On $(\chi,b)$-factors of Cuspidal Automorphic Representations of Unitary Groups I

TL;DR

This paper introduces new period integrals for cuspidal automorphic representations of unitary groups, linking their occurrence to specific global Arthur parameters using theta correspondence, refining existing theories on $L$-functions and automorphic forms.

Contribution

It develops a new family of period integrals for unitary groups and relates them to $( ext{chi},b)$-parameters via theta correspondence, advancing the endoscopic classification framework.

Findings

Established a connection between period integrals and $( ext{chi},b)$-parameters.

Refined the understanding of poles of $L$-functions in relation to automorphic representations.

Enhanced the theta correspondence approach in the context of unitary groups.

Abstract

Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations $\sigma$ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur parameter $\psi_\sigma$ associated to $\sigma$, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the $(\chi,b)$-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of certain $L$-functions, which was outlined in [Ral91].