Let be a reductive algebraic group, a parabolic subgroup of with unipotent radical , and a closed connected subgroup of which is normalized by . We show that acts on with finitely many orbits provided is abelian. This generalizes a well-known finiteness result, namely the case when is central in . We also obtain an analogous result for the adjoint action of on invariant linear subspaces of the Lie algebra of which are abelian Lie algebras. Finally, we discuss a connection to some work of Mal’cev on maximal abelian subalgebras of the Lie algebra of .
Soient un groupe algébrique réductif, un sous-groupe parabolique de avec radical unipotent , et un sous-groupe fermé connexe de , normalisé par . Nous montrons que opère dans avec un nombre fini d’orbites, lorsque est abélien. Ceci généralise un résultat de finitude bien connu, concernant le cas où est central dans . Nous obtenons aussi un résultat analogue pour l’action adjointe de dans les sous-espaces invariants de l’algèbre de Lie de , qui sont des algèbres de Lie abéliennes. Finalement, nous faisons le lien avec un travail de Mal’cev sur les sous-algèbres abéliennes maximales de l’algèbre de Lie de .
@article{AIF_1998__48_5_1455_0, author = {R\"ohrle, Gerhard}, title = {On normal abelian subgroups in parabolic groups}, journal = {Annales de l'Institut Fourier}, pages = {1455--1482}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {48}, number = {5}, year = {1998}, doi = {10.5802/aif.1662}, mrnumber = {99i:20062}, zbl = {0933.20034}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1662/} }
TY - JOUR AU - Röhrle, Gerhard TI - On normal abelian subgroups in parabolic groups JO - Annales de l'Institut Fourier PY - 1998 SP - 1455 EP - 1482 VL - 48 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1662/ DO - 10.5802/aif.1662 LA - en ID - AIF_1998__48_5_1455_0 ER -
%0 Journal Article %A Röhrle, Gerhard %T On normal abelian subgroups in parabolic groups %J Annales de l'Institut Fourier %D 1998 %P 1455-1482 %V 48 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1662/ %R 10.5802/aif.1662 %G en %F AIF_1998__48_5_1455_0
Röhrle, Gerhard. On normal abelian subgroups in parabolic groups. Annales de l'Institut Fourier, Volume 48 (1998) no. 5, pp. 1455-1482. doi : 10.5802/aif.1662. http://www.numdam.org/articles/10.5802/aif.1662/
[1] On the structure of parabolic subgroups, Com. in Algebra, 18 (1990), 551-562. | MR | Zbl
, , ,[2] Linear Algebraic Groups, GTM 126, Springer Verlag, 1991. | MR | Zbl
,[3] Groupes et algèbres de Lie, Chapitres 4,5 et 6, Hermann, Paris, 1975.
,[4] Quelques propriétés des espaces homogénes sphériques, Man. Math., 99 (1986), 191-198. | MR | Zbl
,[5] Classification des espaces homogénes sphériques, Comp. Math., 63 (1987), 189-208. | Numdam | MR | Zbl
,[6] Proceedings of the International Congress of Mathematicians, Zürich, 1994, 753-760. | Zbl
,[7] Dense Orbits and Double Cosets, Proceedings of the NATO/ASI meeting ”Algebraic groups and their representations“, Kluwer, 1998. | MR | Zbl
,[8] Actions of parabolic subgroups of GL(V) on certain unipotent subgroups and quasi-hereditary algebras, preprint 97-115, SFB 343, Bielefeld, 1997.
