Rotational Splitting of Multiplets

What you are seeing: the oscillating surface of the star and the multiplet it produces. The body bulges out (red) and caves in (blue) in the mode pattern. By default the star shows the full multiplet: all $2\ell+1$ components excited together. With equal amplitudes that superposition is a fixed pattern $F(\theta,\phi-st)$ that pulsates at $\nu_0$ and rigidly rotates in azimuth at the splitting rate $s=(1-C_{n\ell})\Omega$. Set $\Omega=0$ and the $2\ell+1$ components are degenerate (one frequency): the pattern pulsates in place and does not rotate. Turn rotation on and it starts to circle the axis, and that circulation is the splitting itself, $\nu_m=\nu_0+m(1-C_{n\ell})\Omega$. The $m=0$ component stays at $\nu_0$; prograde ($m\gt0$) and retrograde ($m\lt0$) split to opposite sides. The $+m\Omega$ part is rigid advection by rotation; the $-mC_{n\ell}\Omega$ part is the Coriolis force in the corotating frame ($C\to0$ for p modes, $1/\ell(\ell+1)$ for g modes). The peak heights are set by the inclination $i$ between the rotation axis and the line of sight, through the visibility $E_{\ell m}(i)=\frac{(\ell-|m|)!}{(\ell+|m|)!}\,[P_\ell^{|m|}(\cos i)]^2$: pole-on ($i=0$) shows only $m=0$, equator-on ($i=90^\circ$) suppresses it. Switch "show on star" to a single component to isolate one $m$.