Families of twisted $\mathcal{D}$-modules and arithmetic models of Harish-Chandra modules

We develop a theory of tdos and twisted $\mathcal{D}$-modules over general base schemes with a focus on functorial aspects. In particular, we introduce a flat base change functor and establish its compatibility with globalization and direct image functors. We also study forms of closed $K$-orbits of $\theta$-stable parabolic subgroups in the total flag variety. We apply these two developed theories to give a geometric construction of half-integral models of cohomologically induced modules. With a view towards arithmetic applications, we further demonstrate desirable properties of the constructed half-integral models, such as projectivity over the base and torsion-free relative Lie algebra cohomology.