The spectral geometry of the canonical Riemannian submersion of a compact Lie group

Let GG be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xym(x,y)=xy be the multiplication operator. We show the associated fibration m:G×G→Gm:G×G→G is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull-backs of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions for the pull-back of a form on the base to be harmonic on the total space.