On the generic Simplicity of the spectrum for Connection Laplacian and $G$-simplicity on Principal Bundles

In this paper, we prove that, for a residual set of $C^k$ connections defined on a smooth vector bundle $E\to M$, all eigenvalues of the connection Laplacian operator $\mathcal{L}$, acting on the space of sections of $E$, are simple. As an application, we prove that all eigenvalues of the Laplace-Beltrami operator on a compact $G$-principal bundle $P\to M$ are $G$-simple.