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Journal of Lie Theory 12 (2002), No. 1, 217--243
Copyright Heldermann Verlag 2002

Poisson Kernels and Pluriharmonic H2 Functions on Homogeneous Siegel Domains

Bartosz Trojan
Instytut Matematyczny, Uniwersytet Wroclawski, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland


\newcommand{\hp}[1]{$\mathcal{H}^{#1}$} We prove that a real function $F$ defined on a homogeneous not necessarily symmetric Siegel domain satisfying an \hp{2} condition is pluriharmonic if and only if $\mathbf{H} F=0$, $\mathcal{L}F=0$, $L F=0$, where $\mathbf{H}$, $\mathcal{L}$, $L$ are second order differential operators. This generalizes the result of E. Damek, A. Hulanicki, D. M\"uller, and M. Peloso ["Pluriharmonic \hp{^2} functions on symmetric irreducible Siegel domains, Geom. Funct. Anal. 10 (2000) 1090--1117], where symmetric domains were considered. Our approach to study non-symmetric case is based on $T$-algebras introduced by E. B. Vinberg ["The theory of convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963) 340--403].

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