A general nonlinear convexity theorem. (English) Zbl 0888.22005

Neeb, Karl-Hermann

Forum Math. 9, No. 5, 613-640 (1997).

The author gives a direct geometric proof of a generalization of the inclusion part of the convexity theorem to more general symmetric pairs which are not necessarily reductive: Let \({\mathfrak g}\) be an admissible Lie algebra, i.e, \({\mathfrak g} \oplus \mathbb{R}\) contains pointed generating invariant cones. Let \(G_\mathbb{C}\) be the simply connected complex Lie group with Lie algebra \({\mathfrak g}_\mathbb{C}\), \(G\) the real form corresponding to \({\mathfrak g}\), and \(L:GAN \to {\mathfrak a}\) the projection corresponding to a \({\mathfrak k}\)-adapted positive system of roots. Further let \(a\in \exp (\Delta^+_p)^*\). Then \(L(aG) \subseteq \text{con} ({\mathfrak W}_{ \mathfrak k} \cdot \log a)+ iC_{\min}\). The idea of the proof is provided by a construction coming from symplectic geometry. As an application a new proof of the inclusion part of the convexity theorem of K.-H. Neeb [Pac. J. Math. 162, 305-349 (1994; Zbl 0809.53058)] is given. Furthermore the author shows how the direct geometric proof of the convexity theorem is related to the constructions based on representations of certain semigroups and he applies the convexity theorem to semigroup actions on vector bundles, establishes a formula to compute the operator norm for certain \(K_\mathbb{C}\)-modules and investigates in which cases the canonical Kähler metrics on certain coadjoint orbits of reductive Lie groups permit large semigroup actions.

Reviewer: J.Ludwig (Metz)

MSC:

22E15	General properties and structure of real Lie groups
43A65	Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
53C55	Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

symmetric pairs; Lie algebra; complex Lie group; convexity theorem; representations; coadjoint orbits; reductive Lie groups

Citations:

Zbl 0809.53058