Extensions of mod p representations of division algebras

Let F be a local field over Q p or 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 Q p ; otherwise, we need only assume p and d are coprime. For the subgroup I 1 =1+ϖ D 𝒪 D of D × we determine the structure of H 1 (I 1 ,π) as a representation of D × /I 1 for an arbitrary smooth irreducible F ¯ p -representation π of D × . We use this to compute the group Ext D × 1 (π,π ′ ) for arbitrary smooth irreducible representations π and π ′ of D × . In the p-adic case, via Poincaré duality we can compute the top cohomology groups and compute the highest degree extensions.