  For a local field $F$ we consider tamely ramified principal series
representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite
extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in
$G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke
algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module
$V^{I_0}$ admits an integral structure, then such an integral structure can be
chosen in a particularly well organized manner, in particular its modular
reduction can be made completely explicit.
