Representation theory of GL(n) over a p-adic division algebra and unitarity in the Jacquet–Langlands correspondence

Marko Tadić

Vol. 223 (2006), No. 1, 167–200

DOI: 10.2140/pjm.2006.223.167

Abstract

Let F be a local nonarchimedean field of characteristic 0, and let A be an F-central division algebra of dimension d_A over F. In this paper, we first develop some parts of the representation theory of GL(m,A), assuming the conjecture that unitary parabolic induction is irreducible for GL(m,A)’s. Among others, we obtain the formula for characters of irreducible unitary representations of GL(m,A) in terms of standard characters. Then we study the Jacquet–Langlands correspondence on the level of Grothendieck groups of GL(pd_A,F) and GL(p,A). Using this character formula, we get explicit formulas for the Jacquet–Langlands correspondence of irreducible unitary representations of GL(n,F) (assuming the conjecture to hold). As a consequence, we get that the Jacquet–Langlands correspondence sends irreducible unitary representations of GL(n,F) either to zero or to irreducible unitary representations, up to a sign.

Keywords

local nonarchimedean field, division algebra, general linear group, Jacquet–Langlands correspodence, functoriality

Mathematical Subject Classification 2000

Primary: 22E50, 22E35, 11F70, 11S37

Milestones

Received: 19 March 2004

Accepted: 24 July 2004

Published: 1 January 2006