## 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 Rep(G^{v}) of finite-dimensional representations of the Langlands dual group G^{v} with the tensor category Perv_{G(O)} (Gr_{G}) of G(O)-equivariant perverse sheaves on the affine Grassmannian Gr_{G} = G(K )/G(O) of G. (Here K = C((t )) and O = 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 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 affine 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} (O)-orbits inside the closure of another Gaff (O)-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^{v}_{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, and so forth) will be addressed in another publication.

Original language | English |
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Pages (from-to) | 175-206 |

Number of pages | 32 |

Journal | Duke Mathematical Journal |

Volume | 152 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2010 |

Externally published | Yes |

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