Journal of Lie Theory, Vol. 10, No. 1, pp. 81-92 (2000)

Poisson liftings of holomorphic automorphic forms on semisimple Lie groups

Min Ho Lee and Hyo Chul Myung

Min Ho Lee
Department of Mathematics
University of Northern Iowa
Cedar Falls, Iowa 50614
U. S. A.
Hyo Chul Myung
Korea Institute for Advanced Study
and KAIST, 207-43
Chunryanri-dong, Dongdaemoon-ku
Seoul 130-012, Korea

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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