Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations

TL;DR

This paper constructs examples of G-irreducible Galois representations for classical groups that can be lifted using new methods, surpassing the scope of existing potential automorphy techniques.

Contribution

It provides explicit examples of G-irreducible Galois representations for classical groups where new lifting methods are applicable, extending previous results.

Findings

Constructed many examples of G-irreducible representations

Demonstrated applicability of new lifting methods

Showed limitations of potential automorphy theorems

Abstract

In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$ a reductive group, to geometric $\ell$-adic representations. In this note we take $G$ to be a classical group and construct many examples of $G$-irreducible representations to which these new lifting methods apply, but to which the lifting methods provided by potential automorphy theorems do not.