Chapter 4 - Classification of Invariant Cones

Abstract

A more algebraic proof of lemma can be found in this chapter. The convexity theorem in the group case was proved by Paneitz. The linear convexity theorem can be viewed as an infinitesimal version of the convexity theorem of Neeb. It can be derived from general symplectic convexity theorems applied to suitable co-adjoint orbits. The proof presented here is based on the proof of the convexity theorem of Paneitz by Spindler. The classification for simple groups is due to Ol'shanskii, Paneitz, and Vinberg. Their results were generalized to arbitrary Lie groups by Hilgert, Hofmann, and Lawson. The extension theorem for invariant cones was proved for the classical spaces and for the general case using the classification. The idea of the proof given is due to Neeb. The invariant cones in the group case have been described quite explicitly for the classical groups. Thus, the theorem can also be used to obtain explicit descriptions in the general case.