The local Gan-Gross-Prasad conjecture for $U(n+1) \times U(n)$ : a non-generic case

TL;DR

This paper investigates the local Gan-Gross-Prasad conjecture for unitary groups in a non-generic setting, revealing that Gan-Gross-Prasad type formulas still exist depending on specific L-parameters.

Contribution

It extends the Gan-Gross-Prasad conjecture analysis to non-generic L-parameters, showing the persistence of related formulas in this broader context.

Findings

Existence of Gan-Gross-Prasad type formulas for non-generic parameters

Dependence of formulas on the choice of L-parameters

Extension of conjecture analysis beyond generic cases

Abstract

The local Gan-Gross-Prasad conjecture of unitary groups, which is now settled by the works of Plessis, Gan and Ichino, says that for a pair of generic $L$-parameters of $(U(n+1),U(n))$, there is a unique pair of representations in their associated Vogan $L$-packets which produces the Bessel model. In this paper, we examined the conjecture for a pair of $L$-parameters of $\big(U(n+1), U(n)\big)$ as fixing a special non-generic parameter of $U(n+1)$ and varing tempered $L$-parameters of $U(n)$. We observed that there still exist a Gan-Gross-Prasad type formulae depending on the choice of $L$-parameter of $U(n)$.