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We consider separately radial (with corresponding group $\mathbb{T}^n$) andradial (with corresponding group $\mathrm{U}(n))$ symbols on the projectivespace $\mathbb{P}^n(\mathbb{C})$, as well as the associated Toeplitz operatorson the weighted Bergman spaces. It is known that the $C^*$-algebras generatedby each family of such Toeplitz operators are commutative. We present a newrepresentation theoretic proof of such commutativity. Our method is easier andmore enlightening as it shows that the commutativity of the $C^*$-algebras is aconsequence of the existence of multiplicity-free representations. Furthermore,our method shows how to extend the current formulas for the spectra of thecorresponding Toeplitz operators to any closed group lying between$\mathbb{T}^n$ and $\mathrm{U}(n)$.

Separately Radial and Radial Toeplitz Operators on the Projective Space and Representation Theory