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Multiplicity-free spaces. (English) Zbl 0548.58017
This paper continues the study of structural correspondences between classical and quantum mechanics in the framework of symplectic geometry and symmetry groups. Corresponding to the notion of multiplicity free unitary representations a symplectic manifold X with Hamiltonian action of a Lie group G on X is called multiplicity free if the ring of G- invariant functions on X is commutative with respect to the Poisson brackets. First it is shown that for compact connected group G multiplicity free spaces are in a neighbourhood of a coadjoint orbit locally equivalent with cotangent bundles. For this result the construction of symplectic induction, which corresponds to induced representations in the quantum category, is used.
In the second part of the paper the following theorem is proved: Let X be a manifold on which G acts and $$H=L^ 2(X)$$ the space of square integrable half densities on X. Then the representation of G on H is multiplicity free if and only if the action of G on $$T^*X$$ is multiplicity free.
Reviewer: C.Günther

##### MSC:
 53D50 Geometric quantization 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53C30 Differential geometry of homogeneous manifolds
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