Residues on buildings and de Rham cohomology of p-adic symmetric domains

The cohomology of Drinfeld's p-adic symmetric domain was computed by P. Schneider and U. Stuhler in 1991. Here we propose a more explicit and combinatorial approach based on a notion of residue of a closed form along simplices in the Bruhat-Tits building. We identify the cohomology with a certain space of harmonic cochains on the building. We also answer a few questions left open in the original approach.