On normal abelian subgroups in parabolic groups

Röhrle, Gerhard

Annales de l'Institut Fourier, Tome 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 $G$.

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR | Numdam

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR | Numdam

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

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

[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). | Zbl | MR

[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. | Zbl | MR

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