MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Isotropic foliations of coadjoint orbits from the Iwasawa decomposition

William Kirwin


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.

MSC Codes:
51N30, 14L35
coadjoint orbit, Iwasawa decomposition, isotropic foliation, lagrangian fibration, Cartan subalgebra

Related publications

2013 Repository Open Access
William D. Kirwin

Isotropic foliations of coadjoint orbits from the Iwasawa decomposition

In: Geometriae dedicata, 166 (2013) 1, pp. 185-202