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 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.