Document Type: Special Issue of BIMS in Honor of Professor Freydoon Shahidi
We first give bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical $p$-adic groups. Roughly, they can show up only if the central character of the inducing irreducible cuspidal representation is dominated by the square root of the modular character of the minimal parabolic subgroup. For unitarizable subquotients supported by a fixed parabolic subgroup, or in a specific Bernstein component, a more precise bound is given.
For the reductive groups of rank at least two, the trivial representation is always isolated in the unitary dual (D. Kazhdan). Still, we may ask if the level of isolation is higher in the case of the automorphic duals, as it is a case in the rank one. We show that the answer is negative to this question for symplectic $p$-adic groups.