Critical values of moment maps on quantizable manifolds

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Let M be a quantizable symplectic manifold acted on by T=(S¹)ʳ in a Hamiltonian fashion and J a moment map for this action. Suppose that the set Mᵗ of fixed points is discrete and denote by αpj∈Zʳ the weights of the isotropy representation at p. By means of the αpj's we define a partition Q+, Q- of Mᵗ. (When r=1, Q± 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 π±:Q±→Q∓ such that J(q)-J(π±(q))∈ I∓, for all q∈ Q±, where I± is the lattice generated by the αpj's with p∈Q±. We define partition functions Nₚ similar to the ones of Kostant Gui and we prove that ∑p∈Q+Nₚ(l)=∑p∈Q-Nₚ(l), for any l∈Zʳ with |l| sufficiently large.