Hua operators and Poisson transform for bounded symmetric domains

Khalid Koufany (1) , Genkai Zhang

1 IECN - Institut Élie Cartan de Nancy

Let $\Omega$ be a bounded symmetric domain of non-tube type in $\mathbb{C}^n$ with rank $r$ and $S$ its Shilov boundary. We consider the Poisson transform $\mathcal{P}_sf(z)$ for a hyperfunction $f$ on $S$ defined by the Poisson kernel $P_s(z, u)=\left({h(z, z)^{\frac{n}{r}}}/{|h(z, u)^{\frac{n}{r}} |^2}\right)^{s}$, $(z, u)\times \Omega\times S$, $s\in \mathbb{C}$. For all $s$ satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When $\Omega$ is the type $\mathbf{I}$ matrix domain in $M_{n, m}(\mathbb{C})$ ($n\leq m$), we prove that an eigenvalue equation for the second order $M_{n, n}-$valued Hua operator characterizes the image.

Domaines: Théorie des représentations [math.RT]

Khalid Koufany, Genkai Zhang. Hua operators and Poisson transform for bounded symmetric domains. Journal of Functional Analysis, 2006, 236, pp.546-580. ⟨hal-00091271⟩