On the Atkin $U_t$-operator for ${\mathrm{\Gamma }}_1(t)$-invariant Drinfeld cusp forms

We study the diagonalizability of the Atkin $U_t$-operator acting on Drinfeld cusp forms for ${\mathrm{\Gamma }}_1(t)$ and $\mathrm{\Gamma }(t)$ using Teitelbaum’s interpretation as harmonic cocycles. For small weights $k⩽2q$, we prove $U_t$ is diagonalizable in odd characteristic and we point out that non diagonalizability in even characteristic depends on antidiagonal blocks.