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

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arxiv 2409.09358 v2 pith:XHGCAGCS submitted 2024-09-14 math.RT

classification math.RT
keywords groupglin-renardnon-zeroa-packetsadiccaseconditionlambda
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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.

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Cited by 1 Pith paper

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  1. On $A$-parameters containing unitary lowest weight representations of $\mathrm{U}(p, q)$

    math.RT 2025-02 conditional novelty 7.0 of 10

    Every Arthur packet of U(p,q) contains at most one unitary lowest weight representation, with explicit conditions for existence and a formula for the lowest K-type.

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