Isotropic foliations of coadjoint orbits from the Iwasawa decomposition

Let G be a real semisimple Lie group. The regular coadjoint orbits of G (a certain dense family of top-dimensional orbits) can be partitioned into a finite set of types. We show that on each regular orbit, the Iwasawa decomposition induces a left-invariant foliation which is isotropic with respect to the Kirillov symplectic form. Moreover, the dimension of the leaves depends only on the type of the orbit. When G is a split real form, the foliations induced from the Iwasawa decomposition are actually Lagrangian fibrations with a global transverse Lagrangian section. For these orbits, we use the structure of the Iwasawa decomposition to construct completely integrable systems. In order to partition the orbits into types and construct isotropic nilpotent foliations, we make a somewhat detailed study of the conjugacy classes of Cartan subalgebras, starting with the work of Sugiura, which may be of independent interest. In particular, we give a fairly explicit construction a representative of each conjugacy class.