  We study the diagonalizability of the Atkin $U_t$-operator acting on Drinfeld
cusp forms for $\Gamma_0(t)$: starting with the slopes of eigenvalues and then
moving to the space of cusp forms for $\Gamma_1(t)$ to use Teitelbaum's
interpretation as harmonic cocycles which makes computations more explicit. We
prove $U_t$ is diagonalizable in odd characteristic for (relatively) small
weights and explicitly compute the eigenvalues. In even characteristic we show
that it is not diagonalizable when the weight is odd (except for the trivial
cases) and prove some cases of non diagonalizability in even weight as well. We
also formulate a few conjectures, supported by numerical search, about
diagonalizability of $U_t$ and the slopes of its eigenforms.
