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Irreducibility criterion for representations induced by essentially unitary ones (case of non-archimedean GL(n, A))

The fundamental representations for the description of the unitary dual of general linear groups over a local field F are the Speh representations, which are denoted by u(d, m). Let A be a finite dimensional central division algebra over F. One can define in a similar way representations u(d, m) for general linear groups over A, and they play a similar fundamental role for the unitary duals of these groups. In the non-archimedean case, for two such representations u(d, m) and u(d', m'), and for real numbers a and b, we obtain a necessary and sufficient condition that the representation parabolically induced by the tensor product of |det|^a u(d, m) and |\det|^b u(d', m') is irreducible. Our motivation for working on this problem comes from our work on unitary duals of classical (symplectic, orthogonal and unitary) groups, related to constructions of complementary series for these groups.

non-archimedean local fields ; division algebras ; general linear groups ; Speh representations ; parabolically induced representations ; reducibility ; unitarizability

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