## Tohoku Mathematical Journal 2017 March SECOND SERIES VOL. 69, NO. 1

 Tohoku Math. J. 69 (2017), 25-33

Title A REMARK ON JACQUET-LANGLANDS CORRESPONDENCE AND INVARIANT $s$

Author Kazutoshi Kariyama

(Received September 5, 2014, revised April 28, 2015)
Abstract. Let $F$ be a non-Archimedean local field, and let $G$ be an inner form of $\mathrm{GL}_N(F)$ with $N \ge 1$. Let $\boldsymbol{\mathrm{JL}}$ be the Jacquet--Langlands correspondence between $\mathrm{GL}_N(F)$ and $G$. In this paper, we compute the invariant $s$ associated with the essentially square-integrable representation $\boldsymbol{\mathrm{JL}}^{-1}(\rho)$ for a cuspidal representation $\rho$ of $G$ by using the recent results of Bushnell and Henniart, and we restate the second part of a theorem given by Deligne, Kazhdan, and Vignéras in terms of the invariant $s$. Moreover, by using the parametric degree, we present a proof of the first part of the theorem.

Mathematics Subject Classification. Primary 22E50.

Key words and phrases. Non-Archimedean local field, central simple algebra, essentially square-integrable representation, Jacquet--Langlands correspondence, simple type, parametric degree.