Quiver Varieties and Symmetric Pairs

代数几何和表示论是数学的两个高度发展的分支。前者利用代数方程研究几何，其丰富的历史可以追溯到希腊人，例如解决了将立方体加倍的德利安问题。后者通过用矩阵表示抽象代数结构的元素来研究抽象代数结构。它最初是由弗罗贝尼乌斯在大约一个世纪前发展起来的，现在已经普及到数学的所有领域。这项研究项目位于上述两个数学分支的十字路口。该项目的目标是理解两个看似无关但基本的对象之间的错综复杂的关系：Nakajima从代数几何中抖动变量，从表示理论中抖动对称对的表示。Nakajima簇为简单花边复单李代数的几何表示理论提供了一个自然的家。这个项目将解决中岛理论中的以下两个长期存在的基本问题。第一个是发展了非简单花边复单李代数的Nakajima理论。归根结底，Nakajima理论是一种关于辛几何和表示论相互作用的理论。第二个问题从这个角度出发：从最近由Nakajima簇作为辛自同构和反辛自同构的不动点轨迹出现的部分辛分解中推导出表示理论信息。事实证明，这两个问题是硬币的两面，它们相互提供答案。此外，通过Okounkov和他的合作者的几何R-矩阵理论，这两个问题的研究收敛到对称对的Nakajima理论，本项目为该理论制定了研究计划。后者可以被认为是实单李群/代数的Nakajima理论的无穷小版本，本项目旨在阐明这一点。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry and representation theory are two highly-developed branches of mathematics. The former studies geometry by using algebraic equations and its rich history can be traced back to the Greeks, such as solving the Delian problem of doubling the cube. The latter studies abstract algebraic structure by presenting their elements by matrices. It was first developed by Frobenius about a century ago and has become pervasive across all fields of mathematics. This research project sits at the crossroads of the above two branches of mathematics. The goal of the project is to understand the intricate relations between two seemingly unrelated, but fundamental, objects: Nakajima quiver varieties from algebraic geometry and representations of symmetric pairs from the representation theory. Nakajima varieties provide a natural home for geometric representation theory of simply-laced complex simple Lie algebras. This project will address the following two long-standing, fundamental problems in Nakajima theory. The first is to develop a Nakajima theory for the non-simply-laced complex simple Lie algebras. Ultimately, Nakajima theory is a theory about the interaction of symplectic geometry and representation theory. The second problem arises from this perspective: deduce representation theoretic information from the partial symplectic resolutions recently shown to arise from Nakajima varieties as fixed-point loci of symplectic and anti-symplectic automorphisms. It turns out that the two problems are two sides of a coin and they provide answers to each other. Furthermore, via the geometric R-matrix theory of Okounkov and his collaborators, the study of the two problems converges to a Nakajima theory for symmetric pairs, for which a research plan is laid out in this project. The latter can be thought of as an infinitesimal version of a Nakajima theory of real simple Lie groups/algebras, upon which this project aims to shed light.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.