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 $G/\mathcal{ U}$. Building on work of Ollivier--Schneider, we investigate the graded $\textrm{ Ext}$-algebra $E_{ \mathcal{ U}}^* := \textrm{ Ext}_G^*(\mathbf{ X}_{ \mathcal{ U}},\mathbf{ X}_{ \mathcal{ U}})^{ \textrm{ op}}$. 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 $G = \textrm{ SL}_2(\mathbb{ Q}_p)$ with $p \geq 5$ 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 $E_J^*$ as an $E_J^0$-bimodule, obtain an explicit description of the center $\mathcal{ Z}(E_J^*)$ of $E_J^*$, and construct a surjective morphism of algebras $\mathcal{ Z}(E_J^*) \longrightarrow E_K^*$ (analogous to the compatibility between Bernstein and Satake isomorphisms in characteristic 0). From this we deduce the (somewhat surprising) fact that $E_K^*$ is not graded-commutative, contrary to what happens for almost all $\ell$-modular characteristics.
Categories: math.RT math.NT