Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$ -Adic Simple Algebras

Silberger, Allan J. ; Zink, Ernst-Wilhelm

Canadian journal of mathematics, Tome 55 (2003) no. 2, pp. 353-378

Résumé

Let $F$ be a $p$ -adic local field and let $A_{i}^{\times }$ be the unit group of a central simple $F$ -algebra ${{A}_{i}}$ of reduced degree $n\,>\,1\,(i\,=\,1,\,2)$ . Let ${{\mathcal{R}}^{2}}\left( A_{i}^{\times } \right)$ denote the set of irreducible discrete series representations of $A_{i}^{\times }$ . The “Abstract Matching Theorem” asserts the existence of a bijection, the “Jacquet-Langlands” map, $J{{L}_{{{A}_{2}},{{A}_{1}}}}\,:\,{{R}^{2}}\left( A_{1}^{\times } \right)\,\to \,{{R}^{2}}\left( A_{2}^{\times } \right)$ which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map JL, but only for “level zero” representations. We prove that the restriction $J{{L}_{{{A}_{2}},{{A}_{1}}}}\,:\,R_{0}^{2}\left( A_{1}^{\times } \right)\,\to \,R_{0}^{2}\left( A_{2}^{\times } \right)$ is a bijection of level zero discrete series (Proposition 3.2) and we give a parameterization of the set of unramified twist classes of level zero discrete series which does not depend upon the algebra ${{A}_{i}}$ and is invariant under $J{{L}_{{{A}_{2}},{{A}_{1}}}}$ (Theorem 4.1).

Détail

DOI : 10.4153/CJM-2003-016-4

Mots-clés : 22E50, 11R39

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