  Let $G$ be a connected semisimple noncompact real Lie group and let $\rho: G
\longrightarrow \mathrm{SL}(V)$ be a representation on a finite dimensional
vector space $V$ over $\mathbb R$, with $\rho(G)$ closed in $\mathrm{SL}(V)$.
Identifying $G$ with $\rho(G)$, we assume there exists a $K$-invariant scalar
product $\mathtt g$ such that $G=K\exp(\mathfrak p)$, where
$K=\mathrm{SO}(V,\mathtt g)\cap G$, $\mathfrak p=\mathrm{Sym}_o (V,\mathtt
g)\cap \mathfrak g$ and $\mathfrak g$ denotes the Lie algebra of $G$. Here
$\mathrm{Sym}_o (V,\mathtt g)$ denotes the set of symmetric endomorphisms with
trace zero. Using the $G$-gradient map techniques we analyze the natural
projective representation of $G$ on $\mathbb P(V)$.
