The local Langlands conjecture for the metaplectic group Mp_{2n}

Let FF be a local field with FCF\neq\mathbb C and char(F)2\operatorname{char}(F)\neq 2, let Mp2nMp_{2n} be the metaplectic group, and let ψ\psi be a nontrivial continuous additive character of FF. Write Irrϵ(Mp2n)\operatorname{Irr}_\epsilon(Mp_{2n}) for the genuine irreducible representations of Mp2nMp_{2n}, and let Par(ΓF×Sp2n)\operatorname{Par}(\mathbf{\Gamma}_F\times\mathbf{Sp}_{2n}) denote the corresponding set of Langlands parameters for the product of the absolute Galois group with the dual group Sp2n\mathbf{Sp}_{2n}. The local Langlands conjecture for Mp2nMp_{2n}. For each nontrivial continuous additive character ψ\psi of FF, there is a finite-to-one parameterization map

Irrϵ(Mp2n)Par(ΓF×Sp2n).\operatorname{Irr}_\epsilon(Mp_{2n})\longrightarrow \operatorname{Par}(\mathbf{\Gamma}_F\times\mathbf{Sp}_{2n}).

This is the expected local Langlands parameterization for genuine representations of the metaplectic group; the source presents it as a well-known conjecture compatible with its construction of the LL-group. The supplied text gives no evidence that the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Martin H. Weissman, “Split metaplectic groups and their L-groups”, arXiv:1108.1413 (2011).

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