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Asai cube \(L\)-functions and the local Langlands correspondence. (English) Zbl 1506.11073

Summary: Let \(F\) be a non-archimedean locally compact field. We study a class of Langlands-Shahidi pairs \((\mathbf{H},\mathbf{L})\), consisting of a quasi-split connected reductive group H over \(F\) and a Levi subgroup L which is closely related to a product of restriction of scalars of \(\mathrm{GL}_1\)’s or \(\mathrm{GL}_2\)’s. We prove the compatibility of the resulting local factors with the Langlands correspondence. In particular, let \(E\) be a cubic separable extension of \(F\). We consider a simply connected quasi-split semisimple group H over \(F\) of type \(D_4\), with triality corresponding to \(E\), and let L be its Levi subgroup with derived group \(\mathrm{Res}_{E / F} \mathrm{SL}_2\). In this way we obtain Asai cube local factors attached to irreducible smooth representations of \(\mathrm{GL}_2(E)\); we prove that they are Weil-Deligne factors obtained via the local Langlands correspondence for \(\mathrm{GL}_2(E)\) and tensor induction from \(E\) to \(F\). A consequence is that Asai cube \(\gamma\)- and \(\varepsilon\)-factors become stable under twists by highly ramified characters.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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