Classification of Very Cuspidal Representations of $\mathrm{GL}_m(D)$

Abstract

Let $\mathrm{F}$ be a non-Archimedean local field with finite residue field. In this paper, we generalize a classification of {\it generic} elements of the general linear group $\mathrm{GL}_n(\mathrm{F})$, $n \geqq 1$ that generate fields of degree $n$ over $\mathrm{F}$ and are {\it minimal} over $\mathrm{F}$, which was given by Hijikata, to an inner form $\mathrm{G}$ of $\mathrm{GL}_n(\mathrm{F})$, and by using the results of Dott [13], we classify supercuspidal representations of $\mathrm{G}$ that are induced from {\it very cuspidal} representations of maximal compact mod center, open subgroups, which are defined in terms of generic elements. This classification generalizes that of {\it epipelagic} supercuspidal representations of $\mathrm{G}$ which was given by Bushnell and Henniart for $\mathrm{GL}_n(\mathrm{F})$ and by Imai and Tsushima for $\mathrm{G}$.