The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ corresponding to traceless symmetric tensors. Are they also known for tensor irreps of ${\rm SO}(n)$ with mixed symmetry (Young diagrams with multiple rows)?
Alternatively, do you know a reference to an explicit calculation showing that the zonal spherical functions for symmetric irreps on ${\rm SO}(n)/{\rm SO}(n-1)$ are the Gegenbauer polynomials?
Since $({\rm SO}(n), {\rm SO}(n-1))$ is a Gelfand pair, is the zonal spherical function for some ${\rm SO}(n)$ irrep given by the corresponding character integrated over ${\rm SO}(n-1)$?
