Classification of multiplicity free symplectic representations

Details Classification of multiplicity free symplectic representations Classification of multiplicity free symplectic representations

2005-05-12

arxiv.org/abs/math/0505268v3

J. Algebra 301 (2006) 531-553

Mathematics

Symplectic Geometry

25 pages; v2: minor changes; v3: typos fixed, final version

Scientific paper

10.1016/j.jalgebra.2005.07.035

Let G be a connected reductive group acting on a finite dimensional vector space V. Assume that V is equipped with a G-invariant symplectic form. Then the ring C[V] of polynomial functions becomes a Poisson algebra. The ring C[V]^G of invariants is a sub-Poisson algebra. We call V multiplicity free if C[V]^G is Poisson commutative, i.e., if {f,g}=0 for all invariants f and g. Alternatively, G also acts on the Weyl algebra W(V) and V is multiplicity free if and only if the subalgebra W(V)^G of invariants is commutative. In this paper we classify all multiplicity free symplectic representations.

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