Asai cube L-functions an the local Langlands conjecture

Resumen

We work over non-Archimedean local fields F of characteristic 0 as well as characteristic p. Let E/F be a separable cubic extension and let H be an ambient group of type D4, which has triality corresponding to E. We choose a maximal Levi subgroup M isogenous to G=GL2(E). The Langlands-Shahidi method applied to (H,M) attaches an Asai cube γ-factor to an irreducible smooth generic representation π of G. If σ is the Weil-Deligne representation corresponding to π via local Langlands, we prove that the Asai cube γ-factor of π is the γ-factor of the Weil-Deligne representation obtained from σ via cubic tensor induction from E to F. A consequence is that Asai cube γ-factors become stable under twists by highly ramified characters.