The Categories of Lubin-Tate and Drinfeld Bundles

TL;DR

The paper resolves the structure of Lubin-Tate bundles on the Drinfeld symmetric space $\Omega$ by proving a precise equivalence with smooth representations of the division algebra units $H=D^{\times}$, via the functor $\Psi_H$. It develops a robust framework connecting semi-linear representations to equivariant vector bundles across Lubin-Tate and Drinfeld towers, employing ${\mathcal{D}}$-modules to control infinitesimal actions and the tangent sheaf. The main achievement is that $\Psi_H:\mathrm{Rep}_K^{\mathrm{sm}}(H)\xrightarrow{\sim} \mathbf{Vect}^G_{\mathrm{LT}}(\Omega)$ (and its extensions to $G^0$-finite and connection-enhanced categories) provides an explicit, fully faithful, and essentially surjective correspondence, with the global description enhanced by passing to $L_{\infty}$ and $K_{\infty}$. The results illuminate the Jacquet-Langlands correspondence at the level of geometric objects on the Drinfeld space, yielding a precise bridge between $p$-adic representation theory and algebraic/analytic geometry in the Lubin-Tate and Drinfeld contexts, and enabling explicit decompositions and twists through functors like $\Phi_H$ and $\Phi_H^0$.

Abstract

For a finite extension $F$ of $\mathbb{Q}_p$ and $n \geq 1$, we show that the category of Lubin-Tate bundles on the $(n-1)$-dimensional Drinfeld symmetric space is equivalent to the category of finite-dimensional smooth representations of the group of units of the division algebra of invariant $1/n$ over $F$.

Theorems & Definitions (100)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5
• Remark 3.1
• Lemma 3.2
• proof