SEPARATELY RADIAL AND RADIAL TOEPLITZ OPERATORS ON THE PROJECTIVE SPACE AND REPRESENTATION THEORY

We consider separately radial (with corresponding group T-n) and radial (with corresponding group U(n)) symbols on the projective space P-n(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 T-n and U(n).