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 + Pi D O D of D x we determine the structure of H1(I1, 1 (I 1 , pi) as a representation of D x /I 1 for an arbitrary smooth irreducible F p-representation pi of D x . We use this to compute the group Ext1Dx(pi, 1D x (pi, pi ' ) for arbitrary smooth irreducible representations pi and pi ' of D x . In the p-adic case, via Poincar & eacute; duality we can compute the top cohomology groups and compute the highest degree extensions.