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