On the structure of parabolic subgroups

Research output: Contribution to journalArticle

Abstract

Let G be a compact connected semisimple Lie group, G its complexification and let P be a parabolic subgroup of GC. Let P = L.Ru(P) be the Levi decomposition of P, where L is the Levi component of P and Ru(P) is the unipotent part of P. The group L acts by the adjoint representation on the successive quotients of the central series u(p) = u(0)(p) ⊃ u(1)(p) ⊃ . . . ⊃ u(i)(p) ⊃ . . . ⊃ u(r-1)(p) ⊃ u(r)(p) = 0, where u(p) is the Lie algebra of Ru(P). We determine for each 0 ≤ i ≤ r - 1 the irreducible components Vi(n1, ..., nv) of the adjoint action of L on u(i)(p)/u(i+1)(p).

Original languageEnglish
Pages (from-to)521-524
Number of pages4
JournalBulletin of the Belgian Mathematical Society - Simon Stevin
Volume12
Issue number4
Publication statusPublished - Oct 2005

Fingerprint

Adjoint Representation
Complexification
Parabolic Subgroup
Irreducible Components
Semisimple Lie Group
Lie Algebra
Quotient
Decompose
Series

Keywords

  • Central series
  • Irreducible representations
  • Parabolic subgroups

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the structure of parabolic subgroups. / Anchouche, Boudjemaa.

In: Bulletin of the Belgian Mathematical Society - Simon Stevin, Vol. 12, No. 4, 10.2005, p. 521-524.

Research output: Contribution to journalArticle

@article{5f0aa64ffa5042d19ebad70c4b4e5b10,
title = "On the structure of parabolic subgroups",
abstract = "Let G be a compact connected semisimple Lie group, Gℂ its complexification and let P be a parabolic subgroup of GC. Let P = L.Ru(P) be the Levi decomposition of P, where L is the Levi component of P and Ru(P) is the unipotent part of P. The group L acts by the adjoint representation on the successive quotients of the central series u(p) = u(0)(p) ⊃ u(1)(p) ⊃ . . . ⊃ u(i)(p) ⊃ . . . ⊃ u(r-1)(p) ⊃ u(r)(p) = 0, where u(p) is the Lie algebra of Ru(P). We determine for each 0 ≤ i ≤ r - 1 the irreducible components Vi(n1, ..., nv) of the adjoint action of L on u(i)(p)/u(i+1)(p).",
keywords = "Central series, Irreducible representations, Parabolic subgroups",
author = "Boudjemaa Anchouche",
year = "2005",
month = "10",
language = "English",
volume = "12",
pages = "521--524",
journal = "Bulletin of the Belgian Mathematical Society - Simon Stevin",
issn = "1370-1444",
publisher = "Belgian Mathematical Society",
number = "4",

}

TY - JOUR

T1 - On the structure of parabolic subgroups

AU - Anchouche, Boudjemaa

PY - 2005/10

Y1 - 2005/10

N2 - Let G be a compact connected semisimple Lie group, Gℂ its complexification and let P be a parabolic subgroup of GC. Let P = L.Ru(P) be the Levi decomposition of P, where L is the Levi component of P and Ru(P) is the unipotent part of P. The group L acts by the adjoint representation on the successive quotients of the central series u(p) = u(0)(p) ⊃ u(1)(p) ⊃ . . . ⊃ u(i)(p) ⊃ . . . ⊃ u(r-1)(p) ⊃ u(r)(p) = 0, where u(p) is the Lie algebra of Ru(P). We determine for each 0 ≤ i ≤ r - 1 the irreducible components Vi(n1, ..., nv) of the adjoint action of L on u(i)(p)/u(i+1)(p).

AB - Let G be a compact connected semisimple Lie group, Gℂ its complexification and let P be a parabolic subgroup of GC. Let P = L.Ru(P) be the Levi decomposition of P, where L is the Levi component of P and Ru(P) is the unipotent part of P. The group L acts by the adjoint representation on the successive quotients of the central series u(p) = u(0)(p) ⊃ u(1)(p) ⊃ . . . ⊃ u(i)(p) ⊃ . . . ⊃ u(r-1)(p) ⊃ u(r)(p) = 0, where u(p) is the Lie algebra of Ru(P). We determine for each 0 ≤ i ≤ r - 1 the irreducible components Vi(n1, ..., nv) of the adjoint action of L on u(i)(p)/u(i+1)(p).

KW - Central series

KW - Irreducible representations

KW - Parabolic subgroups

UR - http://www.scopus.com/inward/record.url?scp=33744970921&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33744970921&partnerID=8YFLogxK

M3 - Article

VL - 12

SP - 521

EP - 524

JO - Bulletin of the Belgian Mathematical Society - Simon Stevin

JF - Bulletin of the Belgian Mathematical Society - Simon Stevin

SN - 1370-1444

IS - 4

ER -