On twisted lifting

Abstract

If $\sigma$ is a generator of the galois group of a finite cyclic extension $E/F$ of local or global fields, and $\varepsilon$ is a character of ${C_E}( = {E^ \times }\;{\text {or}}\;{E^ \times }\backslash {{\mathbf {A}}^ \times })$ whose restriction to ${C_F}$ has order $n$, then the irreducible admissible or automorphic representations $\pi$ of ${\text {GL}}(n)$ over $E$ with $^\sigma \pi \cong \pi \otimes \varepsilon$ are determined.