Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$. Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-dimensional subspaces in $\mathbb{R}^n$.
It is well known that the representation of $G$ in $L^2(G/K)$ is multiplicity free (since $(G,K)$ is a symmetric pair), and all irreducible $G$-submodules can be parameterized explicitly in terms of highest weights of $G$.
Moreover each such irreducible $G$-submodule contains a unique (up to proportionality) non-zero $K$-invariant vector.
Can one write an explicit formula for this $K$-invariant vector as a function on the Grassmannian?
