Critical values of moment maps on quantizable manifolds

Abstract

Let $M$ be a quantizable symplectic manifold acted on by $T=(S^1{)}^r$ in a Hamiltonian fashion and $J$ a moment map for this action. Suppose that the set $M^T$ of fixed points is discrete and denote by ${\alpha }_{pj}\in {\mathbb{Z}}^r$ the weights of the isotropy representation at $p$. By means of the ${\alpha }_{pj}$'s we define a partition ${\mathcal{Q}}_{+}$, ${\mathcal{Q}}_{-}$ of $M^T$. (When $r=1$, ${\mathcal{Q}}_{\pm }$ will be the set of fixed points such that the half of the Morse index of $J$ at them is even (odd)). We prove the existence of a map ${\pi }_{\pm }:{\mathcal{Q}}_{\pm }\to {\mathcal{Q}}_{\mp }$ such that $J(q)-J({\pi }_{\pm }(q))\in I_{\mp }$, for all $q\in {\mathcal{Q}}_{\pm }$, where $I_{\pm }$ is the lattice generated by the ${\alpha }_{pj}$'s with $p\in {\mathcal{Q}}_{\pm }.$ We define partition functions $N_p$ similar to the ones of Kostant \cite{Gui} and we prove that ${\sum }_{p\in {\mathcal{Q}}_{+}}{N}_p(l)={\sum }_{p\in {\mathcal{Q}}_{-}}{N}_p(l)$, for any $l\in {\mathbb{Z}}^r$ with $|l|$ sufficiently large.

10 pages, comments are wellcome