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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 $Y$?

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