A global slice theorem for proper Hamiltonian actions (Q1291031)

Theorem. Let \(G\) be a connected Lie group and let \(X\) be a proper Hamiltonian \(G\)-space. Let \(K\subseteq G\) be a maximal compact subgroup. If all \(G\)-orbits in \(X\) are of the same dimension, then there exists a \(K\)-slice \(S\subseteq X\), i.e., a \(K\)-invariant symplectic submanifold such that the \(G\)-extension \(Y\) of \(S\) is \(G\)-isomorphic to \(X\) in a neighborhood of their zero moment levels. This is the Hamiltonian analog of a theorem of Abel which reduces the proper action of a non-compact Lie group to a compact transformation group.