Kirillov's orbit method in representation theory establishes a correspondence (which is not exact in general) between irreducible unitary representations of a Lie group $G$ and orbits of the action of $G$ on the dual of its Lie algebra $\mathfrak{g}^{\ast}$. The physical intuition behind the orbit method comes from the notion of quantizing a classical system: the rough idea, as I understand it, is that orbits of the action of $G$ on $\mathfrak{g}^{\ast}$ should be thought of as classical systems with $G$-symmetry, and the corresponding irreducible representations of $G$ should be thought of as the corresponding quantum systems.