Transforms of holomorphic maps from flag manifolds into Grassmannians

Building on the generalized do Carmo-Wallach theory, we construct a functorial framework for holomorphic maps from flag manifolds into Grassmannians and quadrics. Through the direct image sheaf and its inverse, we construct Penrose-type transforms that allow both the domain and target to vary. These transforms provide functors between categories of full holomorphic maps satisfying the gauge condition for semi-positive homogeneous bundles. Within this framework, Einstein-Hermitian holomorphic maps form a natural subcategory, stable under all transforms, and the minimality of L^2-norm of the mean curvature operator is preserved. For Grassmannian targets, the moduli of Einstein-Hermitian maps from a flag manifold are identified with r-tuples of non- positive integers modulo symmetry, where r is the rank of the group of holomorphic isometries. For quadric targets, we obtain a complete geometric description of the moduli space. The center of this moduli space corresponds to the Einstein-Hermitian map into the projective space, and the tower of transforms identifies all intermediate moduli spaces with the moduli of holomorphic isometric embeddings at the bottom level. The results yield a unified categorical and geometric description of holomorphic isometric embeddings of flag manifolds.