Isotropic foliations of coadjoint orbits from the Iwasawa decomposition

TL;DR

This paper studies how the Iwasawa decomposition of a noncompact real semisimple Lie group induces isotropic foliations on its regular coadjoint orbits, revealing geometric structures that depend on orbit types.

Contribution

It demonstrates that the Iwasawa decomposition induces isotropic foliations on coadjoint orbits, which are affine subspaces, and identifies conditions under which these are Lagrangian fibrations.

Findings

Foliations are isotropic with respect to the Kirillov form.

Leaves are affine subspaces of the dual Lie algebra.

In split real forms, foliations are Lagrangian fibrations with a global transverse section.

Abstract

Let G be a noncompact real semisimple Lie group. The regular coadjoint orbits of G 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 leaves are affine subspaces of the dual of the Lie algebra, and 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.