Let K be a finitely generated infinite field over the finite prime field Fp with separable closure Ks and let X be a smooth projective variety over K. By Deligne the cohomology groups Hie´t(XKs,Qℓ) for varying primes ℓ≠p form a (Q-rational) compatible system of Galois representations of Gal(Ks/K) and its restriction to the geometric Galois group GgeoK=Gal(Ks/KFsp) is semisimple. Using mainly algebraic geometry, representation theory and Bruhat-Tits theory, Cadoret, Hui and Tamagawa showed recently that also the family of reductions Hie´t(XKs,Fℓ) is semisimple as a representation of GgeoK for almost all ℓ, the key case being that of a global function field K. This has important consequence for the image of GgeoK for its action on the adelic module~Hie´t(XKs,AQ). In joint work with W. Gajda and S. Petersen, using automorphic methods as a main tool, we prove the analog of the above result for any E-rational compatible system of Galois representations of a global function field. In the talk I shall explain the context, indicate the applications and sketch how automorphic methods come to bear on the problem.