I'm looking for interesting applications of Lie groups for an introductory Lie groups graduate course. In particular I'd like to hear of non-standard examples that at first sight do not seem to be related to Lie groups (so please don't suggest well-known things like Clifford algebras or triality that appear in standard Lie groups texts such as Fulton and Harris). Here are some examples of the sorts of things I'm looking for:
*The cohomology of a compact Kaehler manifold is a representation of SL2, so the Hopf manifold cannot be Kaehler.
*q-binomial coefficients are unimodal, as they are characters of representations of SL2
*Hilbert's theorem on the finite generation of rings of invariants can be proved using invariant integration on compact Lie groups.
*Holomorphic modular forms are really highest weight vectors of discrete series representations of certain Lie groups.
*Most closed 3-manifolds are quotients of SL2(C) by discrete subgroups.
*Bessel functions cannot be expressed using elementary functions and indefinite integration. (Differential Galois theory was one of Lie's original motivations, but seems to have been eliminated from texts on Lie theory.)
*Classifying manifolds up to cobordism uses orthogonal groups.