Tohoku Mathematical Journal, Second Series
Online ISSN : 2186-585X
Print ISSN : 0040-8735
ISSN-L : 0040-8735
A REMARK ON JACQUET–LANGLANDS CORRESPONDENCE AND INVARIANT $s$
Kazutoshi Kariyama
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2017 Volume 69 Issue 1 Pages 25-33

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Abstract

Let $F$ be a non-Archimedean local field, and let $G$ be an inner form of GL$_N (F)$ with $N \ge 1$. Let $\mathbf{JL}$ be the Jacquet–Langlands correspondence between GL$_N (F)$ and $G$. In this paper, we compute the invariant $s$ associated with the essentially square-integrable representation $\mathbf{JL}^{-1}(\rho)$ 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.

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