannian, I: Transversal slices via instantons on $A_k$-singularities

#### Abstract

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

#### Article information

Source
Duke Math. J., Volume 152, Number 2 (2010), 175-206.

Dates
First available in Project Euclid: 31 March 2010

https://projecteuclid.org/euclid.dmj/1270041107

Digital Object Identifier
doi:10.1215/00127094-2010-011

Mathematical Reviews number (MathSciNet)
MR2656088

Zentralblatt MATH identifier
1200.14083

#### Citation

Braverman, Alexander; Finkelberg, Michael. Pursuing the double affine Grassmannian, I: Transversal slices via instantons on $A_k$ -singularities. Duke Math. J. 152 (2010), no. 2, 175--206. doi:10.1215/00127094-2010-011. https://projecteuclid.org/euclid.dmj/1270041107

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