On the pro-$p$ Iwahori Hecke Ext-algebra of ${\mathrm{SL}}_2({\mathbb{Q}}_p)$

Abstract

Let $G={\mathrm{SL}}_2(\mathfrak{F})$ where $\mathfrak{F}$ is a finite extension of ${\mathbb{Q}}_p$. We suppose that the pro-$p$ Iwahori subgroup $I$ of $G$ is a Poincaré group of dimension $d$. Let $k$ be a field containing the residue field of $\mathfrak{F}$. In this article, we study the graded Ext-algebra $E^{\ast }={\mathrm{Ext}}_{\mathrm{Mod}(G)}^{\ast }(k[G/I],k[G/I])$. Its degree zero piece $E^0$ is the usual pro-$p$ Iwahori-Hecke algebra $H$. We study $E^d$ as an $H$-bimodule and deduce that for an irreducible admissible smooth representation of $G$, we have $H^d(I,V)=0$ unless $V$ is the trivial representation. When $\mathfrak{F}={\mathbb{Q}}_p$ with $p\ge 5$, we have $d=3$. In that case we describe $E^{\ast }$ as an $H$-bimodule and give the structure as an algebra of the centralizer in $E^{\ast }$ of the center of $H$. We deduce results on the values of the functor $H^{\ast }(I,{}_{-})$ which attaches to a (finite length) smooth $k$-representation $V$ of $G$ its cohomology with respect to $I$. We prove that $H^{\ast }(I,V)$ is always finite dimensional. Furthermore, if $V$ is irreducible, then $V$ is supersingular if and only if $H^{\ast }(I,V)$ is a supersingular $H$-module.