Equivariant Vector Bundles with Connection on Drinfeld Symmetric Spaces

For a finite extension $F$ of ${\mathbb{Q}}_p$ and $n\ge 1$, let $D$ be the division algebra over $F$ of invariant $1\mathrm{/}n$ and let $G^0$ be the subgroup of ${\text{GL}}_n(F)$ of elements with norm $1$ determinant. We show that the action of $D^{\times }$ on the Drinfeld tower induces an equivalence of categories from finite dimensional smooth representations of $D^{\times }$ to $G^0$-finite ${\text{GL}}_n(F)$-equivariant vector bundles with connection on $\mathrm{\Omega }$, the $(n-1)$-dimensional Drinfeld symmetric space.