Refined Siegel–Weil conjecture for theta lifts

Let $\theta_{\Delta(\tau,\ell)}$ and $\theta_{\Delta(\tau,2n+\ell)}$ be theta functions, and let $\Phi(\theta_{\Delta(\tau,\ell)},\theta_{\Delta(\tau,2n+\ell)},s)$ be the proposed section of $\rho_{\Delta(\tau,2n),s}$. Let $M_s$ and $M_s^*$ be respectively the intertwining and normalized intertwining operators. Refined Siegel–Weil conjecture. For $1\leq\ell\leq n$, the Eisenstein series attached to $\Phi(\theta_{\Delta(\tau,\ell)},\theta_{\Delta(\tau,2n+\ell)},s)$ is holomorphic at $s=\ell-n$ and, up to a possible constant, equals the theta integral at that point; moreover, the stated normalized-intertwining construction gives the residue identity in the intermediate case, while for $n<\ell\leq2n$ the theta integral equals, up to a possible constant, the residue at $s=\ell-n$:

$$\int_{\operatorname{Sp}_{4\ell}(F)\backslash\operatorname{Sp}_{4\ell}(\mathbb A)}\theta_{\Delta(\tau,2n+\ell)}(g,h)\theta_{\Delta(\tau,\ell)}(g)\,dg=\operatorname{Value}_{s=\ell-n}E(\Phi(\theta_{\Delta(\tau,\ell)},\theta_{\Delta(\tau,2n+\ell)},s))(h)$$

when $1\leq\ell\leq n$, with the additional intertwining assertion and residue formulation given in the source, and

$$\int_{\operatorname{Sp}_{4\ell}(F)\backslash\operatorname{Sp}_{4\ell}(\mathbb A)}\theta_{\Delta(\tau,2n+\ell)}(g,h)\theta_{\Delta(\tau,\ell)}(g)\,dg=\operatorname{Res}_{s=\ell-n}E(\Phi(\theta_{\Delta(\tau,\ell)},\theta_{\Delta(\tau,2n+\ell)},s))(h)$$

when $n<\ell\leq2n$. The corresponding normalized section is $\Phi^*=M_s^*(\Phi)|_{s=\ell-n}$ as specified in the source. This remains formal and is not established in the supplied context.