Beilinson-Bernstein localization, equivariant modules

Question

I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a discrete valuation ring $R$ of mixed characteristic $(0,p)$ 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 finitely generated $U(\mathfrak{g})$ modules with trivial central character and $K$-equivariant $D_X$ modules. This true when working over fields of characteristic 0 (One proof can be found in HTT theorem 11.5.3). The proof uses the crucial fact that $X$ is $D_X$- affine.

However, when working over a discrete valuation ring the flag variety is no longer $D_X$-affine. Can we still prove that the localization functor sends $K$-equivariant finitely generated $U(\mathfrak{g})$ modules with trivial central character to $K$-equivariant $D_X$ modules? My problem is that without $D$-affine condition I can not pass to the global sections. I am looking for any hints/references.

Remark: This is very much in the spirit of the question I asked here and got no response. https://math.stackexchange.com/questions/2498085/beilinson-bernstein-localization-equivariant-modules

Answer 1

I think the answer to your question might be partly contained in the paper "On irreducible representations of compact p-adic analytic groups" by Ardakov and Wadsley. You can find a link to a pdf on Simon Wadsley's webpage.

If you look at Proposition 5.15 there (putting $n=0$), you get that the cohomology groups, and so in particular the global sections, of every coherent $D_X$-module are finitely generated over $U(\mathfrak{g})$. By using Belinson-Bernstein over the fraction field of $R$, it should be easy to deduce that even though $X$ may not be $D$-affine, it is "up to bounded torsion" as far as coherent modules are concerned, meaning that the adjunction morphisms have kernels and cokernels which are killed by a fixed power of $\pi$ (where $\pi$ is the uniformizer of your ring $R$).

This paper does not work with $K$-equivariant modules but I don't think it would change much of their arguments.