Automorphic distributions for SL(2,R)

Classical (holomorphic) modular forms on the upper half plane have boundary values along the real axis, as do (non-holomorphic) Maass forms. These boundary values are distributions which transform according to a factor of automorphy under the action of the discrete group which leaves the modular forms invariant. The main result of this paper is a characterization of the degree of regularity of such automorphic distributions. The boundary values of Maass forms, for example, are first derivatives of continuous functions, which are Holder continuous of index depending on the eigenvalue corresponding to the Maass form in question. The existence of a continuous antiderivative makes it possible to prove the functional equation for L-functions attached to Maass forms by integration along the real axis, rather than the imaginary axis, as in the usual argument. This new approach to the functional equation appears likely to generalize to higher rank groups that are beyond the reach of the classical arguments.