Asai cube L-functions and the local Langlands conjecture

Let $F$ be a non-archimedean locally compact field. We study a class of Langlands-Shahidi pairs $({\bf H},{\bf L})$, consisting of a quasi-split connected reductive group $\bf H$ over $F$ and a Levi subgroup $\bf L$ which is closely related to a product of restriction of scalars of ${\rm GL}_1$'s or ${\rm GL}_2$'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 $\bf H$ over $F$ of type $D_4$, with triality corresponding to $E$, and let $\bf L$ be its Levi subgroup with derived group ${\rm Res}_{E/F} {\rm SL}_2$. In this way we obtain Asai cube local factors attached to irreducible smooth representations of ${\rm GL}_2(E)$; we prove that they are Weil-Deligne factors obtained via the local Langlands correspondence for ${\rm GL}_2(E)$ 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.