Homological branching laws for $p$-adic and mod $p$ representations of ${\mathrm{G}\mathrm{L}}_2$ over a $p$-adic field

Let $F$ be a finite extension of ${\mathbb{Q}}_p$ and $p\ge 5$. In this paper, we establish homological branching laws for mod $p$ representations of ${\mathrm{G}\mathrm{L}}_2(F)$ 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–Poincaré characteristics under a $p$-torsion-freeness hypothesis. We also establish corresponding results for a class of unitary Banach space representations.