I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the Euclidean space $\mathbb{R}^n$. 


I was told that for general symmetric spaces there is a Harish-Chandra type theorem saying roughly that the above algebra is isomorphic to the algebra of polynomials invariant under appropriate Weyl group. However in the case of Grassmannians I would like to have a more explicit description not referring to this theorem.


To give an example of what I am looking for, let me mention that the algebra of $SO(n)$- (or $O(n)$-) invariant operators on the sphere $S^{n-1}$ ($n>2$) is precisely equal to the algebra of polynomials in the Laplace operator. 



Invariant differential operators on real Grassmannians
The answer to my original question of description of the algebra of $O(n)$-invariant operators on real Grassmannians is explicitly contained in Proposition 3.2 of the paper by
Fulton B. Gonzalez, Tomoyuki Kakehi; "Pfaffian systems and Radon transforms on affine Grassmann manifolds", Mathematische Annalen 326 (2003).


