  We use spectral embeddings to give upper bounds on the spectral function of
the Laplace--Beltrami operator on homogeneous spaces in terms of the volume
growth of balls. In the case of compact manifolds, our bounds extend the 1980
lower bound of Peter Li for the smallest positive eigenvalue to all
eigenvalues. We also improve Li's bound itself. Our bounds translate to
explicit upper bounds on the heat kernel for both compact and noncompact
homogeneous spaces.
