Lübeck's classification of representations of finite simple groups of Lie type and the inverse Galois problem for some orthogonal groups

Abstract: In this paper we prove that for each integer of the form $n=4\varpi$ (where $\varpi$ is a prime between $17$ and $73$) at least one of the following groups: $P\Omega^+n(\mathbb{F}{\ell^s})$, $PSO^+n(\mathbb{F}{\ell^s})$, $PO_n^+(\mathbb{F}_{\ell^s})$ or $PGO^+n(\mathbb{F}{\ell^s})$ is a Galois group of $\mathbb{Q}$ for almost all primes $\ell$ and infinitely many integers $s > 0$. This is achieved by making use of the classification of small degree representations of finite simple groups of Lie type in defining characteristic of F. L\"ubeck and a previous result of the author on the image of the Galois representations attached to RAESDC automorphic representations of $GL_n(\mathbb{A}_\mathbb{Q})$.