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I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a field of charateristic $0$ and $K$ a closed subgroup of $G$ with corresponding lie algebras $\mathfrak{g}, \mathfrak{k}$ and $X$ be the flag variety. One can define then the notions of $K$-equivariant $U(\mathfrak{g})$ modules and $K$-equivariant $D$-modules.

The Beilinson-Bernstein localization states that there is an equivalence between $K$-equivariant finite generated $U(\mathfrak{g})$ modules with trivial central character and $K$-equivariant $D_X$ modules. (One proof can be found in HTT theorem 11.5.3). The proof uses the crucial fact that $X$ is $D-$ affine.

Suppose for a moment that we ignore the fact that $X$ is $D$-affine. Can we still prove that the localization functor sends $K$-equivariant finite generated $U(\mathfrak{g})$ modules with trivial central character to $K$-equivariant $D_X$ modules and similar for the global section functor? I do not require this to be an equivalence.

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