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The classification of transversal multiplicity-free group actions. (English) Zbl 0877.58022

This paper concerns the classification of torsion-free, transversal, multiplicity-free actions of a Lie group \(G\) on compact, connected symplectic manifolds (Hamiltonian actions). The term multiplicity-free here means that the invariant functions form an Abelian Poisson algebra and comes from the fact that geometric quantization of such actions leads to representations in which irreducibles appear no more than once. It is shown that the set of such actions maps bijectively into the subset of convex, reflective, Delzant polytopes (in a closed positive Weyl chamber). The Delzant polytopes were introduced in T. Delzant [Bull. Soc. Math. Fr. 116, No. 3, 315-339 (1988; Zbl 0676.58029)]in the case of torus actions, and the current paper generalises the result to non-Abelian groups. The transversality criterion refers to the transversality of the moment map and a Cartan subalgebra and the torsion-free condition to the triviality of certain isotropy groups.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57S25 Groups acting on specific manifolds
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics

Citations:

Zbl 0676.58029
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References:

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