# On the images of the Galois representations attached to certain RAESDC automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$

## TL;DR

This paper constructs infinite families of Galois representations linked to automorphic forms, demonstrating their images are not contained in geometric maximal subgroups, and applies these results to the inverse Galois problem for orthogonal groups.

## Contribution

It proves the existence of compatible Galois systems with images outside geometric maximal subgroups for certain automorphic representations, advancing understanding of Galois representations and inverse Galois problem.

## Key findings

- Existence of infinite compatible systems with non-geometric images.
- Images of reductions are not contained in maximal geometric subgroups.
- Provides heuristic evidence for the inverse Galois problem for orthogonal groups.

## Abstract

In the 80's Aschbacher classified the maximal subgroups of almost all of the finite almost simple classical groups. Essentially, this classification divide these subgroups into two types. The first of these consist roughly of subgroups that preserve some kind of geometric structure, so they are commonly called subgroups of geometric type. In this paper we will prove the existence of infinitely many compatible systems $\{ \rho_\ell \}_\ell$ of $n$-dimensional Galois representations associated to regular algebraic, essentially self-dual, cuspidal automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$ ($n$ even) such that, for almost all primes $\ell$, the image of $\overline{\rho}_{\ell}$ (the semi-simplification of the reduction of $\rho_\ell$) cannot be contained in a maximal subgroup of geometric type of an $n$-dimensional symplectic or orthogonal group. Then, we apply this result to some 12-dimensional representations to give heuristic evidence towards the inverse Galois problem for even-dimensional orthogonal groups.