On tensor third $L$-functions of automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$

TL;DR

This paper investigates the properties of tensor third $L$-functions associated with automorphic representations of $ ext{GL}_n$ over global fields, motivated by Langlands' beyond endoscopy approach to functoriality.

Contribution

It explores the representation-theoretic aspects of tensor third $L$-functions in the context of automorphic forms, providing insights relevant to Langlands' functoriality conjectures.

Findings

Analysis of the structure of tensor third $L$-functions.

Connections to Langlands' beyond endoscopy proposal.

Foundational results for future functoriality proofs.

Abstract

Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$ where $F$ is a global field.