Abstract
The data needed for a unitary representation of a Lie group G are a Hilbert space H (finite or infinite dimensional) and a (linear) action of G on H. This action should preserve the inner product in H. In the same spirit one can define a symplectic and a hamiltonian representation of G. The data needed are a symplectic manifold (M, ω) and an action of G on M. For a symplectic representation this action should preserve ω; for a hamiltonian representation it should also posses an equivariant moment map (which takes values in the dual Lie algebra of G). In these terms the procedure of geometric quantization can be described as a procedure to transform a hamiltonian representation of G into a unitary representation.
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