Classification of multiplicity free symplectic representations

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.

Appeared in: Journal of Algebra 301 (2006) 531-553

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Last updated: May 26, 2006