Let F be a finite extension of the p-adic numbers with valuation ring O_F. The Drinfeld upper half plane is a non-archimedean analogue of the complex upper half plane. It is equipped with a natural action of GL2(F) and has become ubiquitous in the study of the representation theory of this group. After introducing this space, in this talk I will use Deligne-Lusztig theory to motivate why we might expect studying equivariant vector bundles with connection on affinoid subdomains of the Drinfeld upper half plane to be helpful for exhibiting p-adic geometric realisations of the irreducible, smooth (complex) representations of GL2(O_F). I will then outline some recent work to this end, with a focus on some concrete examples.
