  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.
