Isotropic foliations of coadjoint orbits from the Iwasawa decomposition

2009/11/16 by William D. Kirwin, Kirwin, William D.
Mathematics · #51N30 (also 14L35) Primary #53C12 #53D12 #57S20 Secondary #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.0911.3039

openalex publication_date 2009/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Symplectic manifolds and their lagrangian submanifolds

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