Homogeneous quantization and multiplicities of group representations

Let B be a compact manifold. A cone over B is a principal R⁺-bundle, X, with base B. Let (a, x) → ?_a(x) be the mapping associated with the action of a? R⁺ on X. X is called a symplectic cone if it possesses a symplectic form, ω, such that $\text{?}{}_{\mathrm{a}}{}^1\text{ω = aω}$. A compact Lie group, G, is said to act in a homogeneous fashion on X if it acts on X in such a way that both ω and the principal bundle structure are preserved. It is known that to such an action one can associate in a fairly canonical way a representation of G on a Hilbert space H. (See 3].) In this paper we propose a symplectic recipe for the multiplicities with which H decomposes into G-irreducibles and show that this recipe is correct “generically”.