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

TL;DR

This paper provides a linear constraint-based criterion to determine non-zero parameters in M{\

Contribution

It introduces a new linear system criterion for non-zero conditions in M{\

Findings

The criterion matches the p-adic case formulation.

It offers a systematic way to identify non-zero A-packets.

Suggests a correspondence between real and p-adic A-packets.

Abstract

M{\oe}glin-Renard parametrized A-packet of unitary group through cohomological induction in good parity case. Each parameter gives rise to an $A_{\mathfrak q}(\lambda)$ which is either $0$ or irreducible. Trapa proposed an algorithm to determine whether a ``mediocre'' $A_{\mathfrak q}(\lambda)$ of $\mathrm U(p, q)$ is non-zero. Based on his result, we present a further understanding of the non-zero condition on M{\oe}glin-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.