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Journal of Lie Theory, Vol. 10, No. 1, pp. 81-92 (2000)
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Poisson liftings of holomorphic automorphic forms on semisimple Lie groups

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Min Ho Lee and Hyo Chul Myung

Min Ho Lee

Department of Mathematics

University of Northern Iowa

Cedar Falls, Iowa 50614

U. S. A.

lee@math.uni.edu

and

Hyo Chul Myung

Korea Institute for Advanced Study

and KAIST, 207-43

Chunryanri-dong, Dongdaemoon-ku

Seoul 130-012, Korea

hm@kias.re.kr

**Abstract:** Let $G$ be a semisimple Lie group of Hermitian type, $K \subset G$ a maximal compact subgroup, and $P \subset G$ a minimal parabolic subgroup associated to $K$. A finite-dimensional representation of $K$ in a complex vector space 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 $\G$-invariant section of the bundle over $G/P$ under the Poisson transform is a holomorphic automorphic form on $G/K$ for $\G$. We also discuss the special case of symplectic groups in connection with holomorphic forms on families of abelian varieties.

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