Matsuki, Toshihiko Orbits on flag manifolds. (English) Zbl 0745.22010 Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 807-813 (1991). [For the entire collection see Zbl 0741.00020.]The paper summarizes recent results by Matsuki, Oshima, Uzawa, Brion, Vinberg on the space \(H\backslash G/P\) of double cosets of a connected real semisimple Lie group \(G\). Here \(P\) is a minimal parabolic subgroup of \(G\), \(G/P\) is the corresponding flag manifold, and \(H\) is (almost) the fixed point subgroup of an involutive automorphism of \(G\). Among the topics discussed here (often without proof) are:— the “symbol” of a double coset, when \(G\) is a complex classical group (\(G=GL(n,\mathbb{C})\), \(SO(n,\mathbb{C})\) or \(Sp(n,\mathbb{C})\)).— Uzawa’s function and vector field on \(G/P\),— spherical subgroups of a complex semisimple Lie group \(G\). Reviewer: F.Rouvière (Nice) Cited in 2 ReviewsCited in 14 Documents MSC: 22E46 Semisimple Lie groups and their representations 14M15 Grassmannians, Schubert varieties, flag manifolds 53C30 Differential geometry of homogeneous manifolds Keywords:double cosets; connected real semisimple Lie group; parabolic subgroup; flag manifold; complex semisimple Lie group Citations:Zbl 0741.00020 PDFBibTeX XMLCite \textit{T. Matsuki}, in: Proceedings of the international congress of mathematicians (ICM), August 21--29, 1990, Kyoto, Japan. Volume II. Tokyo etc.: Springer-Verlag. 807--813 (1991; Zbl 0745.22010)