Lifting $G$-Valued Galois Representations when $\ell \neq p$

TL;DR

This paper develops a framework for lifting local Galois representations valued in arbitrary reductive groups when ll  p, constructing smooth deformation spaces and extending classical notions to more general group schemes.

Contribution

It introduces a canonical smooth component in universal lifting spaces for reductive group schemes, generalizing previous minimally ramified deformation conditions for classical groups.

Findings

Constructed a canonical smooth component in lifting spaces.

Extended isotypic decomposition to general reductive groups.

Applied methods to produce geometric lifts for ll  p Galois representations.

Abstract

In this paper we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\textrm{GL}_n$-valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm{G}_2$-valued representations.