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Tohoku Mathematical Journal
2017
March
SECOND SERIES VOL. 69, NO. 1
Tohoku Math. J.
69 (2017), 2533

Title
A REMARK ON JACQUETLANGLANDS CORRESPONDENCE AND INVARIANT $s$
Author
Kazutoshi Kariyama
(Received September 5, 2014, revised April 28, 2015) 
Abstract.
Let $F$ be a nonArchimedean local field, and let $G$ be an inner form of $\mathrm{GL}_N(F)$ with $N \ge 1$. Let $\boldsymbol{\mathrm{JL}}$ be the JacquetLanglands correspondence between $\mathrm{GL}_N(F)$ and $G$. In this paper, we compute the invariant $s$ associated with the essentially squareintegrable 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.
NonArchimedean local field, central simple algebra, essentially squareintegrable representation, JacquetLanglands correspondence, simple type, parametric degree.


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