2021/07/07 by Gebhard Böckle, Böckle, Gebhard, Sara Arias‐de‐Reyna +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2107.03114
openalex publication_date 2021/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a connected reductive almost simple group over the Witt ring W(\mathbbF) for \mathbbF a finite field of characteristic p. Let R and R' be complete noetherian local W(\mathbbF) -algebras with residue field \mathbbF. Under a mild condition on p in relation to structural constants of G, we show the following results: (1) Every closed subgroup H of G(R) with full residual image G(\mathbbF) is a conjugate of a group G(A) for A⊂ R a closed subring that is local and has residue field \mathbbF . (2) Every surjective homomorphism G(R)\toG(R') is, up to conjugation, induced from a ring homomorphism R→ R'. (3) The identity map on G(R) represents the universal deformation of the representation of the profinite group G(R) given by the reduction map G(R)\toG(\mathbbF). This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of G(R) with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general G, and we study in the case at hand in great detail what conditions on \mathbbF or on p in relation to G are necessary for the above results to hold.