On the local Langlands conejcture $\mathrm{GL}(n)$: the cyclic case


Let $F$ be a locally compact nonarchimedean field, and $p$ its residue characteristic. Let $\overline{F}$ be a separable algebraic closure of $F$, $W_F$ the Weil group of $\overline{F}$ over $F$, $\varepsilon$ a character of $F^\times$ of order $n$, and $\tilde\varepsilon$ the corresponding character of $W_F$. Then there exists a canonical map from the set of equivalence classes $\sigma$ of irreducible degree $n$ representations $W_F$ such that $\tilde\varepsilon \otimes \sigma = \sigma$ to the set of equivalence classes $\pi$ of admissible irreducible supercuspidal representations $\mathrm{GL}(n,F)$ verifying $(\varepsilon\circ \mathrm{det}) \otimes \pi = \pi$. This map is shown to be biejctive and to preserve $\varepsilon$-factors for pairs.


Guy Henniart