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Theta series liftings from orthogonal groups to semi-simple groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  09 April 2009

Min Ho Lee
Min Ho Lee
Affiliation:
Department of Mathematics University of Northern lowaCedar Falls, IA 50614USA e-mail: lee@math. uni.edu
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Abstract

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We study a correspondence between automorphic forms on an orthogonal group and automorpbic forms on a semi-simple Lie group associated to an equivariant holomorphic map of a symmetric domain into a Siegel upper half space. We construct an automorphic form on the symmetric domain thatg corresponds to an automorphic form on an orthogonal group using theta series, and prove that such a correspondence is compatible with the appropriate Hecks operator actions on the corresponding automorphic forms. As an example, we describe the case of spin groups.

MSC classification

Secondary: 11F55: Other groups and their modular and automorphic forms (several variables) 11F27: Theta series; Weil representation; theta correspondences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 1 , August 2000 , pp. 127 - 142
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Howe, R., θ-series and invariant theory, Proc. Sympos. Pure Math. 33, Part 1 (Amer. Math. Soc., Providence, 1979) pp. 275–286.Google Scholar
[2]Kashiwara, K. and Vergne, M., ‘On the Segal-Shale-Weil representations and harmonic polyomials’, Invent. Math. 44 (1978), 1–47.CrossRefGoogle Scholar
[3]Kuga, M., Fiber Varieties over a symmetric space whose fibers are abelian varieties I. II (Univ. of. Chicago, Chicago, 1963/1964).Google Scholar
[4]Lee, M. H., ‘Conjugates of equivariant holomorphic maps of symmetric domains’, Pacific J. Math. 149 (1991), 127–144.CrossRefGoogle Scholar
[5]Lee, M. H., ‘Mixed Siegel modular and Kuga fiber varieties’, Illinois J. Math. 38 (1994), 692–700.CrossRefGoogle Scholar
[6]Lee, M. H., ‘Mixed automorphic forms on semisimple Lie groups’, Illinois J. Math. 40 (1996), 464–478.CrossRefGoogle Scholar
[7]Lee, M. H., ‘Mixed automorphic vector bundles on Shimura varieties’, Pacific J. Math. 173 (1996), 105–126.CrossRefGoogle Scholar
[8]Rallis, S., ‘Langlands' functoriality and the Weil representation’, Amer. J. Math. 104 (1982), 469–515.CrossRefGoogle Scholar
[9]Satake, I., ‘Clifford algebras and families of Abelian varieties’, Nagoya Math. J. 27 (1966), 435–446.CrossRefGoogle Scholar
[10]Satake, I., Algebraic structures of symmetric domains (Princeton Univ. Press Princeton, 1980).Google Scholar
[11]Zhuravlev, V., ‘Spherical theta-series and Hecke operators’, Proc. Steklov Inst. Math. (1995), 87–110.Google Scholar