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.