Multiplicity formula for induced representations: Bessel and Fourier-Jacobi models over Archimedean local fields

TL;DR

This paper establishes a multiplicity formula for induced representations in Bessel and Fourier-Jacobi models over Archimedean fields, extending previous methods and providing a uniform proof of the local Gan-Gross-Prasad conjecture for classical groups.

Contribution

It generalizes the multiplicity formula approach to Archimedean fields and applies it to prove the local Gan-Gross-Prasad conjecture uniformly for all classical groups.

Findings

Proved a multiplicity formula for induced representations in Bessel and Fourier-Jacobi models over Archimedean fields.

Provided a uniform proof of the local Gan-Gross-Prasad conjecture for all classical groups.

Extended the approach of Moeglin and Waldspurger to Archimedean local fields.

Abstract

This article proves a formula relating the multiplicity of an induced representation and that of the inducing datum for the Bessel and the Fourier-Jacobi models over Archimedean local fields by generalizing the approach of C. Moeglin and J.-L. Waldspurger in [MW12], which was successful for Bessel models of special orthogonal groups over non-Archimedean local fields. As an application, we give a uniform proof of the local Gan-Gross-Prasad conjecture for all classical groups over Archimedean local fields for generic local $L$-parameters based on the tempered basic cases.