Fix a finite field k and let X be Drinfeld's Upper Half Space over k of dimension n. Let S be the polynomial ring over k in n+1 variables, let G be the algebraic k-group scheme associated with the general linear group of degree n+1 and let Y=Proj S. Fix an algebraic action of G on Y.
In the first part of this thesis, the global sections F(X) on X of a G-equivariant vector bundle F on Y are considered as a G(k)-representation. Due to a theorem of Orlik (2008), the computation of F(X) essentially reduces to the computation of the local cohomologies of F on Y with support in a k-rational closed subvariety Z. This local cohomology is considered as a representation of a certain parabolic subgroup P of G which stabilizes Z resp. of its Levi subgroup L. Three types of descriptions of these local cohomology modules are given:
In the case where F arises from a representation of the stabilizer of a chosen base point of Y, a very general result on the structure of the local cohomology of F on Y with support in Z as an L-module is translated from Orlik's results. For general F, an even coarser strucutre result is proved by using an affine projection of Y onto Y-Z. In particular, when F equals the structure sheaf or a sheaf of differential forms on Y (resp. a Serre twist of either sheaf), this result is made very precise.
In the case where the graded S-module associated with F is generated in degrees <2, a higher divided power version of the distribution algebra of G is used to for a more conceptual approach to the description of the local cohomologies mentioned in terms of the unipotent radical of P.
In the second part of this thesis, the rigid cohomology of X is computed in two ways.
The first method proceeds by computation of the rigid cohomology of the complement of X in Y (which is projective itself, thus its rigid cohomology is simply the de Rham cohomology of an associated rigid-analytic tube). Then application of the associated long exact sequence for rigid cohomology with proper supports yields the rigid cohomology of X.
The second method proceeds by direct computation of the direct limit of the de Rham cohomologies of a certain cofinal family of strict open neighborhoods of the tube of X in the ambient rigid-analytic projective space.
The resulting cohomology formula has been known since 2007, when Große-Klönne proved that it is the same as the one obtained from l-adic cohomology using different methods.