On the structure of parabolic subgroups

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).