TITLE: Lie group examples
QUESTION [38 upvotes]: 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.

REPLY [6 votes]: What could be a nicer example than Milnor invariants? At first sight, these "higher linkage" invariants seem to have nothing to do with Lie groups at all; you can even define them using Habiro moves to make any connection to Lie algebras even more mysterious. Or, you can speak of the Baguenaudier puzzle, whose solution involves Habiro moves in disguise, as discussed by Przytycki and Sikora.
And yet, the natural home for Milnor invariants is the group $D(H)$, which is the kernel of the left bracketting map $L(H)\otimes H \to L(H)$, where $L(H)$ denotes the free Lie algebra over $H$, the first homology of the link complement.
The group $D(H)$ also comes up in other contexts which at first sight don't seem to have anything to do with free Lie algebras. It's related to the rational homology of the outer automorphism groups of free groups, as first observed by Kontsevich. Morita, and Conant-Vogtmann, took this idea and ran with it. The group $D(H)$ was also used by Dennis Johnson to study the relative weight filtration of the mapping class group of a surface.