On the $I(1)$-invariants: Non-abelian Hecke algebra case

Let $F$ be a finite extension of $\mathbb{Q}_p$. The so-called supersingular representations are the basic building blocks in the theory of mod $p$ representations of ${\rm GL}_2(F)$. The space of pro-$p$-Iwahori invariants of a universal module played a crucial role in the construction of the supersingular representations of ${\rm GL}_2(\mathbb{Q}_p)$. In this paper, we give an explicit description of the pro-$p$-Iwahori invariants of the universal module $\pi_r$ for $r = 0, q - 1$ using the Iwahori-Hecke model. We also determine the action of the pro-$p$-Iwahori-Hecke algebra on these newly found invariants. As an application, we recover $\pi_r$ functorially from its space of $I(1)$-invariants and extend a theorem of Ollivier for any totally ramified extension of $\mathbb{Q}_p$ other than itself.