On the structure of parabolic subgroups

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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

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Keywords

  • Central series
  • Irreducible representations
  • Parabolic subgroups

ASJC Scopus subject areas

  • Mathematics(all)

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