# zbMATH — the first resource for mathematics

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)

##### MSC:
 22E46 Semisimple Lie groups and their representations 14M15 Grassmannians, Schubert varieties, flag manifolds 53C30 Differential geometry of homogeneous manifolds