Some incarnations of Hamiltonian reduction in symplectic geometry and geometric representation theory
Peter Crooks, Xiang Gao, Mitchell Pound, Casen Thompson
TL;DR
This survey develops a unifying view of Hamiltonian reduction across symplectic geometry and geometric representation theory, introducing abelianization techniques via symplectic implosion and Gelfand–Cetlin data, and extending reduction to generalized settings with symplectic groupoids. It demonstrates how nonabelian reductions can be recast as abelian reductions on imploded spaces or through higher-rank torus data, providing a cohesive framework that includes level-zero shifts and orbit-type stratifications. The Moore–Tachikawa conjecture is integrated through scheme-theoretic and groupoid formalisms, with a 2D TQFT lens and Morita-equivalence perspectives that link symplectic reduction to topological quantum field theories. Overall, the article offers concrete tools (shifting trick, pre-Poisson reduction) and a broad generalized reduction program that encapsulates classical Marsden–Weinstein theory within a modern, categorical, and geometric representation-theoretic context.
Abstract
In this expository note, we give a self-contained introduction to some modern incarnations of Hamiltonian reduction. Particular emphasis is placed on applications to symplectic geometry and geometric representation theory. We thereby discuss abelianization in Hamiltonian geometry, reduction by symplectic groupoids, and the Moore--Tachikawa conjecture.