#### Vol. 315, No. 2, 2021

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Iwahori–Hecke model for mod $p$ representations of $\mathrm{GL}(2,F)$

### U. K. Anandavardhanan and Arindam Jana

Vol. 315 (2021), No. 2, 255–283
DOI: 10.2140/pjm.2021.315.255
##### Abstract

For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $\tau$ of ${GL}_{2}\left(F\right)$ is determined in the works of Breuil, Schein, and Hendel. The representation $\tau$ is introduced by Barthel and Livné and is defined in terms of the spherical Hecke operator. In Anandavardhanan and Borisagar [2013; 2015], 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 $\pi$ of $\tau$, making use of the Iwahori–Hecke operators. When $F$ is not totally ramified over ${ℚ}_{p}$, the representation $\pi$ is a nontrivial quotient of $\tau$. We determine a basis for the space of invariants of $\pi$ under the pro-$p$-Iwahori subgroup. A pleasant feature of this “new” representation $\pi$ 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 $\tau$.

##### Keywords
modular representations, supersingular representations, Iwahori–Hecke model
##### Mathematical Subject Classification
Primary: 20G05
Secondary: 11F70, 22E50