Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
复制标题
可因式分解的grassmannian上通过gerbes的扭曲几何Satake等价
DOI：
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发表时间：
2010
期刊：
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通讯作者：
R. Reich
中科院分区：
文献类型：
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作者：
R. Reich
Ginzburg和Mirkovic-Vilonen对复约化群G的几何Satake等价是其Langlands对偶群LG的表示的张量范畴作为仿射格拉斯曼群GrG = G（C（（t）/G（C[[t]]）上的“球面”反常层范畴的实现。自其最初的声明以来，它已被推广到两个方向：首先，由Gaitsgory，到Beilinson-Drinfeld或可因式分解的格拉斯曼，对于光滑复曲线X，它是幂Xn上的空间的集合，其一般纤维同构于GrG，但当它们接近具有相等坐标的点时，因子“融合”，允许更自然地描述甚至Mirkovic-Vilonen等价物的结构和性质。第二个推广，由于最近的芬克尔伯格-李森科，认为反常层扭曲在一个适当的意义上由一个根的单位，并获得了范畴的代表以外的一个群体的朗兰兹对偶。这后一个结果可以被认为是“量子群的朗兰兹对偶性”的一部分。在这项工作中，我们得到了一个结果，同时推广上述所有。我们认为一般的概念扭曲的gerbe和定义的自然类的“factorizable”gerbes，其中一个可以扭曲的上下文中的佐竹等价。这些格几乎完全由G的权格上的二次型描述。我们表明，一个合适的形式主义存在，这样的方法Mirkovic-Vilonen可以直接应用在这个一般的上下文中几乎没有变化，并获得扭曲的反常层的佐竹等价。此外，我们提出了新的证明其结构的性质作为一个交换张量范畴。
The geometric Satake equivalence of Ginzburg and Mirkovic– Vilonen, for a complex reductive group G, is a realization of the tensor category of representations of its Langlands dual group LG as a category of “spherical” perverse sheaves on the affine grassmannian GrG = G(C((t)))/G(C[[t]]). Since its original statement it has been generalized in two directions: first, by Gaitsgory, to the Beilinson–Drinfeld or factorizable grassmannian, which for a smooth complex curve X is a collection of spaces over the powers Xn whose general fiber is isomorphic to GrG but with the factors “fusing” as they approach points with equal coordinates, allowing a more natural description of the structures and properties even of the Mirkovic–Vilonen equivalence. The second generalization, due recently to Finkelberg–Lysenko, considers perverse sheaves twisted in a suitable sense by a root of unity, and obtains the category of representations of a group other than the Langlands dual. This latter result can be considered as part of “Langlands duality for quantum groups”. In this work we obtain a result simultaneously generalizing all of the above. We consider the general notion of twisting by a gerbe and define the natural class of “factorizable” gerbes by which one can twist in the context of the Satake equivalence. These gerbes are almost entirely described by the quadratic forms on the weight lattice of G. We show that a suitable formalism exists such that the methods of Mirkovic–Vilonen can be applied directly in this general context virtually without change and obtain a Satake equivalence for twisted perverse sheaves. In addition, we present new proofs of the properties of their structure as an abelian tensor category.