The symplectic structure of the orbits of the coadjoint representation of Lie algebras of type $E\underset{\rho}\times G$

T. A. Pevtsova

Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 275-286

Résumé

The following theorem is proved. Theorem. Let $G$ be the semidirect sum of a simple Lie algebra $H$ and an Abelian algebra relative to representation $\mu$. Then a complete involutive system of rational functions on $G^*$ is explicitly constructed in the following cases: a) {\it$H=\operatorname{gl}(2n)$ and $\mu=\Lambda^2\rho$;} b) {\it$H=\operatorname{sl}(2n)$ and $\mu=s^2\rho$;} c) {\it$H=\operatorname{sp}(2n)$ and $\mu=\rho+\tau$, where $\rho$ is the minimal representation and $\tau$ is the one-dimensional trivial representation.} Bibliography: 9 titles.