PRV for the Fusion product, the case $\lambda \gg \mu$

Given a complex simple Lie algebra $\mathfrak{g}$ and a positive integer $\mathrm{\ell }$, under the assumption $\lambda \gg \mu$, we show that irreducible representations of $\mathfrak{g}$ of the form $V(\lambda +w\mu )$, $w\in W,$ with level at most $\mathrm{\ell }$ appear in the fusion product of $V(\lambda )$ and $V(\mu )$. This verifies the existence of PRV components for the fusion product in this special case.