A method for computing general automorphic forms on general groups

The purpose of this note is to describe a method for computing general automorphic forms. I have carried out only limited computational tests so far, and have not discovered any new automorphic forms using it. However, the method does identify some lifted cusp forms on GL(3) and until recently was the only general method to compute an automorphic form on a higher rank group. It generalizes the methods of Hejhal (e.g. [7]) and others for Maass forms on GL(2), but does not require Whittaker or Bessel functions. Aside from trying to compute some of the same objects, this method is otherwise unrelated to the cohomological methods developed by Ash and others which use geometric data to compute special types of automorphic forms. I shall begin with a highbrow version of the method, and later explain its concrete manifestations and how they relate to existing methods. Every cuspidal automorphic representation for Γ\G, where G is a split reductive Lie group, has associated automorphic distributions as in [10]. These are viewed as objects in C(N), the distributions on the maximal unipotent subgroup N of G. Furthermore, they satisfy invariance properties, in particular under NΓ = Γ ∩ N . Using Fourier expansions on L (NΓ\N) and C (NΓ\N) (which can be computed, for example, using the method of co-adjoint nilpotent orbits), the distribution can be broken into several natural components. The invariance under the rest of the group Γ then provides many nontrivial