On Hodge--Witt cohomology of Drinfeld's upper half space over a finite field

In this dissertation we study the Hodge-Witt cohomology of the $d$-dimensional Drinfeld’s upper half space $\mathcal{X}\subset {\mathbb{P}}_k^d$ over a finite field $k$. We consider the natural action of the $k$-rational points $G$ of the linear group ${\mathrm{G}\mathrm{L}}_{d+1}$ on $H^0(\mathcal{X},{\mathrm{W}}_n{\mathrm{\Omega }}_{{\mathbb{P}}_k^d}^i)$, making them natural ${\mathrm{W}}_n(k)[G]$-modules. To study these representations, we introduce a theory of differential operators over the Witt vectors for smooth $k$-schemes $X$, through a quasi-coherent sheaf of ${\mathrm{W}}_n(k)$-algebras ${\mathcal{D}}_{{\mathrm{W}}_n(X)}$. We apply this theory to equip suitable local cohomology groups arising from $H^0(\mathcal{X},{\mathrm{W}}_n{\mathcal{O}}_{{\mathbb{P}}_k^d})$ with a $\mathrm{\Gamma }({\mathbb{P}}_k^d,{\mathcal{D}}_{{\mathrm{W}}_n({\mathbb{P}}_k^d)})$-module structure. Those local cohomology groups are naturally modules over some parabolic subgroup of ${\mathrm{G}\mathrm{L}}_{d+1}(k)$, and we prove that they are finitely generated $\mathrm{\Gamma }({\mathbb{P}}_k^d,{\mathcal{D}}_{{\mathrm{W}}_n({\mathbb{P}}_k^d)})$-modules.