m at h . SG ] 2 6 Se p 20 06 GROUP ORBITS AND REGULAR PARTITIONS OF POISSON MANIFOLDS

We study a class of Poisson manifolds for which intersections of certain group orbits give partitions into regular Poisson submanifolds. Examples are the varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras with the Poisson structures defined by Lagrangian splittings. In the special case of g⊕g, where g is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on L defined by arbitrary Lagrangian splittings of g ⊕ g. Such Lagrangian splittings have been classified by Delorme, and they contain the Belavin-Drinfeld splittings as special cases.