# Pullback of homogeneous twisted differential operators

Let $$X,Y$$ be smooth complex varieties and let $$G$$ be an smooth affine algebraic group acting on $$X$$ and $$Y$$ such that $$X,Y$$ are $$G$$-homogeneous spaces (the $$G$$ action is transitive). We also let $$f:Y \to X$$ a smooth $$G$$-equivariant morphism.

Finally, consider $$\mathcal{D}$$ a sheaf of $$G$$-homogeneous twisted differential operators(htdo) on $$X$$. Since, $$\mathcal{D}$$ is in particular a tdo, one may define $$f^{\#} \mathcal{D}:= \text{Dif}_{f^{-1} \mathcal{D}}(f^* \mathcal{D},f^*\mathcal{D})$$

a tdo on $$Y$$. Is it true that $$f^{\#} \mathcal{D}$$ is a $$G$$-htdo on $$Y$$?