Invariance of an endo-class under the essentially tame Jacquet-Langlands correspondence

Abstract

Let $F$ be a non-Archimedean local field with a finite residue field. We prove that the conjecture, presented by Broussous, Sécherre, and Stevens, is verified in the essetially tame case, that is, that the Jacquet-Langlands correspondence, which was explicitly described by Bushnell and Henniart, preserves an endo-class for irreducible essentially tame representations of inner forms of $\mathrm{GL}_n(F), n \ge 1$, of parametric degree $n$. Moreover we give explicitly a parameter set for such representations of an inner form $G$ of $\mathrm{GL}_n(F)$ which contain simple characters belonging to an endo-class.