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

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.