Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces

Let $X=G\mathrm{/}H$ be a homogeneous space of a Lie group $G$. When the isotropy subgroup $H$ is non-compact, a discrete subgroup $\mathrm{\Gamma }$ may fail to act properly discontinuously on $X$. In this article, we address the following question: in the setting where $G$ and $H$ are reductive Lie groups and $\mathrm{\Gamma }\mathrm{\backslash }X$ is a standard quotient, to what extent can one deform the discrete subgroup $\mathrm{\Gamma }$ while preserving the proper discontinuity of the action on $X$? We provide several classification results, including conditions under which local rigidity holds for compact standard quotients $\mathrm{\Gamma }\mathrm{\backslash }X$, when a standard quotient can be deformed into a non-standard quotient, a characterization of the largest Zariski-closure of discontinuous groups under small deformations, and conditions under which Zariski-dense deformations occur.