Geometric quantization and multiplicities of group representations

The Heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum-mechanical particle. This can be rephrased as follows: the smallest subsets of classical phase space in which the presence of a quantum-mechanical particle can be detected are its Lagrangian submanifolds. For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true "points"; see Weinstein [17]. Now let G be a compact Lie group and G x X ~ X a Hamiltonian action of G on X (see w for definitions). It is well-known that the fixed points of this action form a symplectic submanifold of X. (See for instance Guillemin and Sternberg [5].) However, what can one say about the fixed "points" of G? We will show that they are also the "points'" of a symplectic manifold, Xc. This manifold is the Marsden-Weinstein reduction of X with respect to the zero orbit in g*, and will be described in Sect. 2. (It was introduced in a completely different context fl'om ours by Marsden and Weinstein [12].) Problems in classical mechanics can often be reduced to the study of Hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on Hilbert space. This fact has inspired a number of efforts to "quantize" symplectic geometry by devising schemes for associating Hilbert space to symplectic manifolds. The "no-go" theorems of Groenwald and Van Hove impose some embarrassing limitations on all such schemes; however, it seems to be a useful idea heuristically to think of every symplectic manifold, X~a~ic, 1, as being symbiotically associated with a Hilbert space, Xq,a, t ..... in such a way that the classical observables on the first space correspond to quantum observables on the second space. The heuristics further suggests that if G is a group of symmetries of X~la~,i~, j, it should also be a group of symmetries of Xq,a,,tum. In this heuristic spirit, we will state the main conjecture of this paper: