Parahoric Hecke Ext-algebras in characteristic $p$

Let $\mathfrak{F}$ be a nonarchimedean local field of residual characteristic $p$, and let $G$ denote the group of $\mathfrak{F}$-points of a connected reductive group over $\mathfrak{F}$. For an open compact subgroup $\mathcal{U}$ of $G$ and a unital commutative ring $k$, we let ${\mathbf{X}}_{\mathcal{U}}$ denote the space of compactly supported $k$-valued functions on $G\mathrm{/}\mathcal{U}$. Building on work of Ollivier–Schneider, we investigate the graded $\text{Ext}$-algebra $E_{\mathcal{U}}^{\ast }:={\text{Ext}}_G^{\ast }({\mathbf{X}}_{\mathcal{U}},{\mathbf{X}}_{\mathcal{U}}{)}^{\text{op}}$. In particular, we describe the Yoneda product, an involutive anti-automorphism, and (when $k$ is a field of characteristic $p$ and $\mathcal{U}$ has no $p$-torsion) a duality operation. We allow for the reductive group to be non-split, and for the open compact subgroup $\mathcal{U}$ to be non-pro-$p$. Specializing further to the case $G={\text{SL}}_2({\mathbb{Q}}_p)$ with $p\ge 5$ and a coefficient field of characteristic $p$, we obtain more precise results when $\mathcal{U}$ is equal to an Iwahori subgroup $J$ or a hyperspecial maximal compact subgroup $K$. In particular, we compute the structure of $E_J^{\ast }$ as an $E_J^0$-bimodule, obtain an explicit description of the center $\mathcal{Z}(E_J^{\ast })$ of $E_J^{\ast }$, and construct a surjective morphism of algebras $\mathcal{Z}(E_J^{\ast })⟶E_K^{\ast }$ (analogous to the compatibility between Bernstein and Satake isomorphisms in characteristic 0). From this we deduce the (somewhat surprising) fact that $E_K^{\ast }$ is not graded-commutative, contrary to what happens for almost all $\mathrm{\ell }$-modular characteristics.