Quiver Varieties and Symmetric Pairs

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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.