Asai cube L-functions and the local Langlands correspondence

Let F be a non-archimedean locally compact field. We study a class of Langlands-Shahidi pairs (H,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 GL₁'s or GL₂'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 D₄, with triality corresponding to E, and let L be its Levi subgroup with derived group Res_E/FSL₂. In this way we obtain Asai cube local factors attached to irreducible smooth representations of GL₂(E); we prove that they are Weil-Deligne factors obtained via the local Langlands correspondence for GL₂(E) and tensor induction from E to F. A consequence is that Asai cube γ- and ε-factors become stable under twists by highly ramified characters.