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

Abstract

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 $\mathrm{Mod}(G)$ of all smooth $k$-representations of $G$. We study the graded Ext-algebra $E^{\ast }={\mathrm{Ext}}_{\mathrm{Mod}(G)}^{\ast }(\mathbf{X},\mathbf{X})$. Its degree zero piece $E^0$ is the usual pro-$p$ Iwahori-Hecke algebra $H$. We describe the product in $E^{\ast }$ and provide an involutive anti-automorphism of $E^{\ast }$. When $I$ is a Poincaré group of dimension $d$, the $\mathrm{Ext}$-algebra $E^{\ast }$ 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.