Critical values of moment maps on quantizable manifolds

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 ${\alpha}{pj}\in{\mathbb Z}^r$theweightsoftheisotropyrepresentationat$p$\mathrm{.}Bymeansofthe$\alpha{pj}${}^{\mathrm{\prime }}swedefineapartition${\mathcal Q}+$,${\mathcal Q}-$of$M^T$\mathrm{.}(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 $p\in {\mathcal{Q}}_{\pm }\mathrm{.}$ We define partition functions $N_p$ similar to the ones of Kostant \cite{Gui} and we prove that $\sum{p\in{\mathcal Q}+}N_p(l)=\sum{p\in{\mathcal Q}_-}N_p(l)$,forany$l\in{\mathbb Z}^r$with$|l|$ sufficiently large.