Gelfand–Cetlin abelianizations of symplectic quotients

authors: Peter Crooks, Jonathan Weitsman year: 2024 See in Zotero

Definition: Gelfand-Cetlin datum

The main definition used to obtain the main results of the paper:

Define the quantities

A Gelfand-Cetlin datum is a pair , which consists of a continuous map

and an open dense subset that satisfy the following conditions:

1. are -invariant on and smooth on
2. is a basis for the lattice for all
4. is a smooth submersion and moment map for a Poisson Hamiltonian -space structure on
5. is a principal -bundle.
6. if is a Hamiltonian -space with moment map , then

is a moment map for a Hamiltonian -space structure on

Main result

Let be a compact connected Lie group, and a Hamiltonian G-space with moment map . Suppose that is a Gelfand-Cetlin datum, and consider a point . The torus acts freely on if and only if acts freely on . In this case, there is a canonical symplectomorphism

Main idea behind proof