Separately radial and radial Toeplitz operators on the projective space and representation theory

summary:We consider separately radial (with corresponding group ${\mathbb {T}}^n$) and radial (with corresponding group ${\rm U}(n))$ symbols on the projective space ${\mathbb {P}^n({\mathbb {C}})}$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^*$-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the $C^*$-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between ${\mathbb {T}}^n$ and ${\rm U}(n)$