Laplacians on spheres

Spheres can be written as homogeneous spaces $G\mathrm{/}H$ for compact Lie groups in a small number of ways. In each case, the decomposition of $L^2(G\mathrm{/}H)$ into irreducible representations of $G$ contains interesting information. We recall these decompositions, and see what they can reveal about the analogous problem for noncompact real forms of $G$ and $H$.