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The theta correspondence and harmonic forms. II. (English) Zbl 0618.10022

Let \(V,(\,,\,)\) be a vector space over \({\mathbb R}\), \({\mathbb C}\), or \({\mathbb H}\) endowed with a (hermitian) form of signature \((p,q)\); the corresponding isometry group \(\mathrm{O}(p,q)\), \(\mathrm{U}(p,q)\) or \(\mathrm{Sp}(p,q)\) will be denoted by \(G\). The paper under review contains a careful study of the relationship between two types of cohomology classes of arithmetic quotients of the symmetric spaces \(D\) (or products of these) attached to \(G\).
The first type of cohomology class arises in a geometric way from a certain cycle, the second one arises from the continuous cohomology of the Schwartz space \(S(V^ n)\). Later on the connection between these classes and certain automorphic classes constructed by means of the theta-correspondence for a dual reductive pair is discussed.
More precisely, given a subspace \(U\) of \(V\) there are (in the adelic language) special cycles in \(M_ K=\underline G(k)^+\setminus D\times G({\mathbb A}_ f)/K_ f\) (\(k\) a totally real algebraic number field) which give rise to geometric cohomology classes in \(H^*(M_ K)\); the space of special cycles in \(H^*(M_ K)\) associated to \(U\) will be denoted by \(SC^*_ H(M_ K)\) where \(H\) is the stabilizer of \(U\oplus U^{\perp}\). Passing to the limit over the open compact subgroups \(K_ f\subset G({\mathbb A}_ f)\) one obtains a \(G({\mathbb A}_ f)\)-submodule \(SC^*_ H(M)\) of \(H^*(M)\). Using the intertwining map (with respect to the \(G({\mathbb A}_ f)\)-module structure) \[ H^*({\mathfrak g},K_{\infty},S(V({\mathbb A})^ n_{\alpha}))\quad \to \quad H^*({\mathfrak g},K_{\infty},C^{\infty}(G(k)^+\setminus G({\mathbb A}))) \] one obtains classes constructed out of Schwartz classes. The first main result (Theorem 3.1) compares these two types of cohomology classes.
Its proof relies on some explicit growth estimates and a characterization of fast decreasing differential forms with respect to fiberings of the form \(\pi: \Gamma\setminus D\to \Gamma \setminus D_ H\), \(\Gamma\) discrete subgroup of \(H\), \(\Gamma\) cocompact, which admits to prove a key formula \[ \int_{\Gamma \setminus D}\phi \wedge \eta =\int_{\Gamma \setminus D_ H}\pi_*\phi \wedge i^*\eta \] with \(\phi\) a closed \(r\)-form satisfying a certain growth condition, \(\eta\) a closed bounded (\(\dim D-r\))-form, \(i: \Gamma\setminus D_ H\to \Gamma \setminus D\) the inclusion. This result is of independent interest.
Using the theory of dual reductive pairs due to R. Howe the authors obtain in the case that \(V,(\,,\,)\) is anisotropic a general qualitative version of the result of F. Hirzebruch and D. B. Zagier [cf. Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36, 57–113 (1976; Zbl 0332.14009)] that the “generating function for intersection numbers” is an automorphic form. Later on a more explicit form of this identity for certain types of Schwartz classes is given. A large supply of such classes was constructed by the authors in the previous paper [Part I, cf. Math. Ann. 274, 353–378 (1986; Zbl 0594.10020)].
Combining these explicit forms with the first main result obtained the authors draw several consequences out of it with respect to automorphic forms. In this unified approach they generalize results of Hirzebruch-Zagier (loc. cit.), of their own [“Geodesic cycles and the Weil representation. I: Quotients of hyperbolic space and Siegel modular forms”, Compos. Math. 45, 207–271 (1982; Zbl 0495.10016)] and Y. L. Tong and S. P. Wang [Theta functions defined by geodesic cycles in quotients of \(\mathrm{SU}(p,1)\), Invent. Math. 71, 467–499 (1983; Zbl 0506.10024)] and S. P. Wang [Correspondence of modular forms to cycles associated to \(\mathrm{O}(p,q)\), J. Differ. Geom. 22, 151–213 (1985; Zbl 0594.10021)] at least in the anisotropic case.
Reviewer: J. Schwermer

MSC:

11F12 Automorphic forms, one variable
22E40 Discrete subgroups of Lie groups
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
57T10 Homology and cohomology of Lie groups
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References:

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