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$${\mathcal {W}}$$ W -algebras and Whittaker categories
Selecta Mathematica ( IF 1.4 ) Pub Date : 2021-06-11 , DOI: 10.1007/s00029-021-00641-6
Sam Raskin

This article is concerned with Whittaker models in geometric representation theory, and gives applications to the study of affine \({\mathcal {W}}\)-algebras. The main new innovation connects Whittaker models to invariants for compact-open subgroups of the loop group. This method, which has a counterpart for p-adic groups, settles a conjecture of Gaitsgory in the categorical setting. This method shows that Whittaker sheaves in geometric representation theory admit t-structures, as had previously been observed in some special cases. We then apply this method to the setting of affine \({\mathcal {W}}\)-algebras. We study a new family of modules for affine \({\mathcal {W}}\)-algebras, which can be thought of as analogues of certain tautological (“generalized vaccuum”) modules over the Kac-Moody algebra. Using the above t-structure, we obtain an affine analogue of Skryabin’s theorem that connects affine \({\mathcal {W}}\)-algebras and Whittaker models. This theorem allows various geometric methods to be used to study affine \({\mathcal {W}}\)-algebras. As one such application, we offer a new proof of one of Arakawa’s foundational results in the theory.



中文翻译:

$${\mathcal {W}}$$ W -代数和惠特克范畴

本文关注几何表示理论中的 Whittaker 模型,并将其应用于仿射\({\mathcal {W}}\) -代数的研究。主要的新创新将 Whittaker 模型与循环群的紧开子群的不变量联系起来。这种方法与p- adic 群对应,解决了分类环境中 Gaitsgory 的猜想。这种方法表明几何表示理论中的惠特克滑轮承认t结构,正如之前在一些特殊情况下观察到的那样。然后我们将此方法应用于仿射\({\mathcal {W}}\) -代数的设置。我们研究了一系列新的仿射模块\({\mathcal {W}}\)-代数,它可以被认为是 Kac-Moody 代数上某些重言式(“广义真空”)模块的类似物。使用上述t结构,我们获得连接仿射\({\mathcal {W}}\) -代数和 Whittaker 模型的 Skryabin 定理的仿射类似物。该定理允许使用各种几何方法来研究仿射\({\mathcal {W}}\) -代数。作为这样的一个应用,我们提供了荒川在该理论中的一项基础结果的新证明。

更新日期:2021-06-13
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