Deformations of G-Valued Pseudocharacters

We define a deformation space of Lafforgue’s G -valued pseudocharacters of a profinite group Γ Γ for a possibly disconnected reductive group G . We show that this definition generalizes Chenevier’s construction. We show that the universal pseudodeformation ring is noetherian and that the functor of continuous G -pseudocharacters on affinoid Qₚ Q p -algebras is represented by a quasi-Stein rigid analytic space, whenever Γ Γ is topologically finitely generated. We also show that the pseudodeformation ring is noetherian when Γ Γ satisfies Mazur’s condition Φ ₚ Φ p and G satisfies a certain invariant-theoretic condition. For G = Sp₂ₙ G = Sp 2 n , we describe three types of obstructed loci in the special fiber of the universal pseudodeformation space of an arbitrary residual pseudocharacter and give upper bounds for their dimension.