SIMPLE SUPERCUSPIDAL REPRESENTATIONS OF $\operatorname{GL}(n)$

Abstract

Following a construction of Gross and Reeder, we define simple supercuspidal representations of $\operatorname{GL}(n)$ over a $\mathfrak{p}$-adic field. We show that they have conductor $\mathfrak{p}^{n+1}$. We then give a general formula for the matrix coefficient attached to a new vector, and make it completely explicit when $n=2$.