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Cycles and Harmonic Forms on Locally Symmetric Spaces

Published online by Cambridge University Press:  20 November 2018

John J. Millson
John J. Millson*
Affiliation:
UCLA, Los Angeles, California, 90024
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Abstract

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Two constructions of cohomology classes for congruence subgroups of unit groups of quadratic forms over totally real number fields are given and shown to coincide. One is geometric, using cycles, and the other is analytic, using the oscillator (Weil) representation. Considerable background material on this representation is given.

Keywords

32N10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 1 , 01 March 1985 , pp. 3 - 38
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Borel, A., Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), pp. 111—122.Google Scholar
2. Borel, A. and Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Ann. of Math. Studies 94, Princeton Univ. Press, 1980.Google Scholar
3. Bott, R. and Tu, L.W., Differential Forms in Algebraic Topology, Graduate Texts in Math., 82, Springer.Google Scholar
4. Hirzbruch, F. and Zagier, D., Intersection numbers of curves on Hilbert modular surfaces and modular forms of nebentypus, Invent. Math., 36 (1976), pp. 57—113.Google Scholar
5. Howe, R., Remarks on classical invariant theory, preprint.Google Scholar
6. Howe, R. and Piatetski-Shapiro, I. I., Some examples of automorphic forms on Sp4 , Duke Mathematical Journal, 50 (1983), pp. 55–106.Google Scholar
7. Kazhdan, D., Connection of the dual space of a group with the structure of its closed subgroups, J. Func. Anal. Appl. (1967), pp. 63–65.Google Scholar
8. Kudla, S. and Millson, J., Geodesic cycles and the Weil representation I; quotients of hyperbolic space and Siegel modular forms, Compositio Mathematica 45 (1982), p. 207, p. 271.Google Scholar
9. Kudla, S. and Millson, J., The theta-correspondence and harmonic forms, preprint.Google Scholar
10. Kudla, S. and Millson, J., in preparation.Google Scholar
11. Lion, G. and Vergen, M., The Weil representation, Maslov index and Theta series, Progress in Mathematics 6, Birkhauser, 1980.Google Scholar
12. Millson, J., On the first Betti number of a constant negatively curved manifold, Annals of Mathematics 104, (1976), pp. 235–247.Google Scholar
13. Millson, J. and Raghunathan, M.S., Geometric Construction of cohomology for arithmetic groups I, Papers Dedicated to the Memory of V. K. Patodi, Indian Academy of Sciences, Bangalore 560080.Google Scholar
14. Mumford, D., Tata Lectures on Theta I, Birkhauser, 1983.Google Scholar
15. Oda, T., Periods of Hilbert modular surfaces, preprint.Google Scholar
16. Rallis, S., Injectivity properties of liftings associated to Weil representations, preprint.Google Scholar
17. Rossman, W., The structure of semisimple symmetric spaces, Can. J. Math., Vol. XXXI, No. 1, (1979), pp. 157–180.Google Scholar
18. Shapiro, A., Group extensions of compact Lie groups, Annals of Math., Series 2, 50 (1949), pp. 581–586.Google Scholar
19. Shintani, T., On the construction of holomorphic cusp forms of half-integral weight, Nagoya Math. J. 58 (1975), pp. 83–126.Google Scholar
20. Siegel, C.L., Über die analytische thorie der quadratischen formen. Annals of Math, Series 2, 36 (1935), pp. 527–606.Google Scholar
21. Zuckerman, G., Continuous cohomology and unitary representations of real reductive groups, Annals of Math. 107 (1978), pp. 495–516.Google Scholar