What is Rigid rotor, spherical harmonics?

A rigid rotor is a body of fixed shape turning freely; for a linear molecule, $\hat H = \hat L^2/2I$, where $\hat L^2$ is the operator of the square of the angular momentum. Its eigenfunctions, which depend only on the angles $\theta$ and $\phi$, are the spherical harmonics $Y_{l,m}(\theta,\phi)$, labelled by $l = 0, 1, 2, \dots$ and $m = -l, \dots, l$; for a molecule $l$ is written $J$.