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

TL;DR

This paper establishes bounds on where unitarizable subquotients can appear in classical p-adic groups' induced representations and investigates the isolation level of the trivial representation in automorphic duals, especially for symplectic groups.

Contribution

It provides new bounds for unitarizability domains in classical p-adic groups and analyzes the isolation properties of the trivial representation in automorphic duals, particularly for symplectic groups.

Findings

Bounds are given for unitarizable subquotients in induced representations.

The trivial representation's isolation level in automorphic duals is not higher than in the unitary dual for symplectic p-adic groups.

Some bounds are proven to be optimal.

Abstract

In the first part of the paper we give some bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical p-adic groups. Roughly, it can show up only if the central character of the inducing irreducible cuspidal representation is dominated in an appropriate way by the square root of the modular character of minimal parabolic subgroup. For representations supported by fixed parabolic subgroup, a more precise bound is given. There are also bounds for specific Bernstein components. A number of these upper bounds are best possible. The second part of the paper addresses a question how far is the trivial representation from the rest of the unramified automorphic dual. By a result of L. Clozel, trivial representation is isolated in the automorphic dual of a split rank one semisimple group over a completion of a global field, but it is very seldom isolated in the unitary dual (it can happen only in the archimedes cases). Further, the level of isolation in the case of SL(2) is important for the number theory. For the higher rank groups, the trivial representation is always isolated in the unitary dual by an old result of D. Kazhdan. Still, we may ask if the level of isolation is higher in the case of the automorphic duals. We show that the answer is negative to this question for symplectic p-adic groups.