Classifying representations of finite classical groups of Lie type of dimension up to ${\ell }^4$

Abstract

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to $\ell$, and splits into two cases according to $G$ is of type $A_{\ell }$ or not. Furthermore, we discuss explicit formulas for computing the dimensions of such representations. The motivation for this work arises, in part, from a desire to obtain new results on two classical problems concerning Galois representations: the large image conjecture for automorphic Galois representations and the inverse Galois problem. We conclude the paper by giving some remarks on potential implications in these addresses.

This revised version incorporates several significant changes. Sections 3 and 4 have been merged, the main theorem of this sections has been refined, and its proof simplified. In addition, the results concerning representations of types B_\ell, C_\ell, and D_\ell have been substantially modified, improving the precision regarding the primes p (the characteristics) for which our results hold