Nearly holomorphic sections on compact Hermitian symmetric spaces

TL;DR

This paper studies nearly holomorphic sections on compact Hermitian symmetric spaces, revealing their structure and decompositions, and applies these findings to the holomorphic tangent space.

Contribution

It characterizes nearly holomorphic sections as U-finite vectors and provides new U-type decompositions for square integrable sections on these spaces.

Findings

N(X,E) equals U-finite vectors in C^(X,E) for homogeneous E

New U-type decomposition results for square integrable sections

Decomposition explicitly determined for the holomorphic tangent space

Abstract

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.