  Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's
symmetric space of dimension $d$ over $K$. Let $\Gamma\subset {\rm
SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let
${{X}}_{\Gamma}=\Gamma\backslash {X}$, a smooth projective ${{K}}$-variety. In
this paper we investigate the de Rham and log crystalline (log convergent)
cohomology of local systems on $X_{\Gamma}$ arising from $K[\Gamma]$-modules.
  (I) We prove the monodromy weight conjecture in this context. To do so we
work out, for a general strictly semistable proper scheme of pure relative
dimension $d$ over a cdvr of mixed characteristic, a rigid analytic description
of the $d$-fold iterate of the monodromy operator acting on de Rham cohomology.
  (II) In cases of arithmetical interest we prove the (weak) admissibility of
this cohomology (as a filtered $(\phi,N)$-module) and the degeneration of the
relevant Hodge spectral sequence.
