L^p-Poisson integral representations of the generalized Hua operators on line bundles over SU(n,n)/S(U(n)xU(n))

Abstract

Let $\tau_\nu$ ($\nu \in \mathbb{Z}$) be a character of $K=S(U(n)\times U(n))$, and $SU(n,n)\times_K\mathbb{C}$ the associated homogeneous line bundle over $\mathcal{D}=\{Z\in M(n,\mathbb{C}): I-ZZ^* > 0\}$. Let $\mathcal{H}_\nu$ be the Hua operator on the sections of $SU(n,n)\times_K\mathbb{C}$. Identifying sections of $SU(n,n)\times_K\mathbb{C}$ with functions on $\mathcal{D}$ we transfer the operator $\mathcal{H}_\nu$ to an equivalent matrix-valued operator $\widetilde{\mathcal{H}}_\nu$ which acts on $\mathcal{D}$ . Then for a given ${\mathbb{C}}$-valued function $F$ on $\mathcal{D}$ satisfying $\widetilde{\mathcal{H}}_\nu F=-\frac{1}{4}(\lambda^2+(n-\nu)^2) F.(\begin{smallmatrix} I&0 0&-I \end{smallmatrix})$ we prove that $F$ is the Poisson transform by $P_{\lambda,\nu}$ of some $f\in L^p(S)$, when $1 n-1$. This generalizes the result in \cite{B1} which corresponds to $\tau_\nu$ the trivial representation.