For reductive algebraic group defined over a finite field, we can associate Deligne-Lusztig varieties, coming with a natural action of the (finite) group of rational points of G, such that this action on the cohomology of the varieties realizes the various representations of said finite group. (See my notes for an exposition of a basic example)
This construction can be generalized to find varieties (or limits of varieties) that produce interesting representations of a parahoric subgroup of a reductive group defined over a local field. My research focuses on understanding the geometry of these varieties. To that end, I'm interested in results and techniques from the study of schemes over finite fields