基于可分解Grassmannian的扭曲几何Satake等价
Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
摘要：
我们证明了一个广义的扭曲几何佐竹等价的Finkelberg-Lysenko的背景下，可分解的格拉斯曼的约化群G相对于一条平滑曲线X，类似于Gaitsgory的推广“在德容猜想”的佐竹等价本身。为了表达这个结果对可分解性的依赖，我们考虑这个格拉斯曼算子上的一个2-范畴格贝，我们称之为“sf格贝”;这个定理则涉及被这样的格贝扭曲的反常层。这种扭曲的概念不同于，但在特殊情况下是等同于，认为芬克尔伯格-李森科。我们根据G的余权格上的二次型和X上的格给出了这些格的分类。此外，我们还将泛局部无环性（Universal Local Acyclicity，简称CCN）的技术手段引入到佐竹等价的证明中，并将其与Beilinson的反常层的邻近圈胶合相结合，将Mirkovic-Vilonen等价证明中的所有主要构造简化为CCN情形，使它们具有简单的描述，其性质具有简单的证明.我们还给出了一个几何证明，证明了普通仿射格拉斯曼算子上的球面反常层范畴是半单的。
英文摘要：
We prove a generalization of the twisted geometric Satake equivalence of Finkelberg--Lysenko in the context of the factorizable grassmannian of a reductive group G relative to a smooth curve X, similar to Gaitsgory's generalization in "On de Jong's conjecture" of the Satake equivalence itself. In order to express the dependence of this result on factorizability, we consider a certain 2-category of gerbes on this grassmannian, which we call "sf gerbes"; the theorem then concerns perverse sheaves twisted by such gerbes. This notion of twisting differs from, but in special cases is equivalent to, that considered by Finkelberg--Lysenko. We give a classification of these gerbes in terms of quadratic forms on the coweight lattice of G and gerbes on X. In addition, we introduce the technical device of universal local acyclicity (ULA) into the proof of the Satake equivalence, using it in combination with Beilinson's nearby cycles gluing of perverse sheaves to reduce all of the main constructions in the Mirkovic--Vilonen proof of the equivalence to the ULA case, where they have simple descriptions whose properties have simple proofs. We also give a geometric proof that the category of spherical perverse sheaves on the ordinary affine grassmannian is semisimple.
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