  We study Toeplitz operators with separately radial and radial symbols on the
weighted Bergman spaces on the unit ball. The unitary equivalence of such
operators with multiplication operators on $\ell^2$ spaces was previously
obtained by analytic methods elsewhere (see the references). We prove that the
same constructions can be performed with a purely representation theoretic
approach to obtain the same conclusions and formulas. However, our method is
shorter, more elementary and more elucidating.
