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On the theta lift from \(\text{SO}_{2n+1}\) to \(\widetilde{\text{Sp}}_ n\). (English) Zbl 0827.11032

The author studies the theta correspondence between \(\text{SO}_{2n+1}\) (split) and \(\widetilde {\text{Sp}}_n\), the double cover of rank \(n\) symplectic group \(\text{Sp}_n \subset \text{GL}_{2n}\) and computes the Whittaker coefficients of theta lifts in both directions. As the main result of this paper, he obtains the following:
Main Theorem. Let \((\pi, V_\pi)\) be an irreducible cuspidal automorphic representation of \(G_n (\mathbb{A})= \text{SO}_{2n+1} (\mathbb{A})\) and let \(\Theta (\pi, \psi)\) be the theta lift of \(\pi\), with respect to a nontrivial additive character \(\psi\), to \(\widetilde {\text{Sp}}_n (\mathbb{A})\). Assume that \(V_\pi\) is orthogonal to \({\mathcal A}^{ng} (G_n)\), the space of automorphic forms on \(G_n (\mathbb{A})\) whose Whittaker functions are identically zero. Then,
(1) \(\pi\) is generic (i.e., having nonzero Whittaker functions);
(2) \(\Theta (\pi, \psi)\) is cuspidal (including the possibility that \(\Theta (\pi, \psi)= \{0\})\); and
(3) \(\Theta (\pi, \psi)\neq \{0\}\) if and only if \(\pi\) has the split Bessel model of special type.
If we assume the existence of archimedean local factors, then the statement (3) above is stated as:
(4) \(\Theta (\pi, \psi)\neq \{0\}\) if and only if \(L({1\over 2},\pi)\neq 0\), where \(L(s, \pi)\) is the degree \(2n\) standard \(L\)-function for \(\pi\).
Thus this may be regarded as a generalization to \(\text{SO}_{2n+1}\) of Waldspurger’s result on \(\text{PGL}_2 \simeq \text{SO}_3\).

MSC:

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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