On the complementary Arthur representations and unitary dual for p-adic classical groups

TL;DR

This paper characterizes complementary Arthur representations for symplectic and split odd orthogonal groups, providing insights into the structure of the unitary dual and implications for automorphic representations.

Contribution

It offers an explicit description of complementary Arthur representations for specific p-adic classical groups, advancing the understanding of their unitary duals.

Findings

Explicit characterization of complementary Arthur representations.

Constraints on local components of automorphic representations.

Support for the conjecture on the structure of the unitary dual.

Abstract

In [HJLLZ24], we proposed a new conjecture on the structure of the unitary dual of connected reductive groups over non-Archimedean local fields of characteristic zero based on their Arthur representations and verified it for all the known cases on the unitary dual problem. One step towards this conjecture involves the question whether certain complementary Arthur representations are unitary. In this paper, we give an explicit characterization of the complementary Arthur representations for symplectic and split odd special orthogonal groups. As applications, we obtain interesting constraints on local components of irreducible self-dual cuspidal automorphic representations of GL(N), especially when N=2,3.