TITLE: HIgher Homotopy Groups and Representation Theory
QUESTION [11 upvotes]: Let $G$ be a compact Lie group, and $g$ its associated Lie algebra.
In what ways do the higher homotopy groups $\pi_{n}(G)$ with $n>1$ appear in the representation theory of $G$? 
As an example, note that there is at least one place where the fundamental group is significant, namely in distinguishing representations of $g$ from those of $G$ itself.  Indeed, if $\tilde{G}$ is the simply connected universal cover of $G$ then all representations of $g$ can be integrated to representations of $\tilde{G}.$  However, if $G=\tilde{G}/H$ ($H$ a discrete subgroup of $\tilde{G}$) then a necessary and sufficient condition for a representation of $g$ to integrate to a representation of $G$ is that $\pi_{1}(G)=H$ acts trivially in the representation of $\tilde{G}$.

REPLY [2 votes]: When $n = 2,3$:
For any simply-connected finite-dimensional Lie group $G$, $\pi_2(G) = 1$ is trivial.  So it can have no applications.
If $G$ is simply-connected, then $\pi_3(G) = \mathrm{H}_3(G,\mathbb Z)$ by Hurewicz's theorem.  This is a copy of $\mathbb Z$ whenever $G$ is simple.  Classes here can be used to construct central extensions of the loop group of $G$.
I do not know of applications of higher $\pi_n$.