Why are $(O(n), O(k) \times O(n-k))$ and $(U(n), U(k) \times U(n-k))$ (corresponding to the real and complex Grassmann manifolds) symmetric Gelfand Pairs? Is this true for $(Sp(n), Sp(k) \times Sp(n-k))$ (quarternions) as well?
Also, for a general local field $\mathbb F$, letting $K(n)$ denote the maximal compact subgroup of $\text{gl}(n,\mathbb F)$, is $(K(n), K(k) \times K(n-k))$ a symmetric Gelfand pair as well?