Homogeneous polynomial representation of SU2

The irreducible representation $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$ consists of homogeneous degree-$n$ polynomials in two variables, with the action induced from the standard two-dimensional representation. It has dimension $n+1$.

If $g\in S{U}_2$ has eigenvalues $z,z^{-1}$, then