English

Invariant convex sets in polar representations

Complex Variables 2014-11-25 v1 Differential Geometry

Abstract

We study a compact invariant convex set EE in a polar representation of a compact Lie group. Polar rapresentations are given by the adjoint action of KK on p\mathfrak{p}, where KK is a maximal compact subgroup of a real semisimple Lie group GG with Lie algebra g=kp\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}. If ap\mathfrak{a} \subset \mathfrak{p} is a maximal abelian subalgebra, then P=EaP=E\cap \mathfrak{a} is a convex set in a\mathfrak{a}. We prove that up to conjugacy the face structure of EE is completely determined by that of PP and that a face of EE is exposed if and only if the corresponding face of PP is exposed. We apply these results to the convex hull of the image of a restricted momentum map.

Keywords

Cite

@article{arxiv.1411.6041,
  title  = {Invariant convex sets in polar representations},
  author = {Leonardo Biliotti and Alessandro Ghigi and Peter Heinzner},
  journal= {arXiv preprint arXiv:1411.6041},
  year   = {2014}
}
R2 v1 2026-06-22T07:08:03.500Z