Extensions between \(p\)-adic principal series and principal series modulo \(p\) of \(G(F)\) (Q2800011)

Let \(F\) be a finite extension of \({\mathbb Q}_p\) and let \(G\) be a split reductive group over \(F\). The representations of \(G(F)\) considered in the article are continuous unitary representations on a Banach space over a finite extension \(E\) of \({\mathbb Q}_p\) (\(p\)-adic representations) or smooth representations on an \(A\)-module, where \(A\) is an artinian local \({\mathcal O}_E\)-algebra with residue field \(k_E\), \(E\) again a finite extension of \({\mathbb Q}_p\), with ring of integers \({\mathcal O}_E\) and residue field \(k_E\) (mod \(p\) representations). The results are formulated for the \(p\)-adic case, but the mod \(p\) analogues are proved first.NEWLINENEWLINEThe author determines the extensions between representations of \(G(F)\) induced from a Borel group. When \(F={\mathbb Q}_p\), \(G\) is assumed to have a connected centre and a simply connected derived group. When \(F\neq {\mathbb Q}_p\), there is no extra assumption on \(G\). The results on \(\text{Ext}^1\) are deduced from a computation of Emerton's functors \(H^i\text{Ord}_{B(F)}\). The functor \(\text{Ord}_{B(F)} = H^0\text{Ord}_{B(F)}\) is right adjoint to \(\text{Ind}^{G(F)}_{B^-(F)}\). The functors \(H^i\text{Ord}_{B(F)}\) form a \(\partial\)-functor from admissible representations of \(G(F)\) over \(A\) to characters of \(T(F)\), which is conjectured to be isomorphic to the \(\partial\)-functor formed by the derived functors of \(\text{Ord}_{B(F)}\) (\(T\) is a maximal split torus in \(G\) and \(B\supset T\)). Under the assumption that the conjecture is true for \(G\) also higher \(\text{Ext}^n\) are calculated.