Flag manifolds and the Matsuki correspondence for semisimple Lie groups.

Abstract

The original proof of Matsuki duality relies heavily on algebraic methods. In 1992, Mirkovic, Uzawa and Vilonen gave a geometric proof of Matsuki duality for a flag manifold associated with a Borel subgroup, and in the real case for such a manifold associated with a minimal parabolic subgroup. In 2002, Bremigan and Lorch extended this result to flag manifolds associated with general parabolic subgroups. The goal of this thesis is to analyze the geometric proof of Bremigan and Lorch. To make our examination as self contained as possible, a lot of details have been added to the original proof including general results from the structure theory of semisimple Lie algebras and groups. The orbits of SL(C,2) and SL(R,2) on their respective flag manifolds will be studied as examples of Matsuki correspondence.