Modular Representations of GL2 of a Local-Field: The Ordinary, Unramified Case Original Research Article

Let F be a local field of residue characteristic p, and let E be an algebraically closed field of characteristic p. We prove that unramified irreducible representations of PGL2(F) on E- vector spaces admit eigenvectors for an appropriate Hecke operator T. We classify them by the eigenvalue λ of T to one dimensional (λ = ± 1), principal series (λ ≠ 0, ± 1) or supersingular (λ = 0). Any λ can appear, and for λ ≠ 0 we prove uniqueness.