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

For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $\tau$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $\tau$ is introduced by Barthel and Livn\'e 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 $\pi$ of $\tau$, making use of the Iwahori-Hecke operators. When $F$ is not totally ramified over $\mathbb Q_p$, the representation $\pi$ is a non-trivial 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-\`a-vis the description of the space of pro-$p$-Iwahori invariants of $\tau$.