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

  Published:2003-04-01
 Printed: Apr 2003
  • Allan J. Silberger
  • Ernst-Wilhelm Zink
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Abstract

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 $\mathcal{R}^2 (A_i^\times)$ 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, $\mathcal{J} \mathcal{L}_{A_2,A_1} \colon \mathcal{R}^2 (A_1^\times) \to \mathcal{R}^2 (A_2^\times)$ which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map $\mathcal{J} \mathcal{L}$, but only for ``level zero'' representations. We prove that the restriction $\mathcal{J} \mathcal{L}_{A_2,A_1} \colon \mathcal{R}_0^2 (A_1^\times) \to \mathcal{R}_0^2 (A_2^\times)$ 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 $\mathcal{J} \mathcal{L}_{A_2,A_1}$ (Theorem~4.1).
MSC Classifications: 22E50, 11R39 show english descriptions Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]
22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05]
11R39 - Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]
 

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