Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of pAdic Simple Algebras

Let F be a p-adic local field and let Ai× be the unit group of a central simple F-algebra Ai of reduced degree n > 1 (i = 1, 2). Let mathcal{R}2 (Ai ×) denote the set of irreducible discrete series representations of Ai ×. The "Abstract Matching Theorem" asserts the existence of a bijection, the "Jacquet Langlands" map {cal JL}A2A1 : mathcal{R}2 ( A1×) to mathcal{R}2 ( A2 × ) 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}A2,A1 : mathcal{R}0 2 (A1 ×) to mathcal{R}02 (A2×) 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 Ai and is invariant under mathcal{J} mathcal{L}A2,A1 (Theorem 4.1).