Construction of Cohomology of Discrete Groups

A correspondence between Hermitian modular forms and vector valued harmonic forms in locally symmetric spaces associated to U(p,q) is constructed and also shown in general to be nonzero. The construction utilizes Rallis-Schiffmann type theta functions and simplified arguments to circumvent differential geometric calculations used previously in related problems. Introduction. In our previous papers [T.W.2-4, W.l, 2] and that of KudlaMillson [K.M.I, 2] various cases are proved about correspondences of harmonic forms on locally symmetric spaces dual to geodesic cycles on the one hand, and Siegel, Hermitian, or quaternionic modular forms on the other hand. The existence of such correspondences is based on R. Howe's theory of decomposition of the oscillator representation on reductive dual pairs [Ho.l, 2]. However, while the general theory has a simple representation theoretic description, cf. [H.-P.S., §2], this description is pretty much lost in the previous works on correspondences of cycles and modular forms. This is mainly due to technicalities in constructing harmonic forms and then passing from harmonic forms to cycles. The purpose of this paper is twofold. First of all in terms of results we generalize [T.W.4] to U(p,q), and in fact we generalize to harmonic forms with coefficients in locally constant vector bundles. We also prove these harmonic forms can be nonzero. Secondly, in terms of methodology, and this is perhaps an equally significant point, we show that the correspondence of cycles and modular forms has a straightforward representation theoretic description. This description consists of the following ingredients. 1. Construction of some functions on the matrix space Mp+q,r(C) which are in the discrete spectrum of the reductive dual pair U(p,q) x U(r,r) and are (U(p) x U(q)) x (U(r) x U(r)) finite. These functions are of highest weight for U(r,r) and are generalizations of the Rallis-Schiffmann functions [R.S.I, 2, L.V.]. These functions determine some pieces of the representation correspondence. If convergence holds, the theta distribution applied to such functions then gives the theta kernels used for "global" correspondence. It is an easy argument [B.W., VIII] to show that the theta kernels are in general nonzero. 2. Construction of an intertwining map ux(Z), X £ Mp+a,r(C), Z £ D = U(p,q)/(U(p) x U(q)), from the representation space of U(p,q) in 1 above to a space of harmonic forms on D with coefficients in a locally constant bundle E\. 3. For an arithmetic discrete subgroup T C G = U(p,q) determination of the cohomology class of ûx on T \ D where ûix = X^erx\r l*ûx, Tx = T n Gx, and Received by the editors February 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32N10, 22E41. The authors are supported in part by N.S.F. Grants DMS-8500995 and DMS-8502364. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 735 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 736 Y. L. TONG AND S. P. WANG Gx C G fixes X. Thus for any E^ valued harmonie form cp on T \ D we prove / ûxAx (Theorem 4.8) and a simple pairing argument (Lemma 5.6). The interpretation of liftings of higher weight modular forms as cohomology classes represented (in the language of currents) by sections of bundles supported on cycles have also appeared in Hilbert modular surfaces [T]. In that case, another interpretation [G] is that the higher weight forms correspond to cycles in certain Kuga fiber spaces. It is likely that similar interpretations are possible for the harmonic forms constructed here. Due to convergence problems our nonvanishing results do not include the case E\ = C. If convergence were to hold we note that 0¿OJx(Z)=irz()l(Cx)) where M(CX) is the harmonic form Poincaré dual to the algebraic cycle Cx and ttz is the pointwise projection to the Kz (isotropy subgroup at Z) irreducible subspace of weight (q_-q,0--0,-q----q)(0_-0). r r 1 It would then be an interesting question to decide if ùx (Z) (or other similar harmonic forms with a more complicated K type) is dual to an algebraic cycle. In [T.W.5] we showed a much wider range of nonvanishing cohomology on T\D. In the dual pair correspondence and on the level of representations these are essentially all the ones that correspond to holomorphic discrete series of U(r, r). It is interesting to see if geometric representatives for these cohomology can also be found. As representations of SU(p,q) the spaces of harmonic forms considered here and their natural generalizations are Flensted-Jensen representations [F.l] for a semisimple symmetric space G/H. (H = S(U(r) x U(p r,q)) for this paper.) In particular, the uniqueness in Theorem 4.8 is closely related to the fact that these representations should have multiplicity 1 in L2(G/H) [F.2]. We are, however, unable to make use of the integral formula of Flensted-Jensen functions for present purposes since it is given in the dual G°/H° while we need the functions on the matrix space Mp+,ir(C). It may be relevant to mention that in [K.M.2] by quite different techniques certain representatives <p of continuous cohomology classes are constructed which in the present context of unitary groups give [<p] £ H*t(U(p,q), S(Mp+q,r(C))a), where S indicates the Schwartz space and the subscript a means twisting by a character. A natural question (a question to that effect is posed in [K.M.2]) is to determine the representation spaces of U(p, q) and U(r, r) spanned by the translates License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use COHOMOLOGY OF DISCRETE GROUPS 737 of <p. By comparing the cohomology degrees it is easy to see that the space discussed in 1 above which is spanned by wx(Z)e~2wtT(x'x's (cf. §1.1 for notation) should give a distinguished subspace in that of <p. The methods used here also work for the reductive dual pairs (0(p, q), Sp(r, R)), (Sp(p,q),0*(2r)). Our results also extend to compact quotients, the readers may consult [T.W.2, 5, W.l] for the details to formulate such results. We should emphasize that in these papers as well as in [K.M.I, 2] the theta kernel is always gotten by summing a Schwartz function, but the Schwartz functions are not in the discrete spectrum. The Rallis-Schiffmann functions are not Schwartz functions, thus some care is needed in summing, but it has considerable advantages in representation questions since it is in the discrete spectrum. We are indebted to [L.V] for a thorough exposition of the approach of Rallis-Schiffmann. 1. Some functions in the discrete spectrum of U(p,q) x U(r,r). 1.1. We define certain functions in the discrete spectrum of the dual pair U(p, q) x U(r,r) and compute the eigenvalues of Casimir operators acting on them. The functions are generalizations of the Rallis-Schiffmann functions studied in [R.S.I, 2] for the pair 0(p,q) x SL(2,R). Let G — U(p, q) (p + q — n) and let g be its Lie algebra. G leaves invariant the Hermitian form on Cn given by (x,y) = *x E„ -Ea Let V = Mnr(C) denote the space of n x r complex matrices. Let X = (Xij) £ V be the usual row and column coordinates and let Xj denote the jth column of X. We let