Global sections of equivariant line bundles on the $p$-adic upper half plane

Let $F$ be a finite extension of ${\mathbb{Q}}_p$, let ${\mathrm{\Omega }}_F$ be Drinfeld’s upper half-plane over $F$ and let $G^0$ the subgroup of $G{L}_2(F)$ consisting of elements whose determinant has norm $1$. Let $\mathcal{L}$ be a torsion $G^0$-equivariant line bundle with connection on ${\mathrm{\Omega }}_F$. We show that the strong dual of $\mathcal{L}({\mathrm{\Omega }}_F)$ is an admissible locally $F$-analytic representation of $G^0$ of topological length at most $2$. It is topologically irreducible if and only if the underlying connection on $\mathcal{L}$ is non-trivial. We give an explicit formula for the length of the strong dual of the space of globally-defined rigid analytic functions on a $G^0$-equivariant finite étale rigid analytic covering of ${\mathrm{\Omega }}_F$ with abelian Galois group as an admissible locally $F$-analytic representation of $G^0$.