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


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


[1] Ahiezer: Sur les opérations avec un nombre fini d’orbites (en russe). Funct. Anal.19, 1-5 (1985) · Zbl 0576.14045
[2] Bialynicki-Birula, A., Hochschild, G., Mostow, G.D.: Extension of representations of algebraic linear groups. Am. J. Math.85, 131-144 (1963) · Zbl 0116.02302
[3] Borel, A.: Linear algebraic groups, Amsterdam: Benjamin 1969 · Zbl 0206.49801
[4] Brion, M.: Quelques propriétés des espaces homogènes sphériques. Prépublication n{\(\deg\)}24 de l’Institut Fourier, Grenoble (1985)
[5] DeConcini, C., Procesi, C.: Complete symmetric varieties. Proc. Montecatini conf. on invariant theory. Lect. Notes Math.996, 1-44 (1983)
[6] Elashvili, A.G.: Stationary subalgebras of points of general position for the Borel subgroups of simple linear Lie groups (en russe) · Zbl 0252.22015
[7] Guillemin, V., Sternberg, S.: Multiplicity-free spaces. J. Differ. Geom.19, 31-56 (1984) · Zbl 0548.58017
[8] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings. Lect. Notes Math.339 (1974)
[9] Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compos. Math.38, 129-153 (1979) · Zbl 0402.22006
[10] Luna, D.: Slices étales. Bull. Soc. Math. Fr. Mémoire33, 81-105 (1973) · Zbl 0286.14014
[11] Luna, D.: Sur la structure locale des opérations algébriques des groupes réductifs. Prépublication n{\(\deg\)}12 de l’Institut Fourier, Grenoble (1984)
[12] Luna, D., Vust, Th.: Plongements d’espaces homogènes. Comment. Math. Helv.58, 186-245 (1983) · Zbl 0545.14010
[13] Mumford, D., Fogarty, J.: Geometric invariant theory. Second enlarged edition. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0504.14008
[14] Pauer, F.: Plongements normaux de l’espace homogèneSL (3)/SL(2). C.R. du 108e Congrès nat. Soc. sav., Grenoble (1983)
[15] Pauer, F.: Caractérisation valuative d’une classe de sous-groupes d’un groupe algébrique. C.R. du 109e Congrès nat. Soc. Sav., Dijon (1984)
[16] Pauer, F.: Sur les espaces homogènes de complication nulle. Bull. Soc. Math. Fr.112, 377-385 (1984) · Zbl 0576.20029
[17] Raynaud, M.: Anneaux locaux henséliens. Lect. Notes Math.169, 1970 · Zbl 0203.05102
[18] Richardson, R.W.: Deformations of Lie subgroups and the variation of isotropy subgroups. Acta Math.129, 35-73 (1972) · Zbl 0242.22020
[19] Richardson, R.W.: Principal orbit types for algebraic transformation spaces in characteristic zero. Invent. Math.16, 6-14 (1972) · Zbl 0242.14010
[20] Servedio, F.J.: Prehomogeneous vector spaces and varieties. Trans. Am. Math. Soc.176, 421-444 (1973) · Zbl 0266.20043
[21] Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ.14, 1-28 (1974) · Zbl 0277.14008
[22] Vinberg, E.B., Kimel’feld, B.N.: Homogeneous domains on flag manifolds and spherical subgroups. Funct. Anal. Appl.12, 168-174 (1978) · Zbl 0439.53055
[23] Vust, Th.: Opération de groupes réductifs dans un type de cônes presque homogènes. Bull. Soc. Math. Fr.102, 317-334 (1974) · Zbl 0332.22018
[24] Bialynicki-Birula, A.: On action ofSL(2) on complete algebraic varieties. Pac. J. Math.86, 53-58 (1980) · Zbl 0461.14012
[25] Rosenlicht, M.: On quotient varieties and the affine embedding of certain homogeneous spaces. Trans. Am. Math. Soc.101, 211-223 (1961) · Zbl 0111.17902
[26] Rosenlicht, M.: Toroidal algebraic groups. Proc. Am. Math. Soc.12, 984-988 (1961) · Zbl 0107.14703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.