# 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 $$Y$$?

This is true.

One definition of htdo is analogues to that of equivariant sheaf: a htdo is a tdo $$\mathcal D$$ equipped with an isomorphism $$act^\sharp \mathcal{D} \cong pr_X^\sharp \mathcal{D}$$ satisfying the cocycle condition, where $$G \times X \xrightarrow{act} X$$ and $$G \times X \xrightarrow{pr_X} X$$ are the action map and projection. Then one can prove homogeneity of $$f^\sharp \mathcal{D}$$ in the same way one proves the analogues statement for equivariant sheaves.

In fact, we can know more. Recall that if $$X = G/K$$, then htdo's on X are parameterized by the subset $$I(\mathfrak{k}^\ast) \subset \mathfrak{k}^\ast$$ consisting of elements vanishing on $$[\mathfrak{k},\mathfrak{k}]$$. Let $$\mathcal{D}_{X,\lambda}$$ be the htdo corresponding to $$\lambda \in I(\mathfrak{k}^\ast)$$.

Claim. Let $$L \subset K \subset G$$ be closed subgroups. Suppose $$Y = G/L \xrightarrow{f} G/K = X$$ is $$G$$-equivariant and $$\lambda \in I(\mathfrak{k}^*)$$, then $$f^\sharp \mathcal{D}_{X,\lambda} \cong \mathcal{D}_{Y,\lambda|{\mathfrak{l}}}$$.

Here's an outline of a ''down-to-earth'' proof. One constructs a chart $$(V \subseteq X, \mathcal{D}_X|_V \xrightarrow{\rho} \mathcal{D}_{X,\lambda}|_V)$$ and translate it to obtain an open cover of $$X$$ by charts. $$\mathcal{D}_{X,\lambda}|_V$$ is then determined by the transition functions, each of which is determined by a closed 1-form $$\omega$$ on the intersection. By functoriality of $$f^\sharp$$, the preimages of translates of $$(V,\rho)$$ are charts for $$f^\sharp \mathcal{D}_{X,\lambda}$$ whose transition functions are determined by the $$f^* \omega$$'s.

If the construction of $$(V,\rho)$$ is ''equivariant enough'', we can use the same construction to obtain charts of $$\mathcal{D}_{Y,\lambda|\mathfrak{l}}$$, and the 1-forms determining the transition functions will be equal to the $$f^* \omega$$'s. Therefore $$f^\sharp \mathcal{D}_{X,\lambda}$$ and $$\mathcal{D}_{Y,\lambda|\mathfrak{l}}$$ have the same transition functions, whence isomorphic.

For the construction of such a chart, see [M] proof of Proposition 2.3, Chapter 1.

[M] $${}$$ Localization and Representation Theory of Reductive Lie Groups, Dragan Milicic.