Let $G$ be a compact connected semisimple Lie group, $G^{\mathbb{C}}$ 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
\begin{equation*}
\mathfrak{u}(\mathfrak{p})\,=\,\mathfrak{u}^{(0)}(\mathfrak{p})\,\supset \,
\mathfrak{u}^{(1)}(\mathfrak{p})\,\,\supset \,\cdots \,\supset \,\mathfrak{u}
^{(i)}(\mathfrak{p})\,\supset \,\cdots \,\supset \,\mathfrak{u}^{(r-1)}(
\mathfrak{p})\supset \mathfrak{u}^{(r)}(\mathfrak{p})\,=\,0\,,
\end{equation*}%
where $\mathfrak{u}(\mathfrak{p})$ is the Lie algebra of $R_{u}(P)$. We
determine for each $0\leq i\leq r-1$ the irreducible components $
V_{i}^{(n_{1},\text{ }...,\text{ }n_{\nu })}$ of the adjoint action of $L$
on $\mathfrak{u}^{(i)}(\mathfrak{p})/\mathfrak{u}^{(i+1)}(\mathfrak{p})$.