The spherical Laplace equation and the spherical harmonics are a beautiful example of a differential equation dominated by the representation theory of the Lie group of rotations $SO_3$. I find much about the history of spherical harmonics and I am aware that Lie's original intention was to develop a Galois theory of differential equations.

But does anyone has historical reference, when concretely (implicitly or explicitly) the representation theory of $SO_3$ or $\mathfrak{so}_3$ was used to understand the structure of spherical harmonics. That is: dimensions of the Eigenspaces $1,3,5,\ldots$, eigenvalues, first order differential equation for highest weight vectors etc. - as it is the standard way to do in physics as well.

Thanks, Simon

$\DeclareMathOperator\SO{SO}$The spherical Laplace equation and the spherical harmonics are a beautiful example of a differential equation dominated by the representation theory of the Lie group of rotations $\SO_3$. I find much about the history of spherical harmonics and I am aware that Lie's original intention was to develop a Galois theory of differential equations.

But does anyone have a historical reference, when concretely (implicitly or explicitly) the representation theory of $\SO_3$ or $\mathfrak{so}_3$ was used to understand the structure of spherical harmonics. That is: dimensions of the eigenspaces $1,3,5,\dotsc$, eigenvalues, first order differential equation for highest weight vectors etc. — as it is the standard way to do in physics as well.