Deformations of G-Valued Pseudocharacters

Abstract We define a deformation space of Lafforgue’s G -valued pseudocharacters of a profinite group $$\Gamma $$ 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 $${\mathbb {Q}}_p$$ Q p -algebras is represented by a quasi-Stein rigid analytic space, whenever $$\Gamma $$ Γ is topologically finitely generated. We also show that the pseudodeformation ring is noetherian when $$\Gamma $$ Γ satisfies Mazur’s condition $$\Phi _p$$ Φ p and G satisfies a certain invariant-theoretic condition. For $$G = \operatorname {Sp}_{2n}$$ 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.