Deformations of G-Valued Pseudocharacters

Abstract We define a deformation space of Lafforgue’s G -valued pseudocharacters of a profinite group $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> 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 $${\mathbb {Q}}_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -algebras is represented by a quasi-Stein rigid analytic space, whenever $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is topologically finitely generated. We also show that the pseudodeformation ring is noetherian when $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> satisfies Mazur’s condition $$\Phi _p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> and G satisfies a certain invariant-theoretic condition. For $$G = \operatorname {Sp}_{2n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>Sp</mml:mo> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , 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.