Finite orbit decomposition of real flag manifolds (Q294205)

Let \(G\) be a connected real semisimple Lie group and \(P\) a minimal parabolic subgroup. Let \(H\) be a closed and connected subgroup of \(G\). The aim of the present paper is to give a proof of the following conjecture of \textit{T. Matsuki} [in: Proc. Int. Congress of Mathematicians, Vol II. Tokyo: Math. Soc. 807--813 (1991; Zbl 0745.22010)]:NEWLINENEWLINEIf there exists an open \(H\)-orbit on the real flag variety \(G/P\) then the double coset space \(H\setminus G/P\) is finite.NEWLINENEWLINEThe proof proceeds in two steps. In the first step, the assertion is reduced to the case where the real rank of \(G\) is one. In the second step, there are two cases treated, namely \(H\) is reductive or non-reductive in \(G\). In case \(H\) is non-reductive, it is shown that \(H\) is contained in a conjugate of \(P\) and that are two, three or four \(H\)-orbits on \(G/P\). For reductive \(H\) the following refined statement is proved:NEWLINENEWLINELet \(G\) be of real rank one and \(H\) is a connected reductive subgroup with an open orbit on \(G/P\). Then there is a symmetric subgroup \(H'\supset H\) such that the \(H'\)-orbit decomposition of \(G/P\) equals the \(H\)-orbit decomposition.