Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences

TL;DR

This paper develops a general framework for automorphic integral transforms related to endoscopy correspondences in classical groups, based on Arthur's classification, aiming to unify and extend existing theories of L-functions and theta correspondence.

Contribution

It introduces the $( au,b)$-theory, a new principle to organize and expand the study of automorphic forms, L-functions, and endoscopy for classical groups.

Findings

Framework connects automorphic integral transforms with endoscopy.

Reorganizes and extends prior work on L-functions and theta correspondence.

Proposes the $( au,b)$-theory to unify classical automorphic theories.

Abstract

A general framework of constructions of endoscopy correspondences via automorphic integral transforms for classical groups is formulated in terms of the Arthur classification of the discrete spectrum of square-integrable automorphic forms. This suggests a principle, which is called the $(\tau,b)$-theory of automorphic forms of classical groups, to reorganize and extend the series of work of Piatetski-Shapiro, Rallis, Kudla and others on standard $L$-functions of classical groups and theta correspondence.