Equivariant Vector Bundles with Connection on Drinfeld Symmetric Spaces

TL;DR

This paper establishes an equivalence between smooth representations of a division algebra's units and equivariant vector bundles with connection on Drinfeld symmetric spaces, deepening the understanding of their geometric and algebraic structures.

Contribution

It introduces a novel categorical equivalence linking representations of $D^ imes$ to vector bundles with connection on Drinfeld spaces.

Findings

Equivalence of categories between $D^ imes$ representations and vector bundles with connection.

Construction of $G^0$-finite $ ext{GL}_n(F)$-equivariant vector bundles.

Insight into the geometric realization of algebraic representations.

Abstract

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 $\Omega$, the $(n-1)$-dimensional Drinfeld symmetric space.