Moduli Stacks of Vector Bundles and Frobenius Morphisms
  We describe the action of the different Frobenius morphisms on the cohomology ring of the moduli stack of algebraic vector bundles of fixed rank and determinant on an algebraic curve over a finite field in characteristic p and analyse special situations like vector bundles on the projective line and relations with infinite Grassmannians. 
Frank Neumann, Ulrich Stuhler