The orbit decomposition of a flag variety over real and complex numbers (Representation theory and various problems in algebra, analysis, and geometry)

Abstract

Gを実簡約群、HをGの閉部分群、PをGの極小放物型部分群とする。この論文ではG/P上のH軌道は有限個しか存在しないがその複素化H_{C}はG_{C}/P_{mathbb{C}}上に無限個の軌道を持つ例を構成する方法を与える. 具体的にはHとしてPの幕単根基Nをとる. するとブリュワー分解よりG/P上のN軌道の個数は有限であるが、もしPのLevi部分群のリー環が非可換であれば、その複素化N_{mathbb{C}}はG_{C}/P_{C}上に無限個の軌道を持つことを幾何学的に証明する. また別証明として表現論を用いた証明も与える.

Let P be a minimal parabolic subgroup of a real simple Lie group G and G_{mathbb{C}}, P_{mathbb{C}} complexifications of G, P, respectively. In this article, we give a subgroup H of G such that a complexification H_{mathbb{C}} of H has infinite orbits on G_{mathbb{C}}/P_{mathbb{C}} although the number of H-orbits on G/P is finite. More precisely, let P=MAN be its Langlands decomposition. Then #(ANbackslash G/P)<infty by Bruhat decomposition. However If the Lie algebra of M is not abelian, #(A_{mathbb{C}}N_{mathbb{C}}backslash G_{mathbb{C}}/P_{mathbb{C}})=infty holds. We give two proofs of this theorem. One is a representation theoretic proof and another is a geometric proof.