Hecke rings for a covering of the symplectic group

Using the standard theta series of genus $n$, the Hecke rings $\hat D=\hat D(\Gamma_0^n(q),S^n(q))$, for a covering $\mathfrak{G}$ of the symplectic group $GSp_n^+(\mathbf R)$ are constructed. The special role of four subrings of $\hat D$ is described, as well as some finitely generated arithmetic subrings $\hat L_p^n(\varkappa)$. The latter are important in the study of multiplicative properties of the Fourier coefficients of Siegel modular forms of half-integral weight.
Bibliography: 11 titles.