On the Fourier-Jacobi model for some endoscopic Arthur packet of $U(3) \times U(3)$ : the non-generic case

TL;DR

This paper investigates the Fourier-Jacobi model for non-generic L-parameters of U(3)×U(3), confirming the conjecture in some cases and providing explicit representations when it holds.

Contribution

It extends the understanding of the Fourier-Jacobi model to certain non-generic cases of U(3)×U(3) and clarifies when the conjecture is valid.

Findings

The conjecture holds for some non-generic L-parameters.

The conjecture fails for other non-generic L-parameters.

Explicit representations are identified when the conjecture holds.

Abstract

For a generic $L$-parameter of $U(n)\times U(n)$, it is conjectured that there is a unique representation in their associated relevant Vogan $L$-packet which produces the unique Fourier-Jacobi model. We investigated this conjecture for some non-generic $L$-parameters of $U(3)\times U(3)$ and discovered that this conjecture is still true for some non-generic $L$-parameter and false for some non-generic $L$-parameter. In the case when it holds, we specified such representation under the local Langlands correspondence for unitary group.