Title: Standard Extension Algebras I: Perverse Sheaves and Fukaya Calculus Show Chinese title

Title: 标准扩张代数 I： perverse 丛与 Fukaya 计算

Abstract: In this first of a series of articles on standard extension algebras we study standard perverse sheaves on varieties with $\mathbb{G}_m$-actions. Based on Braden's hyperbolic localisation, we describe their extension algebra geometrically via a convolution structure on the intersections of attracting with repelling cells. We introduce a multiplicative structure on open Richardson varieties which provides a practical way to compose these extensions in case of flag varieties. For open Richardson varieties in Grassmannians we construct two explicit cell decompositions, of Gauss- and of Deodhar-type. It is shown that the latter is a stratification with the same combinatorics as Deodhar's decomposition. We introduce a calculus of Fukaya diagrams to encode the geometry of the decompositions. It provides a model for the cohomology of open Richardson varieties and thus for standard extensions. The calculus is motivated by the Mak-Smith Fukaya-Seidel category of a natural Lefschetz fibration and should allow to compute morphism spaces in there. We finally discuss the relation of our work to extensions of (parabolic) Verma modules in category O as well as to the computation of R- and R'-polynomials. Show Chinese abstract

Abstract: 在关于标准扩展代数的一系列文章的第一篇中，我们研究了具有$\mathbb{G}_m$-作用的流形上的标准 perverse sheaves。 基于 Braden 的双曲局部化，我们通过吸引与排斥单元的交集上的卷积结构，几何地描述了它们的扩展代数。 我们引入了开放 Richardson 变体上的乘法结构，这提供了一种在旗形变体情况下组合这些扩展的实用方法。 对于格拉斯曼流形中的开放 Richardson 变体，我们构造了两种显式的单元分解，分别是高斯型和 Deodhar 型。 证明了后者是一个具有与 Deodhar 分解相同组合学的分层结构。 我们引入了 Fukaya 图表的演算来编码分解的几何结构。 它为开放 Richardson 变体的上同调提供了一个模型，从而为标准扩展提供了一个模型。 该演算由自然的 Lefschetz 纤维化的 Mak-Smith Fukaya-Seidel 范畴所激发，应该能够计算其中的态射空间。 最后，我们讨论了我们的工作与范畴 O 中（抛物线）Verma 模块的扩展以及 R- 和 R'-多项式的计算之间的关系。