Symplectic implosion

Abstract

Let $K$ be a compact Lie group. We introduce the process of symplectic implosion, which associates to every Hamiltonian $K$-manifold a stratified space called the imploded cross-section. It bears a resemblance to symplectic reduction, but instead of quotienting by the entire group, it cuts the symmetries down to a maximal torus of $K$. We examine the nature of the singularities and describe in detail the imploded cross-section of the cotangent bundle of $K$, which turns out to be identical to an affine variety studied by Gelfand, Vinberg, Popov, and others. Finally we show that ``quantization commutes with implosion''.