Non-Siegel Eisenstein series for symplectic groups (Q1692194)

In this long paper, the author considers the Eisenstein series of the symplectic group \(\mathrm{Sp}_{2n}\) induced from parabolic groups \(P_i\) with Levi quotient \(\mathrm{GL}_i \times \mathrm{Sp}_{2n-i}\) and a unitary Größencharakter \(\chi\) of \(\mathbb{A}^\times / \mathbb{Q}^\times\), inflated to \(\mathrm{GL}_i \times \mathrm{Sp}_{2n-i}(\mathbb{A})\) via \(\det \times 1\). When \(i=0\) we obtain the degenerate Siegel Eisenstein series for \(\mathrm{Sp}_{2n}\), which appeared in the classical Siegel-Weil formula. The case of non-Siegel maximal parabolics appears in the first-term identity of the regularized Siegel-Weil formula of Kudla-Rallis. Such formulae are useful for the study of special \(L\)-values. The main results are: the study of local intertwining operators and their images in terms of \(L\)-parameters, the constant terms of the Eisenstein series alluded to above and their images in the right half-plane. Besides the available information about local degenerate principal series, the arguments also involve some combinatorics related to global normalizing factors. Some assumptions are imposed on the Archimedean place.