Correspondence of modular forms to cycles associated to \(Sp(p,q)\)
\textit{F. Hirzebruch} and \textit{D. Zagier} [Invent. Math. 36, 57-113 (1976; Zbl 0332.14009)] have established that intersection numbers of certain cycles in Hilbert modular surfaces are Fourier coefficients of modular forms. Since then the study of certain liftings of automorphic forms and their relation to the dual forms of certain totally geodesic cycles in locally symmetric spaces has been the main theme of a series of papers of S. S. Kudla and J. J. Millson, resp. Y. L. Tong and the author. On one side the theory of the Weil representation and theta functions for dual reductive pairs is used to construct liftings of automorphic forms. On the other hand, these have a geometric interpretation in terms of harmonic forms arising as the Poincaré duals of totally geodesic cycles for arithmetic quotients attached to arithmetic subgroups of the given groups. The author [J. Differ. Geom. 22, 151-213 (1985; Zbl 0594.10021)] dealt with the case of the dual reductive pairs \((O(p,q), Sp(2r,{\mathbb{R}}))\) with \(r\leq p/2\) and gave a description of the geometric lifting by using representation theory of O(p). In the paper under review the case of the reductive pairs \((Sp(p,q),O^*(2r))\) with \(r\leq p/2\) is taken up in an analogous way. In the spirit of Shintani's construction of modular forms [cf. \textit{T. Shintani}, Nagoya Math. J. 58, 83-126 (1975; Zbl 0316.10016)] one pairs directly a theta function and a harmonic cusp form to obtain a cusp form with period integrals as Fourier coefficients. The proof of the main result involves various intermediate results which will not be summarized here but are of independent methodological interest. Again the representation theory for the group Sp(p) turns out to be a useful technical tool. Note that there is a unified approach to the theta correspondence in this framework in the cases \(O(p,q)\), \(U(p,q)\) or \(Sp(p,q)\) by \textit{S. S. Kudla} and \textit{J. J. Millson} [Math. Ann. 274, 353-378 (1986; Zbl 0594.10020), Part II (to appear)]. There results of R. Howe play an important role.
scientific article; zbMATH DE number 3983273
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