On normal abelian subgroups in parabolic groups
Annales de l'Institut Fourier, Volume 48 (1998) no. 5, pp. 1455-1482.

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical P u , and A a closed connected subgroup of P u which is normalized by P. We show that P acts on A with finitely many orbits provided A is abelian. This generalizes a well-known finiteness result, namely the case when A is central in P u . We also obtain an analogous result for the adjoint action of P on invariant linear subspaces of the Lie algebra of P u 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 G.

Soient G un groupe algébrique réductif, P un sous-groupe parabolique de G avec radical unipotent P u , et A un sous-groupe fermé connexe de P u , normalisé par P. Nous montrons que P opère dans A avec un nombre fini d’orbites, lorsque A est abélien. Ceci généralise un résultat de finitude bien connu, concernant le cas où A est central dans P u . Nous obtenons aussi un résultat analogue pour l’action adjointe de P dans les sous-espaces invariants de l’algèbre de Lie de P u , 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 G.

@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] H. Azad, M. Barry, G. Seitz, On the structure of parabolic subgroups, Com. in Algebra, 18 (1990), 551-562. | MR | Zbl

[2] A. Borel, Linear Algebraic Groups, GTM 126, Springer Verlag, 1991. | MR | Zbl

[3] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4,5 et 6, Hermann, Paris, 1975.

[4] M. Brion, Quelques propriétés des espaces homogénes sphériques, Man. Math., 99 (1986), 191-198. | MR | Zbl

[5] M. Brion, Classification des espaces homogénes sphériques, Comp. Math., 63 (1987), 189-208. | Numdam | MR | Zbl

[6] M. Brion, Proceedings of the International Congress of Mathematicians, Zürich, 1994, 753-760. | Zbl

[7] J. Brundan, Dense Orbits and Double Cosets, Proceedings of the NATO/ASI meeting ”Algebraic groups and their representations“, Kluwer, 1998. | MR | Zbl

[8] T. Brüstle, L. Hille, Actions of parabolic subgroups of GL(V) on certain unipotent subgroups and quasi-hereditary algebras, preprint 97-115, SFB 343, Bielefeld, 1997.

[9] H. Bürgstein, W.H. Hesselink, Algorithmic orbit classification for some Borel group actions, Comp. Math., 61 (1987), 3-41. | Numdam | MR | Zbl

[10] S. Donkin, Rational representations of algebraic groups: Tensor products and filtrations, Springer Lecture Notes in Math., 1140 (1985). | MR | Zbl

[11] E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. Ser. 2, 6 (1957), 111-244. | Zbl

[12] L. Hille, G. Röhrle, 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] L. Hille, G. Röhrle, 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] U. Jürgens, G. Röhrle, Algorithmic Modality Analysis for Parabolic Groups, to appear in Geom. Dedicata. | Zbl

[15] V. Kac, Some remarks on nilpotent orbits, J. Algebra, 64 (1980), 190-213. | MR | Zbl

[16] V.V. Kashin, 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] F. Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv., 70 (1995), 285-309. | MR | Zbl

[18] B. Kostant, Eigenvalues of the Laplacian and commutative Lie subalgebras, Topology, 3, suppl. 2 (1965), 147-159. | MR | Zbl

[19] B. Kostant, 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] M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math., 38 (1979), 129-153. | Numdam | Zbl

[21] A. Mal'Cev, Commutative subalgebras of semi-simple Lie algebras, Translations of the Amer. Math. Soc. Series 1, 9 (1951), 214-227.

[22] O. Mathieu, Filtrations of G-modules, Ann. Sci. École Norm. Sup., 23 (1990), 625-644. | Numdam | MR | Zbl

[23] I. Mikityuk, On the integrability of invariant hamiltonian systems with homogeneous configurations spaces, Math. USSR-Sb., 57 (1987), 527-546. | Zbl

[24] I. Muller, H. Rubenthaler, G. Schiffmann, Structure des espaces préhomogènes associés à certaines algèbres de Lie graduées, Math. Ann., 274 (1986), 95-123. | MR | Zbl

[25] V.L. Popov, A finiteness theorem for parabolic subgroups of fixed modality, Indag. Math. N. S., 8 (1) (1997), 125-132. | MR | Zbl

[26] V. Popov, G. Röhrle, 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] V.S. Pyasetskii, Linear Lie groups acting with finitely many orbits, Funct. Anal. Appl., 9 (1975), 351-353. | Zbl

[28] R.W. Richardson, Finiteness Theorems for Orbits of Algebraic Groups, Indag. Math., 88 (1985), 337-344. | MR | Zbl

[29] R.W. Richardson, G. Röhrle, R. Steinberg, Parabolic subgroups with Abelian unipotent radical, Inv. Math., 110 (1992), 649-671. | MR | Zbl

[30] G. Röhrle, Parabolic subgroups of positive modality, Geom. Dedicata, 60 (1996), 163-186. | MR | Zbl

[31] G. Röhrle, A note on the modality of parabolic subgroups, Indag. Math. N.S., 8 (4) (1997), 549-559. | MR | Zbl

[32] G. Röhrle, On the modality of parabolic subgroups of linear algebraic groups, to appear in Manuscripta Math. | Zbl

[33] M. Rosenlicht, On quotient varieties and the affine embeddings of certain homogeneous spaces, Trans. Amer. Math. Soc., 101 (1961), 211-223. | MR | Zbl

[34] I. Schur, Zur Theorie der vertauschbaren Matrizen, J. reine und angew. Math., 130 (1905), 66-76. | JFM

[35] T.A. Springer, 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] T.A. Springer, Some results on algebraic groups with involutions, Advanced Studies in Pure Math., 6 (1985), 525-543. | MR | Zbl

[37] T.A. Springer, R. Steinberg, Conjugacy classes in Seminar on algebraic groups and related finite groups, Lect. Notes Math., 131, Springer Verlag, Heidelberg (1970). | Zbl

[38] R. Steinberg, Lectures on Chevalley Groups, Yale University, 1968.

[39] R. Steinberg, Conjugacy Classes in Algebraic Groups, Springer Lecture Notes in Math., 366 (1974). | MR | Zbl

[40] N. Vavilov, Weight elements of Chevalley groups, preprint. | Zbl

[41] E.B. Vinberg, The Weyl group of a graded Lie algebra, Math. USSR-Izv., 10 (1976), 463-495. | MR | Zbl

[42] E.B. Vinberg, Complexity of actions of reductive groups, Funct. Anal. Appl., 20 (1986), 1-11. | MR | Zbl

Cited by Sources: