Projective representations of real reductive Lie groups and the gradient map

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