  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 $\mathcal{E}.$
