Poisson liftings of holomorphic automorphic forms on semisimple Lie groups

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Let G be a semisimple Lie group of Hermitian type, K subset of G a maximal compact subgroup, and P subset of G a minimal parabolic subgroup associated to K. If sigma is a finite-dimensional representation of It in a complex vector space, it determines the associated homogeneous vector bundles on the homogeneous manifolds G/P and G/K. The Poisson transform associates to each section of the bundle over G/P a section of the bundle over G/K, and it generalizes the classical Poisson integral. Given a discrete subgroup Gamma of G, we prove that the image of a Gamma-invariant section of the bundle over G/P under the Poisson transform is a holomorphic automorphic form on G/K for Gamma. We also discuss the special case of symplectic groups in connection with holomorphic forms on families of abelian varieties.
Publisher
HELDERMANN VERLAG
Issue Date
2000
Language
English
Article Type
Article
Citation

JOURNAL OF LIE THEORY, v.10, no.1, pp.81 - 91

ISSN
0949-5932
URI
http://hdl.handle.net/10203/70323
Appears in Collection
RIMS Journal Papers
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