%0 Generic
%A Quast, Julian Christian
%C Heidelberg
%D 2023
%F heidok:33231
%K Pseudodarstellung, Pseudocharakter, Determinantengesetz
%R 10.11588/heidok.00033231
%T Deformation theory of G-valued pseudocharacters and symplectic determinant laws
%U https://archiv.ub.uni-heidelberg.de/volltextserver/33231/
%X We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and  Lafforgue. We refer to Taylor’s and Rouquier’s pseudorepresentations as pseudocharacters. They are  very closely related, the main difference being that Taylor’s pseudocharacters are defined for a group,  where as Rouquier’s pseudocharacters are defined for algebras. Chenevier’s pseudorepresentations are  so-called polynomial laws and will be called determinant laws. Lafforgue’s pseudorepresentations are a  generalization of Taylor’s pseudocharacters to other reductive groups G, in that the corresponding notion  of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters.    We survey the known comparison theorems, notably Emerson’s bijection between Chenevier’s determinant  laws and Lafforgue’s GL(n)-pseudocharacters and the bijection with Taylor’s pseudocharacters away from  small characteristics.    We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously,  we obtain that tensor products of determinant laws exist and are compatible with tensor products  of representations. Further the tensor product of Lafforgue’s pseudocharacters agrees with the tensor  product of Taylor’s pseudocharacters.    We generalize some of the results of [Che14] to general reductive groups, in particular we show that  the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely  generated profinite group Γ with values in a finite field (of characteristic p) is noetherian. We also show,  that for specific groups G it is sufficient, that Γ satisfies Mazur’s condition Φ_p.    One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp  be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and  concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater  of Gal(F) into obstructed subloci Xdec(Θ), Xpair(Θ), Xspcl(Θ) of dimension smaller than the expected dimension  n(2n + 1)[F : Qp].    We also prove that Lafforgue’s G-pseudocharacters over algebraically closed fields for possibly nonconnected  reductive groups G come from a semisimple representation. We introduce a formal scheme  and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on  our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition  as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to  conjugacy and Frobenius automorphisms.    At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic  groups, which is based on adding a ’Pfaffian polynomial law’ to a determinant law which is invariant under  an involution. We prove the expected basic properties in that we show that symplectic determinant laws  over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and  that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings  are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgue’s  pseudocharacters and show that it is an isomorphism over reduced rings.