The square-integrability of double descent

Double descent is a method to construct automorphic representations of classical groups. For given A-parameter $\psi$ with certain good properties, double descent constructs a space of functions orthogonal to any cuspidal representation whose A-parameter is not $\psi$ and not orthogonal to any cuspidal representation with A-parameter $\psi$. In this paper, we show that functions constructed by double descent are always square-integrable.