G-actions on graphs

Abstract

Let G be an n-dimensional torus and $τ$ a Hamiltonian action of G on a compact symplectic manifold, M. If M is pre-quantizable one can associate with $τ$ a representation of G on a virtual vector space, Q(M), by $\spin^{\CC}$-quantization. If M is a symplectic GKM manifold we will show that several well-known theorems about this ``quantum action'' of G: for example, the convexity theorem, the Kostant multiplicity theorem and the ``quantization commutes with reduction'' theorem for circle subgroups of G, are basically just theorems about G-actions on graphs.