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

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 $G{L}_{d+1}(k)$-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 $\mathcal{E}.$

20 pages