WEAK EXPLICIT MATCHING FOR LEVEL ZERO DISCRETE SERIES OF UNIT GROUPS OF p-ADIC SIMPLE ALGEBRAS

Let F be a p-adic local eld and let A i be the unit group of a central simple F-algebra A i of reduced degree n > 1 (i = 1; 2). Let R 2 (A i) denote the set of irreducible discrete series representations of A i. The \Abstract Matching Theorem" asserts the existence of a bijection, the \Jacquet-Langlands" map,A 2) 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) is a bijection of level zero discrete series (Proposition 3.2) and we give a parameterization of the set of unramiied twist classes of level zero discrete series which does not depend upon the algebra A i and is invariant under JL A 2 ;A 1 (Theorem 4.1). x0. Introduction. This paper is the third of a series of papers ((SZ], GSZ]) in which the authors are working toward an explicit description of the Jacquet-Langlands correspondence for the level zero case. For the proofs of this paper we depend upon the Abstract Matching Theorem (AMT) (see x0.3), so our results are of the nature, \If AMT is true, then the correspondence has to be this correspondence." x0.1 Some Structure and Notation. This section will be used throughout and is presented here for easy reference. Our notation will be consistent with that of GSZ]. Let F be a p-adic local eld and n > 1 an integer. For d 1 let D := D d denote a central F-division algebra of index d and let A := M m (D) a central simple F-algebra of reduced degree n := dm. Let O := O D denote the ring of integers of D, $ := $ D a prime element of O such that $ d = $ F , and let p := p D = $O be the prime ideal of O. We x a maximal unramiied eld extension F d D which is normalized by $. The residual eld k D := O=p is of order q d and may be identiied with k d , the residual eld of F d. More generally, for`1 we write F ` for an unramiied extension of F of degreè and k ` for a nite eld extension of k of degreè. We write X (k `) …