Equivariant vector bundles on Drinfeld's halfspace over a finite field

TL;DR

This paper investigates the structure of global sections of homogeneous vector bundles on Drinfeld's halfspace over finite fields, describing their infinite-dimensional modular representations via algebraic and differential operator frameworks.

Contribution

It provides two different descriptions of the global sections as $GL_{d+1}(k)$-representations, extending ideas from the $p$-adic case to finite fields.

Findings

Descriptions of $H^0( ext{X}, ext{E})$ as $GL_{d+1}(k)$-representations

Use of crystalline universal enveloping algebra and differential operators

Extension of $p$-adic methods to finite field setting

Abstract

Let $\mathcal{X} \subset \mathbb{P}_k^d$ be Drinfeld's halfspace over a finite field $k$ and let $\mathcal{E}$ be a homogeneous vector bundle on $\mathbb{P}_k^d$. The paper deals with two different descriptions of the space of global sections $H^0(\mathcal{X},\mathcal{E})$ as $GL_{d+1}(k)$-representation. This is an infinite dimensional modular representation. Here we follow the ideas of \cite{O2,OS} treating the $p$-adic case. As a replacement for the universal enveloping algebra we consider both the crystalline universal enveloping algebra and the ring of differential operators on the flag variety with respect to $\mathcal{E}.$