Newforms in the Kohnen plus space

TL;DR

This paper defines the Kohnen plus space for Hilbert modular forms using a representation-theoretic approach, demonstrating its equivalence to Kohnen's classical definition and analyzing Hecke operator actions on new forms.

Contribution

It introduces a new representation-theoretic definition of the Kohnen plus space for Hilbert modular forms and characterizes new forms via Hecke operators.

Findings

The defined plus space matches Kohnen's classical space.

Hecke operators' actions on new forms are characterized.

Results are comparable to the integral weight case.

Abstract

In this paper we want to define the Kohnen plus space for Hilbert modular forms with a odd square-free level and a quadratic character by a representation-theoretic way. We will show that in the classical case the one we defined is the same with the one given by Kohnen. Also, we will interpret the actions of the Hecke operators on the new forms in the plus space and give a characterization for the new forms using Hecke operators. All the results with respect to Hecke operators can are comparable with the integral weight case.