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

SUR UNE CONJECTURE DE TADIĆ

A. I. Badulescu and D. A. Renard

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.