Nearly holomorphic sections on compact Hermitian symmetric spaces

Let X be a K"ahler manifold, and E be a Hermitian vector bundle on X. We investigate the space N(X,E) of nearly holomorphic sections in E, which generalizes the notion of nearly holomorphic functions introduced by Shimura. If X=U/K is a compact Hermitian symmetric space, and E is U-homogeneous, it turns out that N(X,E) coincides with the space of $U$-finite vectors in C^\infty(X,E), and we obtain new results on the U-type decomposition of the Hilbert space of square integrable sections. As an application, we determine this decomposition for the holomorphic tangent space of X.