Take the 2-minute tour ×  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