Let $\mathbb{P}(V) = \mathbb{P}(\mathbb C \oplus \mathbb C)$ be with a $\mathbb C^*$ action : $\lambda (u,v) = (u,\lambda v)$. There are two fixed points of this action, say $0$ and $\infty$. What does this mean to equivariantly lift this action to a line bundle $L$ over $\mathbb P(V)$, by choosing the weights $[l_0, l_{\infty}]$ of the fibre representations $L_0, L_{\infty}$ at the fixed points?
