Galois characters arising from Drinfeld modules

Boston and Ose find a necessary condition for a Galois character ρ to be a Drinfeld character in the sense that it arises from the Galois action on the torsion points of a Drinfeld module over a finite field. We prove that this necessary condition is equivalent to the condition that the fixed field of the kernel of ρ can be identified with that of a Drinfeld character. This shows in particular that surjective characters are Drinfeld up to twist in many cases.