A Beilinson-Bernstein theorem for twisted arithmetic differential operators on the formal flag variety

Let us suppose that ${\mathbb{Q}}_p$ is the field of $p$-adic numbers and $\mathbb{G}$ is a split connected reductive group scheme over ${\mathbb{Z}}_p$. In this work we will introduce a sheaf of twisted arithmetic differential operators on the formal flag variety of $\mathbb{G}$, associated to a general character. We will generalize the arguments of C. Huyghe and T. Schmidt, concerning the ${\mathcal{D}}^{\mathrm{†}}$-affinity of the formal flag variety of $\mathbb{G}$, of certain sheaves of $p$-adically complete twisted arithmetic differential operators associated to an algebraic character, and the calculation of the global sections.