The modular pro-$p$ Iwahori-Hecke ${\operatorname{Ext}}$-algebra

Let $\mathfrak F$ be a locally compact nonarchimedean field of positive residue characteristic $p$ and $k$ a field of characteristic $p$. Let $G$ be the group of $\mathfrak{F}$-rational points of a connected reductive group over $\mathfrak{F}$ which we suppose $\mathfrak F$-split. Given a pro-$p$ Iwahori subgroup $I$ of $G$, we consider the space $\mathbf X$ of $k$-valued functions with compact support on $G/I$. It is naturally an object in the category ${\operatorname{Mod}}{(G)}$ of all smooth $k$-representations of $G$. We study the graded Ext-algebra $E^*={\operatorname{Ext}}_{{\operatorname{Mod}}(G)}^*(\mathbf X, \mathbf X)$. Its degree zero piece $E^0$ is the usual pro-$p$ Iwahori-Hecke algebra $H$. We describe the product in $E^*$ and provide an involutive anti-automorphism of $E^*$. When $I$ is a Poincar\'e group of dimension $d$, the ${\operatorname{Ext}}$-algebra $E^*$ is supported in degrees $i\in\{0\dots d\}$ and we establish a duality theorem between $E^i$ and $E^{d-i}$. Under the same hypothesis (and assuming that $\mathbf G$ is almost simple and simply connected), we compute $E^d$ as an $H$-module on the left and on the right. We prove that it is a direct sum of the trivial character, and of supersingular modules.