Singular cotangent bundle reduction & Spin Calogero-Moser systems

Simon Hochgerner
(University of Vienna)

Abstract: Suppose a Riemannian configuration manifold $Q$ is acted upon by a compact Lie group $K$ such that there is only a single isotropy type. The cotangent lifted action by $K$ on $T^*Q$ is Hamiltonian with momentum map $J$. We study the symplectically reduced space $J^{-1}(\orbit)/K$ where $\orbit$ is a coadjoint orbit. This is a singular space which is stratified into smooth symplectic pieces (in a technical sense). We are able to give a description of the reduced space as a fibered product of $T^*(Q/K)$ and a symplectic reduction of $\orbit$. Moreover, we can give an explicit formula of the reduced symplectic form in terms intrinsic to this fibered product description. The talk will include some informal background on Hamiltonian mechanics and singular symplectic reduction, and, if time permits, I will discuss some examples.