Invertible functions on nonarchimedean symmetric spaces

Let u be a nowhere vanishing holomorphic function on the Drinfeld space Omega(r) of dimension r - 1, where r >= 2. The logarithm log(q)vertical bar u vertical bar of its absolute value may be regarded as an affine function on the attached Bruhat-Tits building BTr. Generalizing a construction of van der Put in case r = 2, we relate the group O(Omega(r))* of such u with the group H(BTr, Z) of integer-valued harmonic 1-cochains on BTr. This also gives rise to a natural Z-structure on the first (l-adic or de Rham) cohomology of Omega(r),