Symplectic slice for subgroup actions

TL;DR

This paper develops a method to relate symplectic slices for a Lie group and its subgroup actions on a symplectic manifold, enhancing understanding of subgroup symmetries in Hamiltonian systems.

Contribution

It introduces a construction for compatible Witt-Artin decompositions of tangent spaces relative to both group and subgroup actions, explicitly relating their symplectic slices.

Findings

Provides explicit relations between symplectic slices for subgroup and group actions.

Enables better analysis of subgroup symmetries in Hamiltonian systems.

Offers a new framework for decomposing tangent spaces in symplectic geometry.

Abstract

Given a symplectic manifold $(M,\omega)$ endowed with a proper Hamiltonian action of a Lie group $G$, we consider the action induced by a Lie subgroup $H$ of $G$. We propose a construction for two compatible Witt-Artin decompositions of the tangent space of $M$, one relative to the $G$-action and one relative to the $H$-action. In particular, we provide an explicit relation between the respective symplectic slices.