Irreducibility and Reducibility for the Energy Representation of the Group of Mappings of a Riemannian Manifold into a Compact Semisimple Lie Group

Abstract The irreducibility of the energy representation of the group of smooth mappings from a Riemannian manifold of dimension d ⩾ 3 into a compact semisimple Lie group is proven. The nonequivalence of the representations associated with different Riemann structures is also proven for d ⩾ 3. The case d = 2 is examined and irreducibility and nonequivalence results are also given. The reducibility in the case d = 1 is pointed out (in this case the commutant contains a representation equivalent with the energy representation).