The paper under review studies the functorial lift from the metaplectic group \(\widetilde{\text{Sp}}_n\) (sometimes denoted as \(\widetilde{\text{Sp}}_{2n}\) in the literature) to \(\text{GL}_{2n}\) via a comparison of relative trace formulas. Such functorial lifts have been studied from different angles, eg. via the Arthur-Selberg trace formula. One of the advantage of the relative trace formulas is that it yields information about the central L-value \(L(1/2, \Pi)\), which generalizes Waldspurger’s celebrated work for the case \(n=1\).
To be precise, let \(k\) be a number field, \(\mathbb{A}\) be its ring of adèles, and \(G\) be a connected reductive \(k\)-group. The relative trace formula associated to a datum \((H_1, \chi_1; H_2, \chi_2)\) concerns the distribution \(f \mapsto I_G(f: H_1, \chi_1; H_2, \chi_2)\), that is, the integration of the kernel function \(K_f(x,y)\) along \(H_1(k) \backslash H_1(\mathbb{A}) \times H_2(k) \backslash H_2(\mathbb{A})\). Here \(f\) lies in the Schwartz space \(\mathcal{S}(G(\mathbb{A}))\); for \(i=1,2\), \(H_i\) is a closed subgroup of \(G\) and \(\chi_i\) is an automorphic character of \(H_i(\mathbb{A})\). We will also need the generalization in which \(\chi_i\) is multiplied by a certain theta function. In the cases encountered in this paper, such integrals are absolutely convergent.
The main result (Theorem 1.1) is a relative trace identity of the form \[ I_{\text{GL}_{2n}}(f: \text{GL}_n \times \text{GL}_n, 1; N, \theta) = I_{\widetilde{\text{Sp}}_n}(\tilde{f}: N', {\theta'}^{-1}; N', \theta') \] for factorizable test function \(f\) and a certain “transfer map” \(f \mapsto \tilde{f}\). Here \(N\), \(N'\) are appropriate maximal unipotent subgroups and \(\theta\), \(\theta'\) are non-degenerate characters. Such an identity is expected to yield some relation between \(|W(\tilde{\varphi})|^2\) (Whittaker functional on \(\widetilde{\text{Sp}}_n\)) and \(W(\varphi)\) (Whittaker functional on \(\text{GL}_{2n}\)) times the \(\text{GL}_n \times \text{GL}_n\)-period \(P(\varphi)\) on \(\text{GL}_{2n}\), under the functorial lift.
This is reduced to another relative trace identity of the form \[ I_{\text{Sp}_{2n}}(f'': \text{Sp}_n \times \text{Sp}_n, 1; N_3, \theta_4 \Theta^{\Phi}_{\psi^{-1}}) = I_{\widetilde{\text{Sp}}_n}(\tilde{f}: N', {\theta'}^{-1}, N', \theta'). \] The latter identity is then proved by matching relative orbital integrals. As usual, one must establish the so-called fundamental lemma for elements in the spherical Hecke algebras. This is done for the unit element first. Then, by using a variant of Plancherel formula together with a computation of Jacquet modules, one deduces the general case. Note that the descent method of Ginzburg-Rallis-Soudry is heavily used in the proofs.