On normal abelian subgroups in parabolic groups

Gerhard Röhrle

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 5, page 1455-1482
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Röhrle, Gerhard. "On normal abelian subgroups in parabolic groups." Annales de l'institut Fourier 48.5 (1998): 1455-1482. <http://eudml.org/doc/75326>.

@article{Röhrle1998,
abstract = {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 $G$.},
author = {Röhrle, Gerhard},
journal = {Annales de l'institut Fourier},
keywords = {reductive algebraic groups; parabolic subgroups; numbers of orbits},
language = {eng},
number = {5},
pages = {1455-1482},
publisher = {Association des Annales de l'Institut Fourier},
title = {On normal abelian subgroups in parabolic groups},
url = {http://eudml.org/doc/75326},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Röhrle, Gerhard
TI - On normal abelian subgroups in parabolic groups
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 5
SP - 1455
EP - 1482
AB - 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 $G$.
LA - eng
KW - reductive algebraic groups; parabolic subgroups; numbers of orbits
UR - http://eudml.org/doc/75326
ER -

References

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