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

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$, let $\Omega_F$ be Drinfeld's upper half-plane over $F$ and let $G^0$ the subgroup of $GL_2(F)$ consisting of elements whose determinant has norm $1$. Let $\mathscr{L}$ be a torsion $G^0$-equivariant line bundle with connection on $\Omega_F$. We show that the strong dual of $\mathscr{L}(\Omega_F)$ 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 $\mathscr{L}$ 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 \'etale rigid analytic covering of $\Omega_F$ with abelian Galois group as an admissible locally $F$-analytic representation of $G^0$.