Deformations of $G$-valued pseudocharacters

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