Cohomology of discrete groups in harmonic cochains on buildings

Modules of harmonic cochains on the Bruhat-Tits building of the projective general linear group over a p-adic field were defined by one of the authors, and were shown to represent the cohomology of Drinfel'd's p-adic symmetric domain. Here we define certain non-trivial natural extensions of these modules and study their properties. In particular, for a quotient of Drinfel'd's space by a discrete cocompact group, we are able to define maps between consecutive graded pieces of its de Rham cohomology, which we show to be (essentially) isomorphisms. We believe that these maps are graded versions of the Hyodo-Kato monodromy operator N.