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The Langlands parameter of a simple supercuspidal representation: symplectic groups. (English) Zbl 1458.11165

Summary: Let \(\pi\) be a simple supercuspidal representation of the symplectic group \(\mathrm{Sp}_{2l}(F)\), over a \(p\)-adic field \(F\). In this work, we explicitly compute the Rankin-Selberg \(\gamma \)-factor of rank-1 twists of \(\pi \). We then completely determine the Langlands parameter of \(\pi \), if \(p \ne 2\). In the case that \(F = \mathbb{Q}_2\), we give a conjectural description of the functorial lift of \(\pi \), with which, using a recent work of C. J. Bushnell and G. Henniart [Math. Ann. 358, No. 1–2, 433–463 (2014; Zbl 1304.22021)], one can obtain its Langlands parameter.

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
11F85 \(p\)-adic theory, local fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 1304.22021
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References:

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