L2-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmannmanifold SU(r, r plus b)/S(U(r) x U(r plus b))

Let E-l = G x(K) C be the associated homogeneous line bundle to a one-dimensional K-representation tau(l) (l is an element of Z) over the noncompact complex Grassmann manifold G/ K; G = SU(r, r + b) and K = S(U(r) x U(r + b)). Let D(E-l) be the algebra of G-invariant differential operators on E-l. Let lambda be a real and regular spectral parameter in a*, and let F be a solution of the system differential equations on E-l: DF = chi(lambda, l) (D)F for all D in D(E-l). In this article, we obtain a necessary and sufficient condition for this F to be represented by the Poisson transform of f in the section space L-2(K x(M) C).