Orthogonal projection of a test configuration to vector fields

Given a polarized complex manifold, projection of a torus-equivariant test configuration to holomorphic vector fields was introduced by G. Sz\'ekelyhidi, as the limit of the associated $\mathbb{C}^*$-actions. We show that there actually holds the moment convergence of the weight distributions. Our analytic approach at the same time specifies the limit in terms of the weak geodesic ray associated with the test configuration. Related to the result, we discuss about the reduced $L^p$-norm of the test configuration in attempt to describe the uniform K-stability of the polarization relative to the automorphism group.