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Metaplectic forms. (English) Zbl 0559.10026
The authors prove a Siegel-Weil formula [A. Weil, Acta Math. 113, 1-87 (1965; Zbl 0161.023)] for the n-fold cover \(^{\sim}_ r({\mathbb{A}})\) of \(GL_ r({\mathbb{A}})\), where \({\mathbb{A}}\) is the ring of adèles of an A- field k containing all the n-th-roots of unity. Here n is a positive integer coprime to the characteristic of k. For \(^{\sim}_ 2({\mathbb{A}})\) and \(n=2\), the formula proves that the classical theta-functions are in fact residues of Eisenstein series [T. Kubota, On automorphic functions and the reciprocity law in a number field (1969; Zbl 0231.10017); and S. Gelbart and P. Sally, Proc. Natl. Acad. Sci. USA 72, 1406-1410 (1975; Zbl 0303.22010)]. In fact, generalized theta-functions are certain functions on \(GL_ r(k)^*\setminus^{\sim}_ r({\mathbb{A}})\). Using the language of group representations they may be considered as certain constituents of \(L^ 2(GL_ r(k)^*\setminus GL_ r({\mathbb{A}})).\)
More precisely, at every place v of k, there is always a reducible principal series representation of \(^{\sim}_ r(k_ v)\) which has a unique distinguished quotient (in the terminology of S. Gelbart and I. I. Piatetski-Shapiro, Invent Math. 59, 145-188 (1980; Zbl 0426.10027), i.e. it has a unique Whittaker model). This is in fact only true if \(r=n\) or n-1. The global representation which has these distinguished representations as its local components is then the generalized theta form, and the main result of this paper, Theorem II.2.2, proves that it is in fact the residue of an Eisenstein series, and consequently is automorphic. When \(r=2\), \(n=3\), and \(k={\mathbb{Q}}(\sqrt{-3})\), a similar result has been proved by the second author [J. Reine Angew. Math. 296, 125-161 (1977; Zbl 0358.10011) and 217-220 (1977; Zbl 0358.10012)] using classical language; and P. Deligne [Sémin. Bourbaki, Exp. No.539, Lect. Notes Math. 770, 244-277 (1980; Zbl 0433.10018)], using group representations.
As mentioned above, the methods used in the paper are those of the representation theory of metaplectic groups. These are developed in Chapter I. Among many interesting results proved in Chapter I is a formula for the dimension of the space of Whittaker functionals of any irreducible admissible representation of \(^{\sim}_ r(k_ v)\) in terms of its character (Theorem I.5.3). Finally at the end of Chapter II, global methods are used to answer a number of local questions whose proofs are otherwise not available (Theorem II.2.5 and Corollary II.2.6).
Reviewer: F.Shahidi

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
11F12 Automorphic forms, one variable
11R56 Adèle rings and groups
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