Equivariant Vector Bundles with Connection on Drinfeld Symmetric Spaces

For a finite extension $F$ of $\mathbb{Q}_p$ and $n \geq 1$, let $D$ be the division algebra over $F$ of invariant $1/n$ 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 $Ω$, the $(n-1)$-dimensional Drinfeld symmetric space.

💡 Research Summary

The paper establishes a precise categorical equivalence between finite‑dimensional smooth representations of the multiplicative group of a division algebra (D) over a (p)‑adic field (F) and certain equivariant vector bundles with connection on the Drinfeld symmetric space (\Omega).

Setting and Motivation.
Let (F/\mathbb{Q}_p) be a finite extension, (n\ge1), and let (D) be the central division algebra over (F) of invariant (1/n). Denote by (G^0\subset\mathrm{GL}_n(F)) the subgroup of matrices whose determinant has norm 1. The Drinfeld tower \