Sur une conjecture de Tadic

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\u27r = GL(r,D). In 1990, M. Tadic gave a conjectural classification of the unitary dual of the G\u27r, 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. Tadic in 1986. Here we prove U1. Thus, the classification of the unitary dual of the G\u27r is now reduced to the conjecture U0, which states that a parabolically induced representation from an irreducible unitary representation is irreducible

Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations

In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).Comment: 49 pages; Appendix by N. Grba

Unitary Dual of GL_n at archimedean places and global Jacquet-Langlands correspondence

In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean places so that this assumption can be removed. Along the way, we collect several results about the unitary dual of general linear groups over \bbR, \bbC or \bbH of independent interest

On unitarizability in the case of classical p-adic groups

In the introduction of this paper we discuss a possible approach to the unitarizability problem for classical p-adic groups. In this paper we give some very limited support that such approach is not without chance. In a forthcoming paper we shall give additional evidence in generalized cuspidal rank (up to) three.Comment: This paper is a merged and revised version of ealier preprints arXiv:1701.07658 and arXiv:1701.07662. The paper is going to appear in the Proceedings of the Simons Symposium on Geometric Aspects of the Trace Formul