The upper-bound conjecture for maximal Fourier coefficients

Let GG be an FF-quasisplit classical group, let ψΨ~2(G)\psi\in\widetilde\Psi_2(G) be a global Arthur parameter, and let Π~ψ(ϵψ)\widetilde\Pi_\psi(\epsilon_\psi) be its automorphic L2L^2-packet. Let p(ψ)\underline p(\psi) be the partition attached to (ψ,G(C))(\psi,G^\vee(\mathbb C)). Upper-bound conjecture for maximal Fourier coefficients. For every πΠ~ψ(ϵψ)\pi\in\widetilde\Pi_\psi(\epsilon_\psi), the maximal Fourier-coefficient partition satisfies

nm(π)ηg,g(p(ψ)).\mathfrak n^m(\pi)\leq\eta_{\mathfrak g^\vee,\mathfrak g}(\underline p(\psi)).

The claim relates Fourier coefficients of Arthur-packet members to Barbasch–Vogan duality. The source states that determining the exact partitions remains open, while proving the bound for all parameters is not asserted there.

Sources & referencesView supporting material

Primary source

Dihua Jiang, “Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences”, arXiv:1212.6525 (2012).

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