The energy function and homogeneous harmonic maps

For equivariant maps from a compact homogeneous space into an adjoint orbit of a compact Lie group, it is shown that the energy function is the restriction of a quadratic function on the Lie algebra, providing the orbit has the metric induced from the Lie algebra. This is related to similar functions studied by R. Bott [2] and by F. C. Kirwan [7]. One obtains a simple version of the harmonic map equation, and an identity relating the energy and the square of the norm of the moment map. Several applications are given, including an example which illustrates how a change of metric in a flag manifold affects the harmonicity of equivariant maps from the two-sphere to the flag manifold.