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

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 $U(n)$.