On Partial Poincare Series

The theory of Poincare series has played a central role in the theory of automorphic forms and their applications. For the analysis of Fourier coefficients, for example, one deals with a Poincare series formed with functions that have a broad spectral "footprint". For the converse theorem, one would like to make a similar construction but beginning with a function having a small spectral footprint. For such functions, one cannot form a full Poincare series, but only what we call a partial Poincare series. In this note we recall the partial Poincare series on GL(n) (A) that play a role in the converse theorem and show that they are rapidly decreasing automorphic functions on the embedded GL(n-1) (A). It is then the purpose of the converse theorem to determine when these partial Poincare series are actually cuspidal automorphic forms Oil GL(n) (A).