Some bounds on unitary duals of classical groups‎ - ‎non-archimeden case

Document Type: Special Issue of BIMS in Honor of Professor Freydoon Shahidi

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Abstract

‎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.

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