, ,[9] Algorithmic orbit classification for some Borel group actions, Comp. Math., 61 (1987), 3-41. | Numdam | MR | Zbl
, ,[10] Rational representations of algebraic groups: Tensor products and filtrations, Springer Lecture Notes in Math., 1140 (1985). | MR | Zbl
,[11] Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. Ser. 2, 6 (1957), 111-244. | Zbl
,[12] On parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, C. R. Acad. Sci. Paris, Série I, 325 (1997), 465-470. | Zbl
, ,[13] A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, to appear in Transformation Groups. | Zbl
, ,[14] Algorithmic Modality Analysis for Parabolic Groups, to appear in Geom. Dedicata. | Zbl
, ,[15] Some remarks on nilpotent orbits, J. Algebra, 64 (1980), 190-213. | MR | Zbl
,[16] Orbits of adjoint and coadjoint actions of Borel subgroups of semisimple algebraic groups, Problems in Group Theory and Homological algebra, Yaroslavl', (Russian), 1997, 141-159. | MR | Zbl
,[17] On the set of orbits for a Borel subgroup, Comment. Math. Helv., 70 (1995), 285-309. | MR | Zbl
,[18] Eigenvalues of the Laplacian and commutative Lie subalgebras, Topology, 3, suppl. 2 (1965), 147-159. | MR | Zbl
,[19] The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations, Internat. Math. Res. Notices, 5 (1998), 225-252. | MR | Zbl
,[20] Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math., 38 (1979), 129-153. | Numdam | Zbl
,[21] Commutative subalgebras of semi-simple Lie algebras, Translations of the Amer. Math. Soc. Series 1, 9 (1951), 214-227.
,[22] Filtrations of G-modules, Ann. Sci. École Norm. Sup., 23 (1990), 625-644. | Numdam | MR | Zbl
,[23] On the integrability of invariant hamiltonian systems with homogeneous configurations spaces, Math. USSR-Sb., 57 (1987), 527-546. | Zbl
,[24] Structure des espaces préhomogènes associés à certaines algèbres de Lie graduées, Math. Ann., 274 (1986), 95-123. | MR | Zbl
, , ,[25] A finiteness theorem for parabolic subgroups of fixed modality, Indag. Math. N. S., 8 (1) (1997), 125-132. | MR | Zbl
,[26] On the number of orbits of a parabolic subgroup on its unipotent radical, Algebraic Groups and Lie Groups, Australian Mathematical Society Lecture Series, 9, ed. G.I. Lehrer, Cambridge University Press, 1997, 297-320. | MR | Zbl
, ,[27] Linear Lie groups acting with finitely many orbits, Funct. Anal. Appl., 9 (1975), 351-353. | Zbl
,[28] Finiteness Theorems for Orbits of Algebraic Groups, Indag. Math., 88 (1985), 337-344. | MR | Zbl
,[29] Parabolic subgroups with Abelian unipotent radical, Inv. Math., 110 (1992), 649-671. | MR | Zbl
, , ,[30] Parabolic subgroups of positive modality, Geom. Dedicata, 60 (1996), 163-186. | MR | Zbl
,[31] A note on the modality of parabolic subgroups, Indag. Math. N.S., 8 (4) (1997), 549-559. | MR | Zbl
,[32] On the modality of parabolic subgroups of linear algebraic groups, to appear in Manuscripta Math. | Zbl
,[33] On quotient varieties and the affine embeddings of certain homogeneous spaces, Trans. Amer. Math. Soc., 101 (1961), 211-223. | MR | Zbl
,[34] Zur Theorie der vertauschbaren Matrizen, J. reine und angew. Math., 130 (1905), 66-76. | JFM
,[35] The unipotent variety of a semisimple group, Proc. of the Bombay Colloq. in Algebraic Geometry (ed. S. Abhyankar), London, Oxford Univ. Press (1969), 373-391. | MR | Zbl
,[36] Some results on algebraic groups with involutions, Advanced Studies in Pure Math., 6 (1985), 525-543. | MR | Zbl
,[37] Conjugacy classes in Seminar on algebraic groups and related finite groups, Lect. Notes Math., 131, Springer Verlag, Heidelberg (1970). | Zbl
, ,[38] Lectures on Chevalley Groups, Yale University, 1968.
,[39] Conjugacy Classes in Algebraic Groups, Springer Lecture Notes in Math., 366 (1974). | MR | Zbl
,[40] Weight elements of Chevalley groups, preprint. | Zbl
,[41] The Weyl group of a graded Lie algebra, Math. USSR-Izv., 10 (1976), 463-495. | MR | Zbl
,[42] Complexity of actions of reductive groups, Funct. Anal. Appl., 20 (1986), 1-11. | MR | Zbl
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