A weighted Plancherel formula II. The case of the ball

The group SU(1,d) acts naturally on the Hilbert space ${\displaystyle L²\left(Bd{\mu }_{\alpha }\right)(\alpha >-1)}$, where B is the unit ball of ${\displaystyle {ℂ}^d}$ and ${\displaystyle d{\mu }_{\alpha }}$ the weighted measure ${\displaystyle {(1-\left|z\right|²)}^{\alpha }dm\left(z\right)}$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic tensor fields.