I hope someone can help me with this.

Let $X$ be a smooth projective variety, say over the complex numbers and let $Z$ be a closed subvariety of $X$. Assume that $X$ is acted upon by a finite group $G$ and that $Z$ is stable under the action of $G$.

Consider the blow-up $Y$ of $X$ along $Z$. The action of $G$ canonically extends to $Y$. Let $E$ be the exceptional divisor.

**Question**: Is sheaf $\mathcal{O}_E(1)$ G-linearized? If so, how does one prove it?

thanks!