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

Let X=G/H be a homogeneous space of a Lie group G. When the isotropy subgroup H is non-compact, a discrete subgroup Γ 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 Γ X is a standard quotient, to what extent can one deform the discrete subgroup Γ 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 Γ 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.