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 $\mathrm{Ext}^1{D^\times}(\pi,\pi')$ for arbitrary smooth irreducible representations $\pi$ and $\pi'$ of $D^\times$. In the $p$-adic case, via Poincar\'{e} duality we can compute the top cohomology groups and compute the highest degree extensions.