Equivariant Vector Bundles with Connection on Drinfeld Symmetric Spaces

For a finite extension F of Qₚ and n ≥ 1, let D be the division algebra over F of invariant $1/n$ and let G⁰ be the subgroup of GLₙ(F) of elements with norm $1$ determinant. We show that the action of D^× on the Drinfeld tower induces an equivalence of categories from finite dimensional smooth representations of D^× to G⁰-finite GLₙ(F)-equivariant vector bundles with connection on Ω, the $(n-1)$-dimensional Drinfeld symmetric space.