Abstract
Let G be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group Ğ in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian GrG = G((t))/G[[t]] of the original group G. In the present paper we perform a first step in realizing the category of representations of the quantum group corresponding to Ğ in terms of the geometry of GrG. The idea of the construction belongs to Jacob Lurie.
Highlights
Let us note that the assertion of the above conjecture is inherently transcendental, due to the appearance of the exponential function relating the parameters on both sides
Should be understood in the same framework as (0.4) and Conjecture 0.4: it is an equivalence of chiral categories, where the corresponding structure on the RHS comes from the braided monoidal structure
Decomposes as an external product. This insures, among the rest, that the forgetful map Zxμ → Wx is not in general smooth, the pull-back functor applied to twisted D-modules that belong to Whitc does not produce higher or lower cohomologies
Summary
Let us note that the assertion of the above conjecture is inherently transcendental, due to the appearance of the exponential function relating the parameters on both sides We will cure this as follows: When q is not a root of unity, i.e., when c is irrational, we will eventually replace the RHS, i.e., Rep(Uq(G)), by another category, namely that of factorizable sheaves, denoted FSc(G), which will be equivalent to Rep(Uq(G)) by a transcendental procedure. The main result of the present paper, Theorem 3.11, is that we have an equivalence for c ∈/ Q: Whitc(GrG)n ≃ FSc(G)n for every n This should be interpreted as an equivalence of chiral categories (0.5). Should be understood in the same framework as (0.4) and Conjecture 0.4: it is an equivalence of chiral categories, where the corresponding structure on the RHS comes from the braided monoidal structure.
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