I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
Below I am making it more precise what kind of description I need. It is a generalization of the famous Harish-Chandra isomorphism which I recall first.
Let $\mathfrak{g}$ be a complex reductive Lie algebra. Fix a Cartan subalgebra $\frak{h}$ and its positive roots. Then there exists an algebra isomorphism
$$\gamma: Z(U(\mathfrak{g}))\mathrel{\tilde{\to}} Sym(\mathfrak{h})^W$$
of the center of the universal enveloping algebra $Z(U(\mathfrak{g}))$ and the algebra $Sym(\mathfrak{h})^W$ of $W$-invariant polynomials on $\mathfrak{h}^*$ where $W$ is the Weyl group, which is uniquely characterized by the property that any element $x\in Z(U(\mathfrak{g}))$ acts on an irreducible representation $V_\lambda$ with highest weight $\lambda\in \mathfrak{h}^*$ by the scalar $\gamma(x)(\lambda+\rho)$ where $\rho$ is the half sum of positive roots.
The algebra $Z(U(\mathfrak{g}))$ can alternatively be viewed as the algebra of bi-invariant differential operators on functions on a Lie group $G$ such that $\mathfrak{g}$ is the complexification of $Lie(G)$, or the algebra of invariant differential operators on the symmetric space $(G\times G)/G$ where $G\subset G\times G$ is the diagonal subgroup.
I have heard that there exists a generalization of the above isomorphism, but in the context of compact symmetric spaces (the relevant description for symmetric spaces of non-compact type is contained in Helgason's book "Groups and Geometric Analysis"). I would be happy to have a precise statement and a reference.
Let $(G,K)$ be a symmetric pair of compact type. Assume that $G$ is connected, but $K$ does not have to be (it is the standard convention in the literature, though I will need the case of disconnected $K=O(n)$). I have heard that the algebra of invariant differential operators on $G/K$ is isomorphic to the algebra of $W'$-invariant polynomial on the Cartan space of the symmetric space $G/K$, where $W'$ is the Weyl group of $G/K$. Moreover this isomorphism is somehow uniquely characterized by the action of the differential operator on all irreducible subspaces of $L^2(G/K)$ in terms of their highest weights (all of them are known exactly for a compact symmetric space $G/K$).
