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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 \(G\).

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