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 $\mathcal{ E}.$
Categories: math.AG math.NT math.RT