Aubert duals of discrete series: the first inductive step

Let Gn denote either symplectic or odd special orthogonal group of rank n over a non-archimedean local field F . We provide an explicit description of the Aubert duals of irreducible representations of Gn which occur in the first inductive step in the realization of discrete series representations starting from the strongly positive ones. Our results might serve as a pattern for determination of Aubert duals of general discrete series of Gn and should produce an interesting part of the unitary dual of this group. Furthermore, we obtain an explicit form of some representations which are known to be unitarizable.