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Theta correspondence associated to \(G_ 2\). (English) Zbl 0723.11026

It seems we are no longer surprised now by the construction of interesting cusp forms on a reductive group using theta-series and Howe’s theory of dual reductive pairs. Still, because they involve not GL(2) or a symplectic group, but rather the exceptional group \(G_ 2\), the examples constructed in the present paper can’t fail to catch our attention. In this paper, the authors exploit two intriguing facts about \(G_ 2:\) the first is the well-known embedding of \(G_ 2\) in the orthogonal group O(Q) of a seven-dimensional quadratic space; the second is the double transitivity of \(G_ 2\) on this seven-dimensional space - i.e., not only does \(G_ 2\) act transitively on each of the quadrics \(\Gamma_ t=\{X: Q(X)=t\},\) \(t\neq 0\), but also \(Stab_{G_ 2}(\xi)\) acts transitively on \(\xi^{\perp}\cap \Gamma_ t\), for \(\xi\) in \(\Gamma_ t\). The first fact makes it possible to restrict the theta- kernel for O(Q)\(\times \overline{SL_ 2}\) to \(G_ 2\times \overline{SL_ 2}\), while the second makes it possible to establish a Howe correspondence between the representations of \(G_ 2\) and \(\overline{SL_ 2}\) even though these groups are not mutually centralizing in \(\overline{Sp_ 7}\) (as is the case for the dual reductive pair O(Q),\(\overline{SL_ 2})\). Thus the authors are able to lift certain cuspidal representations of \(\overline{SL_ 2}\) to nonzero cusp forms on \(G_ 2({\mathbb{A}})\). Interestingly enough, the resulting cuspidal representations of \(G_ 2\) are seen to be shadows of Eisenstein series (CAP representations in the sense of I. Piatetski-Shapiro, cf. Number Theory Related to Fermat’s Last Theorem, Proc. Conf., Prog. Math. 26, 143-151 (1982; Zbl 0505.12017), and the representations of \(\overline{SL_ 2}\) used are precisely those which lift to the Saito- Kurokawa space of O(3,2) [I. Piatetski-Shapiro, Invent. Math. 71, 309-338 (1983; Zbl 0515.10024)].

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
11F32 Modular correspondences, etc.
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