The local Gan-Gross-Prasad conjecture for $U(3) \times U(2)$ : the non-generic case

TL;DR

This paper explores the local Gan-Gross-Prasad conjecture for $U(3) imes U(2)$ involving non-generic representations, revealing deviations from known generic cases and establishing new local theta correspondence results.

Contribution

It demonstrates that the uniqueness of Bessel models fails for certain non-generic $L$-parameters and provides a detailed local theta correspondence for specific unitary groups.

Findings

Uniqueness of Bessel models does not hold for some non-generic $L$-parameters.

Established the local theta correspondence for $(U(1),U(3))$ at the individual representation level.

Proved an analog of the Ichino-Ikeda conjecture in a non-tempered setting.

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.