The local Gan–Gross–Prasad conjecture for U(3)×U(2): The non-generic case

In this paper, we investigate the local Gan–Gross–Prasad conjecture for some pair of representations of U(3)×U(2)U(3)×U(2) involving a non-generic representation. For a pair of generic L  -parameters of (U(n),U(n−1))(U(n),U(n−1)), it is known that there is a unique pair of representations in their associated Vogan L-packets which produces the unique Bessel model of these L-parameters. We showed that this is not true for some pair of L-parameters involving a non-generic one.On the other hand, we give the precise local theta correspondence for (U(1),U(3))(U(1),U(3)) not at the level of L-parameters but of individual representations in the framework of the local Langlands correspondence for unitary group. As an application of these results, we prove an analog of Ichino–Ikeda conjecture for some non-tempered case. The main tools in this work are the see-saw identity, local theta correspondence for (almost) equal rank cases and recent results on the local Gan–Gross–Prasad conjecture both on the Fourier–Jacobi and the Bessel case.