Invertible functions on non-archimedean symmetric spaces

Let $u$ be a nowhere vanishing holomorphic function on the Drinfeld space ${\mathrm{\Omega }}^r$ of dimension $r-1$, where $r\ge 2$. The logarithm ${\log }_q\mid u\mid$ of its absolute value may be regarded as an affine function on the attached Bruhat-Tits building ${\mathcal{B}\mathcal{T}}^r$. Generalizing a construction of van der Put in case $r=2$, we relate the group $\mathcal{O}({\mathrm{\Omega }}^r{)}^{\ast }$ of such $u$ with the group $\mathbf{H}({\mathcal{B}\mathcal{T}}^r,\mathbb{Z})$ of integer-valued harmonic 1-cochains on ${\mathcal{B}\mathcal{T}}^r$. This also gives rise to a natural $\mathbb{Z}$-structure on the first ($\mathrm{\ell }$-adic or de Rham) cohomology of ${\mathrm{\Omega }}^r$.