  Let G be a compact connected Lie group which is equipped with a bi-invariant
Riemannian metric. Let m(x,y)=xy be the multiplication operator. We show the
associated fibration m mapping GxG to G is a Riemannian submersion with totally
geodesic fibers and we study the spectral geometry of this submersion. We show
the pull back 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 that the pull back of a
form on the base is harmonic on the total space.
