Homological branching laws for $p$-adic and mod $p$ representations of ${\rm GL}_2$ over a $p$-adic field

Abstract

Let $F$ be a finite extension of $\mathbb Q_p$ and $p\geq 5$. In this paper, we establish homological branching laws for mod $p$ representations of $\mathrm{GL}_2(F)$ restricted to split and nonsplit maximal tori, treating principal and special series and, when $F=\mathbb Q_p$, supersingular representations. We determine the derived Hom-complexes in both orders for nonexceptional characters. At exceptional characters, we calculate their Euler--Poincar\'e characteristics under a $p$-torsion-freeness hypothesis. We also establish corresponding results for a class of unitary Banach space representations.