Remarks on the inverse Galois problem over function fields

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of $\mathrm{\ell }$-adic Galois representations valued in a semisimple group $G$ using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result of Larsen on Galois images for compatible sytems.