Principal curvatures of isoparametric hypersurfaces in $\mathbb{C}P^{n}$

Let $M$ be an isoparametric hypersurface in $\mathbb{C}P^{n}$, and $\overline{M}$ the inverse image of $M$ under the Hopf map. By using the relationship between the eigenvalues of the shape operators of $M$ and $\overline{M}$, we prove that $M$ is homogeneous if and only if either $g$or $l$ is constant, where $g$ is the number of distinct principal curvatures of $M$ and $l$ is the number of non-horizontal eigenspaces of the shape operator on $\overline{M}$.