Glasnik Matematicki, Vol. 39, No.1 (2004), 49-54.


A. I. Badulescu and D. A. Renard

Departement de mathematiques, Universite de Poitiers, Teleport 2, boulevard Marie et Pierre Curie, BP 30179 86962 Futuroscope Cedex, France

Abstract.   Let F be a non-archimedian field of characteristic zero and D a central division algebra over F of finite dimension d2. For all positive integer r, set G'r = GL(r,D).
In 1990, M. Tadić gave a conjectural classification of the unitary dual of the G'r, and five statements denoted U0, ... , U4, which imply the classification. M. Tadic proved U3 and U4. Also, U0 and U1 imply U2. These statements, and the resulting classification are the natural generalization of the case D = F completely solved by M. Tadić in 1986. Here we prove U1. Thus, the classification of the unitary dual of the G'r is now reduced to the conjecture U0, which states that a parabolically induced representation from an irreducible unitary representation is irreducible.

2000 Mathematics Subject Classification.   22E50, 20G05.

Key words and phrases.   Representations of p-adic groups, unitary dual.

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