Hodge-type decomposition for de Rham cohomology of $p$-adically uniformized varieties

We prove a Hodge-type decomposition for the de-Rham cohomology of $p$-adically uniformized varieties by the product of Drinfeld’s symmetric spaces. It is based on work of Schneider, Stuhler, Iovita and Spiess on the cohomology of Drinfeld’s symmetric space.