On the Rapid Decay of Cuspidal Automorphic Forms

Abstract

Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets. We also state and prove the decay for smooth but not necessarily K-finite cusp forms. We shall use this to establish new results about the analytic continuation of L-functions. Our decay statements also close a gap in the literature on L-functions.