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

Representations modulo p of the p-adic group GL(2, F)

“…In [BL95], [BL94] and [Bre01], the functor of I(1)-invariant vectors plays an important but hidden role. Instead of the Hecke algebra of the trivial F p -representation of the pro-p-Iwahori, they use the Hecke algebra of an irreducible F p -representation of GL(2, O F )F * which is always a commutative algebra isomorphic to F p [T ].…”

Speculations on the mod p representation theory of p-adic groups

“…The mod p representation theory of p-adic groups began with the papers [5,6] that treated the case of G = GL(2, K), where K is a non-archimedean local field. Those papers already revealed an interesting dichotomy that continues to dominate the subject.…”

Central Characters for Smooth Irreducible Modular Representations of GL$_2$ ($Q_p$)

“…This theorem is proved in [BL95] and [BL94], which treat the case l T 0, and in [Bre03], which treats the case l 0.…”

“…We say that P admits a central character, if every z P Z(GL 2 (Q p )), the center of GL 2 (Q p ), acts on P by a scalar. The smooth irreducible representations of GL 2 (Q p ) over an algebraically closed field of characteristic p, admitting a central character, have been studied by Barthel-Livne  in [BL94,BL95] and by Breuil in [Bre03]. The purpose of this note is to prove the following theorem.…”