Quiver Varieties and Symmetric Pairs

代数几何和表示理论是数学中两个高度发展的分支。前者通过使用代数方程来研究几何，其丰富的历史可以追溯到希腊人，比如解决了将立方体加倍的Delian问题。后者研究抽象代数结构，用矩阵表示其元素。它最初是由Frobenius在大约一个世纪前提出的，并在数学的各个领域都很普遍。这个研究项目处于上述两个数学分支的交叉点。该项目的目标是理解两个看似无关但基本的对象之间的复杂关系：代数几何中的中岛颤栗变体和表示理论中的对称对表示。中岛变异体为单列复单李代数的几何表示理论提供了一个天然的家园。本项目将解决中岛理论中两个长期存在的基本问题。首先是发展了非单列复单李代数的中岛理论。最终，中岛理论是关于辛几何和表征理论相互作用的理论。第二个问题从这个角度产生：从最近显示的作为辛和反辛自同构的不动点轨迹的Nakajima变异体中产生的部分辛分辨中推导出表示理论信息。事实证明，这两个问题是一枚硬币的两面，它们相互提供答案。此外，通过Okounkov及其合作者的几何r矩阵理论，将这两个问题的研究收敛到对称对的Nakajima理论，并为此制定了研究计划。后者可以被认为是真正简单李群/代数的中岛理论的无穷小版本，该项目旨在阐明这一点。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.