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 $1<p<\infty$ or $F=P_{ \lambda,\nu}\mu$ for some Borel measure $\mu$ on the Shilov boundary $S$, when $p=1$ if and only if \[ \sup_{ 0\leq r < 1}(1-r^2)^{ \frac{ -n(n-\nu-\Re(i\lambda))}{ 2}}\left( \int_S |F(rU)|^p { \rm d}U\right) ^{ \frac{ 1}{ p}} < \infty, \] provided that the complex parameter $\lambda$ satisfies $i\lambda \notin 2\mathbb{ Z}^- +n-2\pm \nu$ and $\Re(i\lambda)>n-1$. This generalizes the result in \cite{ B1} which corresponds to $\tau_\nu$ the trivial representation.
Categories: math.RT