Hypergeometric Sheaves and General Linear Groups

We find all irreducible hypergeometric sheaves whose geometric monodromy group is finite, almost quasisimple and has the projective special linear group $PS{L}_n(q)$ with $n\ge 3$ as a composition factor. We use the classification of semisimple elements with specific spectra in irreducible Weil representations to prove that if an irreducible hypergeometric sheaf has such geometric monodromy group, then it must be of certain form. Then we extend results of Katz and Tiep on a prototypical family of such sheaves to full generality to show that these hypergeometric sheaves do have such geometric monodromy groups, and that they have some connection to a construction of Abhyankar.