Intertwining operator associated to symmetric groups and summability on the unit sphere

Abstract: An integral representation of the intertwining operator for the Dunkl operators associated with symmetric groups is derived for the class of functions of a single component. The expression provides a closed form formula for the reproducing kernels of $h$-harmonics associated with symmetric groups when one of the components is a coordinate vector. The latter allows us to establish a sharp result for the Ces`aro summability of $h$-harmonic series on the unit sphere.