  For a local non-Archimedean field $K$ we construct ${\rm
GL}_{d+1}(K)$-equivariant coherent sheaves ${\mathcal V}_{{\mathcal O}_K}$ on
the formal ${\mathcal O}_K$-scheme ${\mathfrak X}$ underlying the symmetric
space $X$ over $K$ of dimension $d$. These ${\mathcal V}_{{\mathcal O}_K}$ are
${\mathcal O}_K$-lattices in (the sheaf version of) the holomorphic discrete
series representations (in $K$-vector spaces) of ${\rm GL}_{d+1}(K)$ as defined
by P. Schneider \cite{schn}. We prove that the cohomology $H^t({\mathfrak
X},{\mathcal V}_{{\mathcal O}_K})$ vanishes for $t>0$, for ${\mathcal
V}_{{\mathcal O}_K}$ in a certain subclass. The proof is related to the other
main topic of this paper: over a finite field $k$, the study of the cohomology
of vector bundles on the natural normal crossings compactification $Y$ of the
Deligne-Lusztig variety $Y^0$ for ${\rm GL}_{d+1}/k$ (so $Y^0$ is the open
subscheme of ${\mathbb P}_k^d$ obtained by deleting all its $k$-rational
hyperplanes).
