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On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups. (English) Zbl 1291.11083

Summary: We derive a (weak) second term identity for the regularized Siegel–Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the “second term range”. As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let \(\pi\) be a cuspidal automorphic representation of an orthogonal group \(O(V)\) with \(\dim V = m\) even and \(r+1 \leq m \leq 2r\). Assume further that there is a place \(v\) such that \(\pi_v \simeq \pi_v \otimes \det\). Then the global theta lift of \(\pi\) to \(\text{Sp}_{2r}\) does not vanish up to twisting by automorphic determinant characters if the (incomplete) standard \(L\)-function \(L^{S}(s,\pi)\) does not vanish at \(s = 1+(2r-m)/2\). Note that we impose no further condition on \(V\) or \(\pi\). We also show analogous non-vanishing results when \(m > 2r\) (the “first term range”) in terms of poles of \(L^{S}(s,\pi)\) and consider the “lowest occurrence” conjecture of the theta lift from the orthogonal group.

MSC:

11F27 Theta series; Weil representation; theta correspondences
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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