Symplectic implosion and the Grothendieck-Springer resolution

TL;DR

This paper demonstrates that the Grothendieck-Springer resolution is a Lagrangian correspondence in shifted symplectic geometry, linking Hamiltonian G-spaces to torus spaces via symplectic implosion, with extensions to quasi-Hamiltonian and elliptic cases.

Contribution

It establishes the Lagrangian property of the Grothendieck-Springer resolution in shifted symplectic structures and connects it to symplectic implosion, extending the framework to quasi-Hamiltonian and elliptic spaces.

Findings

Grothendieck-Springer resolution is Lagrangian in shifted symplectic sense.

Reduction of Hamiltonian G-spaces to H-spaces via this correspondence.

Generalizations to quasi-Hamiltonian and elliptic spaces.

Abstract

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to reduce a Hamiltonian $G$-space to a Hamiltonian $H$-space where $H$ is the maximal torus of $G$. We show that this procedure coincides with an algebraic version of symplectic implosion of Guillemin, Jeffrey and Sjamaar. We explain how to obtain generalizations of this picture to quasi-Hamiltonian spaces and their elliptic version.