On the structure of parabolic subgroups

Abstract

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 $\mathfrak{u}^{(i)}(\mathfrak{p})/\mathfrak{u}^{(i+1)}(\mathfrak{p})$.