Doubling Constructions and Tensor Product ${L}$-Functions: the linear case

TL;DR

This paper introduces a new integral representation for tensor product L-functions involving classical and general linear groups, applicable to all cuspidal representations without requiring genericity, using innovative models and inducing data.

Contribution

It provides a uniform construction for tensor product L-functions across all classical groups, utilizing generalized Speh representations and a novel global and local model.

Findings

First integral representation for tensor product L-functions of classical and GL groups

Applicable to all cuspidal representations, not requiring genericity

Introduces generalized Speh representations and new models

Abstract

We present an integral representation for the tensor product $L$-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. This is the first in a series of papers, treating symplectic and even orthogonal groups. Subsequent papers (in preparation) will treat odd orthogonal and general spin groups, the metaplectic covering version of these integrals, and applications to functoriality coming from combining this work with the converse theorem (and independent of the trace formula).