I want to let $G$ be a finite group and $X$ be an algebraic variety. We define a holomorphic $G$-bundle on $X$ to be a holomorphic vector bundle $V$ on $X$ where the action of $G$ on $X$ lifts to $V$. My first question is quite simply, how is this lift of the action defined? I'm guessing if $g \cdot x = x'$ with $x,x' \in X$ and $g \in G$, then we need to give a rule which assigns a vector in the fiber over $x$ to a vector in the fiber over $x'$ in a compatible way with the action. But this is extra data we need to provide, right? This seems far from unique.
My main question has to do with fixed points of the action. Let $X^{g}$ be the points in $X$ which are fixed by $g \in G$ (I'm happy to let this be a point, for concreteness). I then want to consider the following decomposition into line bundles:
$$TX\big|_{X^{g}} = \oplus_{\lambda}V_{\lambda},$$
for all $\lambda \in \mathbb{Q} \cap [0,1)$ such that $\lambda$ acts on $V_{\lambda}$ by $e^{2 \pi i \lambda}$. I'm incredibly confused by these $V_{\lambda}$. For concreteness, assume $X^{g}$ is a point. So we're given $X$, we're given an action of $G$ on $X$, and we have some isolated fixed point. Do we get a summand $V_{\lambda}$ for every single rational $\lambda$ between 0 and 1, or does the existence of such, depend on the action? Actually we clearly can only have as many summands $V_{\lambda}$ as the complex dimension of $X$. I'm failing to see how the $\lambda$ relate to the underlying $G$-action on $X$.