Classification of multiplicity free symplectic representations
Abstract
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.
Cite
@article{arxiv.math/0505268,
title = {Classification of multiplicity free symplectic representations},
author = {Friedrich Knop},
journal= {arXiv preprint arXiv:math/0505268},
year = {2024}
}
Comments
25 pages; v2: minor changes; v3: typos fixed, final version; v3: As reported by Wen Chan, the table entries (1.8) and (11.2) contained mistakes and were corrected. The classification itself is not affected