On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms

Abstract

We study the diagonalizability of the Atkin $U$-operator acting on Drinfeld cusp forms for $\Gamma_1(t)$ and $\Gamma(t)$ using Teitelbaum's interpretation as harmonic cocycles. We prove $U$ is diagonalizable for small weights and explicitly compute the eigenvalues. We also formulate a conjecture, supported by numerical search and proofs in some special cases, about non diagonalizability of $U$ in even characteristic.