Put $X := \mathbb A^{n+1}\!-\lbrace0\rbrace$. Let $G=\mathbb C^*$ act on $X$ with (positive) weights $w_0,\dots,w_n$. The quotient stack $[X/G]$ is called the weighted projective stack.

Each vector bundle on $[X/G]$ corresponds to a $G$-equivariant vector bundle $E$ on $X$, and vice versa. So I want to understand the latter (because it seems to be easier than the former).

**Question:** is every complex $G$-equivariant vector bundle $E$ on $X$ of the form $X \times V$ for any representation $V$ of $G$?

Notes:

This question comes from this paper by Prof. Edidin. In §4.2 he says that the statement of my question holds when $n=1$. Because I can not understand whether the dimension $n$ is important or not, I wrote my question with a general settings.

If we forget the $G$-action, then $E$ is trivial. (Read the comments of my question at SE for details.)

If a holomorphic structure of the bundle $E$ is important, then please assume it.