Unitarizability in generalized rank three for classical p-adic groups

TL;DR

This paper advances the understanding of unitarizability for classical p-adic groups of rank up to three by analyzing representations supported on specific cuspidal lines, building on a strategy involving reducibility points.

Contribution

It solves the unitarizability problem for irreducible subquotients of certain induced representations in classical p-adic groups of rank up to three, extending prior partial results.

Findings

Unitarizability depends on reducibility exponents for rank ≤ 3.

Provides a solution for classical groups of split rank up to three.

Supports the proposed approach's potential for general unitarizability analysis.

Abstract

In an earlier paper we propose an approach to the unitarizability problem in the case of classical groups over a p-adic field of characteristic zero based on cuspidal reducibility points. We have reduced earlier the unitarizability for these groups to the case of so called weakly real representations. Following C. Jantzen, to an irreducible weakly real representation $\pi$ of a classical group one can attach a sequence ($\pi_1,\dots,\pi_k)$ of irreducible representations of classical groups, each of them supported by a line of cuspidal representations $X_\rho$ of general linear groups containing a selfcontragredient representation $\rho$, and an irreducible cuspidal representation $\sigma$ of a classical group. The first question is if $\pi$ is unitarizable if and only if all $\pi_i$ are unitarizable. Further, the pair $\rho,\sigma$ determines the non-negative reducibility exponent $\alpha_{\rho,\sigma}\in\frac12\mathbb Z$ among $\rho$ and $\sigma$. The question is if the unitarizability of irreducible representations supported by $X_\rho\cup \sigma$ depends only on $\alpha_{\rho,\sigma}$. Following the above proposed strategy, in this paper we solve the unitarizability problem for irreducible subquotients of representations Ind$_P^G(\tau)$, where G is a classical group over a p-adic field of characteristic zero, P is a parabolic subgroup of G of the generalized rank (at most) 3 and $\tau$ is an irreducible cuspidal representation of a Levi factor M of P. As a consequence, this gives also a solution of the unitarizability problem for classical p-adic groups of the split rank (at most) three. This paper also provides some very limited support for the possibility of the above approach to the unitarizability could work in general.