Tournaments and parabolic almost complex structures on flag manifolds

Abstract

For a complex semisimple Lie group G and a real form G 0 we define a Poisson structure on the variety of Borel subgroups of G with the property that all G 0-orbits in X as well as all Bruhat cells (for a suitable choice of a Borel subgroup of G) are Poisson submanifolds. In particular, we show that every non-empty intersection of a G 0-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.