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.
Best Answer
Here are three examples.
A finite field extension of $\mathbb R$ must be quadratic. For if $\mathbb R^n$ carries the structure of a field, then its group of units $\mathbb R^n - \{0\}$ is an abelian Lie group. But $\mathbb R^n - \{0\}$ is simply connected if $n>2$, which means that $\exp$ gives us an isomorphism of groups $\mathbb R^n - \{0\} \cong \text{Lie}(\mathbb R^n - \{0\}) = \mathbb R^n$, which is absurd.
The complex structure on the complex grassmannian $Gr(d,\mathbb C^n)$ is locally rigid. Roughly what this means is that if you have a smoothly varying family $X_t$ of complex manifolds, where the index $t$ takes values in an open connected subset of $\mathbb{C}^N$ that contains $0$, and if $X_0 = Gr(d,\mathbb C^n)$, then one can find a neighborhood $U$ of $0$ such that $X_t$ is isomorphic to $X_0$ as a complex manifold for all $t \in U$. A theorem of Frölicher and Nijenhuis states that the complex structure of a compact complex manifold $X$ is locally rigid if $H^1(X,T_X)=0$, where $T_X$ is the holomorphic tangent bundle of $X$. Using representation theory, Bott showed that $H^q(X,T_X)=0$ for all $q\geq1$ if $X=G/P$ for $G$ a complex semisimple Lie group and $P$ a parabolic subgroup, i.e., if $X$ is a "generalized flag variety." This establishes the local rigidity of the complex structure of generalized flag varieties, such as $Gr(d,\mathbb C^n)$.
It's easy to believe that the combinatorics of integer partitions and Young diagrams is related to the representation theory of the symmetric group $S_n$ (over $\mathbb C$, say). Schur--Weyl duality relates the latter to the representation theory of $GL_m(\mathbb C)$. This in turn can be related to the geometry of the flag varieties of $GL_m(\mathbb C)$. With these observations one can, for example, relate the combinatorics of Young diagrams to the multiplication in the cohomology ring of the grassmannian, which of course carries some kind of geometric information. This is quite remarkable, in my opinion.