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

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 28 p.

MSC 2000

22E50 Representations of Lie and linear algebraic groups over local fields

11S37 Langlands-Weil conjectures, nonabelian class field theory

Let F be a p-adic local field and let A_i^x be the unit group of a central simple F-algebra A_i of reduced degree n>1 (i=1,2). Let R²(A_i^x) denote the set of irreducible discrete series representations of A_i^x. The ``Abstract Matching Theorem" asserts the existence of a bijection, the ``Jacquet-Langlands" map, JL_A2,A1: R²(A₁^x) -> R²(A₂^x) 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 JL_A2,A1: R₀²(A₁^x) -> R₀²(A₂^x) 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 JL_A2,A1 (Theorem 4.1).

Keywords:Representations, p-adic groups, Langland's functionality