Pullback of homogeneous twisted differential operators

Question

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 $Y$?

Answer 1

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.

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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.