Asai cube L-functions and the local Langlands conjecture

Abstract

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 $\mathbf{H}$ over $F$ and a Levi subgroup $\mathbf{L}$ 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 $\mathbf{H}$ over $F$ of type $D_4$, with triality corresponding to $E$, and let $\mathbf{L}$ 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 $\epsilon$-factors become stable under twists by highly ramified characters.

References in corpus (1)

• On automorphic L-functions in positive characteristic

Cited by in corpus (1)

• Unitary dual of quasi-split $PGS{O}_8^E$