On Automorphic Duals and Isolated Representations;New Phenomena

In the paper we study automorphic duals of split classical groups in the non-archimedean case (defined in [10]), and a relation between isolated points in the unitary and automorphic duals. Particular attention is devoted to the unramified unitary representations ([26]). In the unramified case, we study relation between the property of being automorphic (and isolated there), and intrinsic properties of representations. In the case of split classical groups we give combinatorial formulas for the number of isolated representations in the unramified unitary duals (these representations are also isolated representations in the automorphic duals), and for the number of so called strongly negative representations, which can be expected to be sets of isolated representations in the unramified automorphic duals. For the difference of special linear groups, we have plenty of both of these representations. We also discuss the case of automorphic duals of general linear groups.