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

Abstract

In this paper, we investigate the local Gan-Gross-Prasad conjecture for some pair of representations of U(3) x U(2) involving a non-generic representation. For a pair of generic L-parameters of (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)) 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. (C) 2016 Elsevier Inc. All rights reserved