Lifts of automorphic representations and their associated \(L\)-functions are constructions where specific cases of Langlands functoriality can be studied. In this case, the author considers a number field \(F\) and a quadratic extension \(E\) over \(F\). Let \(X\) be a skew-Hermitian space over \(E\) and \(Y\) a Hermitian space over \(E\). Consider the unitary groups \(G(X)\) and \(G(Y)\) associated to \(X\) and \(Y\), respectively. The groups \(G(X)\) and \(G(Y)\) form a dual reductive pair and given a cuspidal automorphic representation \(\pi\) of \(G(X)(\mathbb A_F)\) one can define a global theta lift of \(G(Y)(\mathbb A_F)\). However, lifts can be also defined in the other direction. The basic ideas come from several results starting with the Rallis inner product formula [S. S. Kudla and S. Rallis, Ann. Math. (2) 140, No. 1, 1–80 (1994; Zbl 0818.11024)] relating the poles certain \(L\)-functions to the vanishing of theta lifts. Ramified local factors are present in the Rallis inner product formula and therefore the relation between the poles of these \(L\)-functions is not exact.
Let \(\sigma\) be a cuspidal representation of \(G(X)(\mathbb A_F)\) and \(\chi\) a conjugate self-dual character of \(\mathbb A^\times_E\). The author follows an idea of C. Mœglin [J. Lie Theory 7, No. 2, 201–229 (1997; Zbl 0885.11032)] considering the poles of Eisenstein series associated to the cuspidal datum \(\sigma\otimes \chi\). These Eisenstein series are closely related to the \(L\)-functions \(L(s,\sigma\times \chi^{-1})\). In the paper [Mœglin, loc. cit.] the cases of orthogonal and symplectic/metaplectic dual reductive pairs were already investigated. The author of the present paper and D. Jiang started the study of unitary dual pairs in [J. Number Theory 161, 88–118 (2016; Zbl 1400.11102)]. Here we state the main result of the author:
Theorem. Let \(\sigma\) be a cuspidal representation of \(G(X)(\mathbb A_F)\). Then \(s_0\) is the maximal pole of the Eisenstein series \(E(g,f_s\) for \(f_s\) running over \(\mathcal A_1(s, \chi, \sigma)\) if and only if \(\mathrm{LO}_{\psi, \chi}(\sigma) = \dim X -1 -2s_0\). Here \(\mathcal A_1(s, \chi, \sigma)\) denotes the space of automorphic forms on some quotient space and the Eisenstein series is defined for each section \(f_s\) in \(\mathcal A_1(s, \chi, \sigma)\).
The proof of this theorem relies on an intricate result relating period integrals of residues of Eisenstein series twisted by a theta series and a the fist occurrence index \(\mathrm{FO}^Y_{\psi, \chi}(\sigma)\) for the Witt tower of \(Y\).