Weyl invariant polynomial and deformation quantization on Kähler manifolds

Given a polynomial PP of partial derivatives of the Kähler metric, we prove a simple criterion in terms of the coefficients for PP to be an invariant polynomial, i.e. invariant under the transformation of coordinates. As applications, we prove an explicit composition formula for covariant differential operators under a canonical basis, also known as invariant differential operators in the case of bounded symmetric domains. We also prove a general explicit formula of star products on Kähler manifolds.