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Espaces homogènes sphériques. (Spherical homogeneous spaces). (French) Zbl 0604.14047
This is an important paper on the local structure of embeddings of spherical homogeneous spaces.
Let G be a connected reductive group over an algebraically closed field of characteristic zero, and let H be a closed subgroup of G. The homogeneous space G/H is ”spherical”, iff the action of a Borel subgroup of G has a dense orbit in G/H. For example, symmetric spaces are spherical homogeneous spaces. - Let G/H be spherical and let B be a Borel subgroup such that B.H is dense in G. Let P be the set of elements \(s\in G\) such that \(s.B.H=B.H\). Then P is a parabolic subgroup. Denote by \(P^ u\) its unipotent radical.
Theorem: There is a Levi subgroup L of P such that \((1)\quad L\cap H=P\cap H,\) and this group is reductive. \((2)\quad H\quad contains\) the commutator subgroup (L,L). \((3)\quad Let\) Z be an algebraic G-variety, \(z\in^ HZ\) and let C be the connected center of L. Then \(P^ u.\overline{C.z}\) contains a dense open subset of each G-orbit in \(\overline{G.z}\).
Reviewer: F.Pauer

MSC:
14M17 Homogeneous spaces and generalizations
20G15 Linear algebraic groups over arbitrary fields
14L30 Group actions on varieties or schemes (quotients)
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