Double Descent in Classical Groups

TL;DR

This paper introduces a new automorphic descent method called double descent, which constructs inverse images of certain automorphic representations via generalized doubling integrals, advancing the understanding of Langlands functoriality.

Contribution

It presents the double descent method, a novel automorphic descent technique based on recent doubling integrals, for constructing inverse images under Langlands functorial lifts.

Findings

Constructs inverse images of automorphic representations for classical groups.

Extends results to double covers of symplectic groups.

Validates the method using generalized doubling integrals.

Abstract

Let ${\bf A}$ be the ring of adeles of a number field $F$. Given a self-dual irreducible, automorphic, cuspidal representation $\tau$ of $\GL_n(\BA)$, with trivial central characters, we construct its full inverse image under the weak Langlands functorial lift from the appropriate split classical group $G$. We do this by a new automorphic descent method, namely the double descent. This method is derived from the recent generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan \cite{CFGK17}, which represent the standard $L$-functions for $G\times \GL_n$. Our results are valid also for double covers of symplectic groups.