Platitude du pro-$p$-module universel de GL$_{2}(F)$ en caractéristique $p$

Abstract

Let $F$ be a $p$-adic field with uniformizer $\pi$. We consider $G={\rm GL}_{2}(F)/\pi$ and $I(1)$ the pro-$p$-Iwahori subgroup of $G$. The exploration of the smooth mod $p$ representations of $G$ motivates the study of the space of functions with values in $\overline{\mathbb{F}}_p$ and compact support in the set of right cosets $I(1)\backslash G$. We show that this universal module is flat over the pro-$p$-Hecke algebra if and only if the cardinal of the residue field of $F$ is equal to $p$.