So i am trying to identify a relation between the Casimir operator and the Laplacian on the sphere $S = S^{n-1} \in \mathbb{R}^n$.
We consider the Lie group $SO(n)$ with Lie algebra $\mathfrak{g}$. I use the invariant inner product
$$\beta(X,Y). = -\frac{1}{2}tr(XY), \quad X, Y \in \mathfrak{g}.$$
G acts on $\mathcal{H} = L^2(S^{n-1})$ by
$$ [\pi(g)f](x) = f(g^{-1}x), \quad g \in G, x \in S. $$
For $(\pi, \mathcal{H})$ unitary.
For a function $f \in C^\infty(S)$ we define
$$[d\pi(X)f](x) := \frac{d}{dt}|_{t=0} [\pi(e^{tX})f](x) $$
So the Casimir operator is defined
$$ \Omega_\pi = \sum_{i=1}^{n}(d\pi(X_i))^2 $$
For some orthonormal basis $\{X_i\}$ for $\mathfrak{g}$.
The laplacian on the sphere is defined as $$\Delta_S f = (\Delta \tilde{f})|_S$$ Where $\tilde{f} : \mathbb{R}^n \backslash \{0\} \to \mathbb{C}$ is the homogeneous expansion of degree 0 of $f : S \to \mathbb{C}$ : $$ \tilde{f}(x) = f\left(\frac{x}{||x||}\right). $$
So we let $X = E_{ij} - E_{ji}, (1 \leq i < j \leq n$. The first step i did was finding an expression for $d\pi(X)f$ by using $\tilde{f}$.
I get that $$d\pi(X)f = x_i \frac{\partial \tilde{f}}{\partial x_j} - x_j \frac{\partial \tilde{f}}{\partial x_i}$$.
Next step is trying to compute $\Omega_\pi f$ by using $\tilde{f}$, but here i get in some troubles. So what i get it i substitute in the first expression i found. And we let $\{ X_{ij}\}_{1 \leq i < j \leq n}$ be the orthonormal basis for $\mathfrak{g}$. \begin{align*} \Omega_\pi f= \sum_{1 \leq i < j \leq n} \left( x_i \frac{\partial \tilde{f}}{\partial x_j} - x_j \frac{\partial \tilde{f}}{\partial x_i} \right) ^2 \tilde{f} \end{align*} But cant figure out a way to get a "nicer" expression.
Edit: So I think i would like the expression to be over all i's and j's. So i would have to subtract something from the sum. But i get lost in all the indexes :)