Drinfeld discriminant function and Fourier expansion of harmonic cochains

Let $$F_{\infty }={{\mathbb {F}}_q}\left( \!\left( {1/T}\right) \!\right) $$ be the completion of $${\mathbb {F}}_q(T)$$ at 1/T. We develop a theory Fourier expansions for harmonic cochains on edges Bruhat–Tits building $${{\,\textrm{PGL}\,}}_r(F_{\infty })$$ , $$r\ge 2$$ generalizing an earlier construction Gekeler $$r=2$$ . then apply this to study modular units Drinfeld symmetric space $$\Omega ^r$$ over }$$ and cuspidal divisor groups Satake compactifications certain varieties. In particular, we obtain higher dimensional analogue result Ogg classical curves $$X_0(p)$$ prime level.