Diagonalizing Eisenstein Series. I

Abstract

In this paper we consider the action of Hecke operators T _n (n ∈ IN), and their adjoint operators T* _n , on Eisenstein series belonging to the group Γ₀(N) and having integral weight k > 2 and arbitrary character χ modulo N. It is shown that the space ɛ _k (x) spanned by these Eisenstein series splits up into a number of subspaces ɛ _k (x,t)> where t is a divisor of N, each being invariant under the operators T _n and T* _n with (n, N) = 1. If x is a primitive character modulo N, this holds also for T _n with (n, N) > 1, but this need not be true for general x modulo N. A basis of modular forms that are eigenfunctions for T _n with (n, N) = 1 is constructed for each appropriate t and explicit evaluations of G _L \T _n are given for each Eisenstein series G _L (L ∈ Γ(l)) and any positive integer n prime to N, or any n that is a prime divisor of N, the results being particularly simple when N is squarefree. The corresponding results for G _L \T* _n when (n, N) > 1 will be given in a subsequent paper.