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

### SUR UNE CONJECTURE DE TADIĆ

### 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

*e-mail:* `badulesc@mathlabo.univ-poitiers.fr`

*e-mail:* `renard@mathlabo.univ-poitiers.fr`

**Abstract.** Let *F* be a non-archimedian field of
characteristic zero and *D*
a central division algebra over *F* of finite dimension
*d*^{2}. 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.

**Full text (PDF)** (free access)

*Glasnik Matematicki* Home Page