Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications

For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace $S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\subset S_{\kappa+\frac{1}{2}}(N),\,\,{\text{and}}\,\,S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\,\,{\text{and}}\,\,S_{2k}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product $a_{g}(m)\overline{a_{g}(n)}$ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem he deduced the formula in [2, Theorem 3]. In this paper we will prove that there is a Hecke equivariant isomorphism between the spaces $S_{2k}^{+}(p)\,\,{\text{and}}\,\,\mathbb{S}_{k+\frac{1}{2}}(p).$ We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].