Non-zero condition on Mœglin-Renard's parametrization for Arthur packets of $\mathrm{U}(p,q)$

Abstract

Mœglin-Renard parametrized A-packet of unitary group through cohomological induction in good parity case. Each parameter gives rise to an $A_{\mathfrak{q}}(\hat{I}»)$ which is either $0$ or irreducible. Trapa proposed an algorithm to determine whether a ``mediocre'' $A_{\mathfrak{q}}(\hat{I}»)$ of $\mathrm{U}(p,q)$ is non-zero. Based on his result, we present a further understanding of the non-zero condition on Mœglin-Renard's parametrization. Our criterion comes out to be a system of linear constraints, and has the same formulation as $p$-adic case. This suggests a map from A-packets of real unitary group to A-packets of $p$-adic symplectic group or special orthogonal group.