Given the change of variables from the independent entries of an $N\times N$ Hermitian matrix $X$ to the coordinates of the eigenvalues and eigenvectors
$$\label{1}\tag{1} X=U\Lambda U^{-1} $$
The Laplacian transforms as follows:
\begin{equation} \Delta_X =\sum_{1\leq i<j\leq N}\frac{\partial^2}{\partial X^2_{ij}}=\frac{1}{J}\sum_{i=1}^N \frac{\partial}{\partial \lambda_i}\Big(J\frac{\partial}{\partial \lambda_i}\Big)+O_U \end{equation}
where $J$ is the jacobian determinant of the transformation \eqref{1} (Vandermonde determinant) and $O_U$ is the operator involving derivatives with respect to variables related to $U$. (see https://arxiv.org/pdf/1711.10691 for $X$ real anti-symmetric)
My question: Is the explicit form of the operator $O_U$ known?
(Given matrices $U\in G$ where $G$ is a Lie group, is $O_U$ related to the Laplace-Beltrami operator on $G$?)
