New models for the action of Hecke operators in spaces of Maass wave forms

Abstract

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_{\lambda}(N)$ of Maass forms with eigenvalue $1/4-\lambda^2$ on a congruence subgroup $\Gamma_1(N)$. We introduce the field $F_{\lambda} = {\mathbb Q} (\lambda ,\sqrt{n}, n^{\lambda /2} \mid \~n\in {\mathbb N})$ so that $F_{\lambda}$ consists entirely of algebraic numbers if $\lambda = 0$. The main result of the paper is the following. For a packet $\Phi = (\nu_p \mid p\nmid N)$ of Hecke eigenvalues occurring in $M_{\lambda}(N)$ we then have that either every $\nu_p$ is algebraic over $F_{\lambda}$, or else $\Phi$ will - for some $m\in {\mathbb N}$ - occur in the first cohomology of a certain space $W_{\lambda,m}$ which is a space of continuous functions on the unit circle with an action of $\mathrm{SL}_2({\mathbb R})$ well-known from the theory of (non-unitary) principal representations of $\mathrm{SL}_2({\mathbb R})$.