The square-integrability of double descent

TL;DR

This paper proves that functions constructed via the double descent method in automorphic representation theory are always square-integrable, confirming a key property of these functions.

Contribution

It establishes the square-integrability of functions obtained through double descent, a significant step in understanding their analytical properties.

Findings

Functions from double descent are always square-integrable.

Supports the analytical foundation of double descent in automorphic forms.

Enhances understanding of the spectral properties of automorphic representations.

Abstract

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.