Hecke operators on Drinfeld cusp forms

In this paper, we study the Drinfeld cusp forms for Gamma(1)(T) and Gamma(T) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Gamma(1)(T) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for Gamma(1) (T) of large weights, and not for Gamma(T) even of small weights. The Hecke eigenvalues on cusp forms for Gamma(T) with small weights are determined and the eigenspaces characterized. (C) 2008 Elsevier Inc. All rights reserved.