Geodesic
Poisson Kernels and Pluriharmonic H2 Functions on Homogeneous Siegel Domains
Journal of Lie theory, Tome 12 (2002) no. 1, pp. 217-243.

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\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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     author = {B. Trojan },
     title = {Poisson {Kernels} and {Pluriharmonic} {H\protect\textsuperscript{2}} {Functions} on {Homogeneous} {Siegel} {Domains}},
     journal = {Journal of Lie theory},
     pages = {217--243},
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     year = {2002},
     url = {http://geodesic.mathdoc.fr/item/JLT_2002_12_1_JLT_2002_12_1_a10/}
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B. Trojan . Poisson Kernels and Pluriharmonic H2 Functions on Homogeneous Siegel Domains. Journal of Lie theory, Tome 12 (2002) no. 1, pp. 217-243. http://geodesic.mathdoc.fr/item/JLT_2002_12_1_JLT_2002_12_1_a10/