ON REPRESENTATION THEORY OF GL(n) OVER p-ADIC DIVISION ALGEBRA AND UNITARITY IN JACQUET-LANGLANDS CORRESPONDENCE

Let F be a p-adic field of characteristic 0, and let A be an F -central division algebra of dimension dA over F . In the paper we first develop 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, 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(pdA, F ) and GL(p,A). Using the above character formula, we get explicite formulas for the JacquetLanglands correspondence of irreducible unitary representations of GL(n, F ) (assuming the conjecture to hold). As a consequence, we get 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. Introduction A key aspect of Langlands program is functoriality ([L]). One of the first examples of functoriality which were studied in the local case, was the connection between representations of various inner forms of GL(n) (see [KnRg]). The first example was studied already in [JL], the connection of irreducible representations of GL(2) over a local field F and irreducible representations of the multiplicative group of the quaternion algebra over F (the L groups of these two groups are both GL(2,C) × Gal(F̄ /F ), and the functoriality considered here corresponds to the identity mapping). Let F be a p-adic field of characteristic 0 and let A be an F -central division algebra of rank dA over F . P. Deligne, D. Kazhdan and M.-F. Vigneras established bijections LJpdA between irreducible essentially square integrable representations of groups GL(pdA, F ) and GL(p,A). The crucial requirement which holds for these bijection, and which characterizes them uniquely, is that characters Θδ and ΘLJpdA (δ) of representations δ and LJpdA(δ) satisfy (−1)Θδ(g) = (−1)pΘLJpdA (δ)(g ′) 1991 Mathematics Subject Classification. Primary 22E50, 22E35, 11F70, 11S37.