Endo-class and the Jacquet–Langlands correspondence

Abstract

Let $F$ be a non-archimedean local field. Recently, Broussous, Sécherre, and Stevens extended the notion of an endo-class, introduced by Bushnell and Henniart for ${GL}_N\left(F\right)$ with $N\ge 1$, to an inner form of ${GL}_N\left(F\right)$ over $F$, and conjectured that this endo-class for discrete series representations is preserved by the Jacquet–Langlands correspondence. Explicit realizations of the correspondence are given by Silberger and Zink for level-zero discrete series representations and by Bushnell and Henniart for totally ramified ones. In this paper, we show that these realizations confirm the conjecture.