The following is from page 3 of the article Matsuki correspondence for sheaves:
Let $G$ be a semi-simple Lie group with a maximal compact subgroup $K$. Let $\mathfrak{g}= \mathfrak{k} \oplus \mathfrak{s} $ be the corresponding Cartan decomposition. Let $V$ be a finite dimensional representation of $G$ over $\mathbb{R}$. We denote by $P(V)$ the projective space associated to $V$. There exists a $K$-invariant inner product $(,)$ on $V$ such that any $\alpha \in \mathfrak{s} $ acts as a symmetric operator. A choice of such inner product gives a moment map $m_V: P(V) \rightarrow \mathfrak{s}^*$ by the formula $$<m_V(v),\alpha>= \frac{(\alpha v,v)}{(v,v)}, \alpha \in \mathfrak{s}.$$
My question is why is this moment map defined from $P(V)$ to $\mathfrak{s}^*$ not to $\mathfrak{g}^*$, because I know that in general if $G$ is a Lie group acting on a manifold $M$, where the action is Hamiltonian, then by definition the moment map has values in $\mathfrak{g}^*$?