Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations

Consider the space of $C^{\mathrm{\infty }}$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus {\mathcal{H}}_n$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus {\mathcal{H}}_n$ as functions of two variables, a nonnegative integer variable $n$ and a complex variable $u$ (a restriction of such function to the set $n=k$ is a polynomial in $u$ of degree $\le 2k$). For operators of multiplication by $x_1$, $x_2$, $x_3$ in $C^{\mathrm{\infty }}(S^2)$ we obtain the corresponding operators in $\oplus {\mathcal{H}}_n$, they are differential-difference operators in the variables $u$, $n$ (including second derivatives in $u$ and shifts $n\mapsto n\pm 1$). We obtain the similar correspondence for operators of differentiation along conformal vector fields on $S^2$.