### Abstract

Let G be a compact connected semisimple Lie group, G^{ℂ} its complexification and let P be a parabolic subgroup of G^{C}. Let P = L.R_{u}(P) be the Levi decomposition of P, where L is the Levi component of P and R_{u}(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 R_{u}(P). We determine for each 0 ≤ i ≤ r - 1 the irreducible components V_{i}^{(n1, ..., nv)} of the adjoint action of L on u^{(i)}(p)/u^{(i+1)}(p).

Original language | English |
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Pages (from-to) | 521-524 |

Number of pages | 4 |

Journal | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Volume | 12 |

Issue number | 4 |

Publication status | Published - Oct 2005 |

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### Keywords

- Central series
- Irreducible representations
- Parabolic subgroups

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Belgian Mathematical Society - Simon Stevin*,

*12*(4), 521-524.

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

Research output: Contribution to journal › Article

*Bulletin of the Belgian Mathematical Society - Simon Stevin*, vol. 12, no. 4, pp. 521-524.

}

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 -