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On the representation theory of GL(n) over a p-adic division algebra and unitarity in the Jacquet-Langlands correspondences

Let F be a p-adic field of characteristic 0, and let A be an F- central division algebra of dimension d over F. In the paper, first are developed some parts of the representation theory of GL(m, A), assuming that holds the conjecture which claims that unitary parabolic induction is irreducible for GL(m, A)'s. Among others, the formula for characters of irreducible unitary representations of GL(m, A) is obtained in terms of standard characters. The Jacquet-Langlands correspondence on the level of Grothendieck groups of GL(pd, F) and GL(p, A) is then studied. Using the above character formula, an explicit formulas for the Jacquet-Langlands correspondence of irreducible unitary representations of GL(n, F) are obtained (assuming the conjecture to hold). As a consequence, it is shown that Jacquet-Langlands correspondence sends irreducible unitary representations of GL(n, F) either to zero, or to the irreducible unitary representations, up to a sign (assuming the conjecture).

general linear groups; p-adic fields; division algebras; irreducible unitary representations; Jacquet-Langlands correspondences

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