Iwahori–Hecke model for mod p representations of GL(2,F)

Abstract

For a p-adic field F, the space of pro-p-Iwahori invariants of a universal supersingular mod p representation τ of GL₂(F) is determined in the works of Breuil, Schein, and Hendel. The representation τ is introduced by Barthel and Livné and this is defined in terms of the spherical Hecke operator. In earlier work of Anandavardhanan-Borisagar, an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient π of τ, making use of the Iwahori-Hecke operators. When F is not totally ramified over Q_n, the representation π is a non-trivial quotient of τ. We determine a basis for the space of invariants of π under the pro-p Iwahori subgroup. A pleasant feature of this "new" representation π is that its space of pro-p-Iwahori invariants admits a more uniform description vis-à-vis the description of the space of pro-p-Iwahori invariants of τ.