Extensions of mod p representations of division algebras over non-Archimedean local fields

Abstract

Let $F$ be a local field over ${\mathbf{Q}}_p$ or ${\mathbf{F}}_p((t))$, and let $D$ be a central simple division algebra over $F$ of degree $d$. In the $p$-adic case, we assume $p>de+1$ where $e$ is the ramification degree over ${\mathbf{Q}}_p$; otherwise, we need only assume $p$ and $d$ are coprime. For the subgroup $I_1=1+{\varpi }_D{\mathcal{O}}_D$ of $D^{\times }$ we determine the structure of ${\mathrm{H}}^1(I_1,\pi )$ as a representation of $D^{\times }/I_1$ for an arbitrary smooth irreducible representation $\pi$ of $D^{\times }$. We use this to compute the group ${\mathrm{Ext}}_{D^{\times }}^1(\pi ,{\pi }^{\prime })$ for arbitrary smooth irreducible representations $\pi$ and ${\pi }^{\prime }$ of $D^{\times }$. In the $p$-adic case, via Poincaré duality we can compute the top cohomology groups and compute the highest degree extensions.