The Kohnen plus space for Hilbert-Siegel modular forms

The Kohnen plus space, roughly speaking, is a space consisting of modular forms of half integral weight with some property in Fourier coefficients. For example, the $n$-th coefficient of a normal modular form of weight $k+1/2$ in the plus space is $0$ unless $(-1)^kn$ is congruent to some square modulo $4$. The concept of plus space was initially introduced by Kohnen in 1980. Eichler and Zagier showed that the plus space is isomorphic to the space of Jacobi forms in the one variable case. Later, Ibukiyama generalized these results to the cases for Siegel modular forms in 1992. Also, Hiraga and Ikeda generalized these results to the cases for Hilbert modular forms in 2013. In this paper, we continue to consider the case of Hilbert-Siegel modular forms. An analogue of the previous results will be given.