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 $\tilde{G}$. 



HIgher Homotopy Groups and Representation Theory
I don't know if this is the sort of thing you are looking for, but the higher *rational* homotopy groups appear naturally in representation theory. For example, if $G$ is a simply connected compact Lie group, then the algebra 


$Ext^\ast\_{\mathcal U (\mathfrak g)}(\mathbb C, \mathbb C) = H^\ast(\mathfrak g)$ 


is a model for the cochain algebra, so it encodes the rational homotopy groups by rational homotopy theory. Using this, one can show that such group is rationally homotopic to a product of odd spheres.


EDIT: I thought maybe I should add some extra detail. 


The cohomology algebra $H^\ast(G; \mathbb C) = H^\ast(\mathfrak g; \mathbb C)$ is a graded commutative Hopf algebra, and by a theorem of Hopf must be a symmetric algebra $S(V)$ on a graded vector space $V$ in odd degrees ("symmetric algebra" here is meant in the graded sense; as an ungraded algebra it looks like an exterior algebra). Sullivan's rational homotopy theory says that


$\pi\_n(G) \otimes \mathbb C = V^\ast$.


Moreover, the vector space $V$ can be explicitly computed as follows. The cohomology of the classifying space $H^\ast(BG)$ is given by the $W$ invariants in $H^\ast(BT)$, where $T$ is a maximal torus and $W=N\_G(T)/T$ is the Weyl group. The classifying space $BT$ is a product of $\mathbb CP^\infty$'s, thus $H^\ast(BT)$ is a polynomial algebra on the vector space $\mathfrak t^\ast$ in degree $2$. By a theorem of Chevalley, $H^\ast(BT)^W$ is also polynomial algebra on some graded vector space in even degrees. This vector space precisely $V^\ast[-1]$.


In particular, this shows that $\pi\_2(G)$ is torsion. On the other hand, consider the flag variety $G/T$. This is a compact complex manifold with a complex cell structure. Thus its cohomology is only in even degrees. In particular $\pi\_2(G/T)$ is free, and by the LES of a fibration $\pi\_2(G)$ injects into $\pi\_2(G/T)$ and is torsion free, so zero.


