Orbifold Bergman Kernels

Let $({X}, \omega)$ be a compact $n$-dimensional K\"ahler orbifold, the stabilizer groups of which are abelian and have rank at most two. Let ${E}$ be an orbi-ample vector bundle of rank $2$ over ${X}$ and let $H$ be a Hermitian metric on ${E}$ such that the curvature form of $\det H$ is $-2\pi \sqrt{-1} \omega$. We show that a certain weighted sum of Bergman kernels for ${Sym}^i {E} \otimes \det({E})^{k+j}$ as $i$ and $j$ vary over a finite set admit an asymptotic expansion. This extends a similar result for cyclic K\"ahler orbifolds.