Projective representations of real reductive Lie groups and the gradient map

Let $G$ be a connected semisimple noncompact real Lie group and let $ρ: G \longrightarrow \mathrm{SL}(V)$ be a representation on a finite dimensional vector space $V$ over $\mathbb R$, with $ρ(G)$ closed in $\mathrm{SL}(V)$. Identifying $G$ with $ρ(G)$, 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 $\mathbb P(V)$.

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