### 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 |

### Keywords

- Central series
- Irreducible representations
- Parabolic subgroups

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Anchouche, B. (2005). On the structure of parabolic subgroups.

*Bulletin of the Belgian Mathematical Society - Simon Stevin*,*12*(4), 521-524.