The Strichartz conjecture for the Poisson transform on homogeneous line bundles over Noncompact Complex Grassmann manifolds

Let $X=G/K$ be a noncompact complex Grassmann manifold of rank $r$. Let $\tau_l$ be a character of $K$, $G\times_P{\C}$ and $G\times_K{\C}$ the homogeneous line bundles associated with the representations $\sigma_{\lambda,l}=\tau_l\otimes a^{\rho-i\lambda}\otimes 1$ of $P=MAN$ and $\tau_l$ of $K$. We give an image characterization for the Poisson transform $P_{\lambda,l}$ of \\$L^2$-sections of the unitary principal series representations of $G$ parametrized by $\sigma_{\lambda,l}$. More precisely for real and regular parameter $\lambda$ in $\mathfrak{a}^\ast$ we prove that $P_{\lambda,l}$ is an isomorphism from $L^2(K\times_M{\C})$ onto the space of joint eigensections $F$ of the algebra of $G$-invariant differential operators on $G\times_K{\C}$ that satisfy the following growth condition \begin{eqnarray*} \sup_{R>1}\frac{1}{R^r}\int_{B(R)}\mid F(g)\mid^2\, {\rm d}g<\infty. \end{eqnarray*} This generalizes a conjecture by Strichartz which corresponds to $\tau_l$ trivial.