A remark on the simple cuspidal representations of GL(n, F)

Abstract

Let $F$ be a non-archimedean local field of residue characteristic $p$, $G$ be the group $GL(n, F)$. In this note, under the assumption $(n, p)=1$, we show a simple cuspidal representation $\pi$ (that with normalized level $\frac{1}{n}$) of $G$ is determined uniquely up to isomorphism by the local constants of $\chi\circ \text{det}\otimes \pi$ for all characters $\chi$ of $F^\times$.