Multiplicative character sums and Kloosterman sheaves

We are given a prime p, a power q of p, and a prime to p integer a with q>a≥2. For a nontrivial multiplicative character χ, we consider the one parameter family of character sums t↦−∑xχ(x^q+x^a−t), which are the traces of a local system on the G_m/F_p(χ) of nonzero t's. We show that this local system is the pullback of a Kloosterman sheaf K_q,a,ρ (any ρ with ρ^q−a=χ), and determine the geometric monodromy group G_geom of this K. We also determine G_geom for the universal family F_χ,e of sums −∑_xχ(f_e(x)), as f_e runs over degree e polynomials with all distinct roots. These local systems F_χ,e were the main focus of [14, Chapter 4], and our new results for F_χ,e are the complete determination of G_geom in the cases where G_geom is finite.