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Classification of spherical homogeneous spaces. (Classification des espaces homogènes sphériques.) (French) Zbl 0642.14011
Let $$G$$ be a reductive connected algebraic group over an algebraically closed field $$k$$ of characteristic $$0$$. Suppose $$H$$ is an algebraic subgroup of $$G$$. The homogeneous space $$G/H$$ of the pair $$(G,H)$$ is said to be spherical if (among other equivalent conditions) there exists a Borel subgroup $$B$$ of $$G$$ such that $$BH$$ is open in $$G$$. M. Krämer [Compos. Math. 38, 129–153 (1979; Zbl 0402.22006)] described all spherical pairs $$(G,H)$$ where $$G$$ is simple and simply connected and $$H$$ is connected and reductive.
The author determines those pairs $$(G,H)$$ with $$G$$ semisimple and simply connected, $$H$$ connected and reductive. He remarks that I. V. Mikityuk [Math. Sb., Nov. Ser. 129 (171), 514–534 (1986; Zbl 0621.70005)] determined simultaneously these spherical pairs. If $$(G',H')$$, $$(G'',H'')$$ are spherical pairs, then $$(G'\times G'', H'\times H'')$$ is a spherical pair. So one may suppose that the pair is indecomposable. Furthermore, Krämer [loc. cit.] showed that one may suppose that $$G$$ is semisimple. Then the author reduces the study to the case where $$H$$ is reductive, handles the case where $$H$$ is connected and each subgroup of $$G$$ containing $$H$$ is reductive, and then finishes the classification. The examination is often a case-by-case study.
Reviewer: R.Fossum

##### MSC:
 14M17 Homogeneous spaces and generalizations 20G15 Linear algebraic groups over arbitrary fields
##### Keywords:
spherical homogeneous space; spherical pair
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##### References:
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