The local Gan-Gross-Prasad conjecture for $U times U$ : the non-generic case

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Abstract

In this paper, we investigate the local Gan-Gross-Prasad conjecture for some pair of representations of $U(3)\times U(2)$ 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 associateed Vogan $L$-packets which produces the unique Bessel model of these $L$-parameters. We showed that this is not ture 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 applicaiton of these results, we prove an analog of Ichino-Ikeda local conjecture for some non-tempered case.

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URL

https://arxiv.org/abs/1501.00885

PDF

https://arxiv.org/pdf/1501.00885