Polynomial algebras on coadjoint orbits of semisimple Lie groups (Q1602660)

Let \(\mathcal O\) be a coadjoint orbit of a real (finite-dimensional) semisimple Lie algebra \(g\) and \(P({\mathcal O})\) be the Poisson algebra of polynomial functions on \(\mathcal O\). The first main result of this paper is that \(P({\mathcal O})= {\mathbb R}\oplus \{P({\mathcal O}), P({\mathcal O})\}\) where \(\{.,.\}\) denotes the Poisson bracket on \(P({\mathcal O})\). The second main result asserts that the Poisson algebra \(P({\mathcal O})\) is essentially simple if and only if the orbit \(\mathcal O\) is semisimple.