## General Idea

The *orbit method* relates unitary representations of Lie group $G$ with the $G$-action on the space of sections of prequantum bundles over the coadjoints orbits of $G$ that foliates $\mathfrak{g}^\ast$. Essentially this is the geometric quantization of a classical system that can be described by a coadjoint orbit $\mathfrak{g}^\ast$.

The orbit method, is still a *method*, not a theorem relating coadjoint orbits and unitary representations. For *some* Lie groups, *e.g.*, simply connected nilpotent Lie groups, there is a perfect bijection between orbits and their irreducible unitary representations.

Vogan’s review says that while there are examples of representations *not* corresponding to orbits, these ‘complementary series’ representations are sometimes the least interesting: the orbit method conjecturally provides all the presentations needed to study *automorphic forms*. This is the content of a version of the Ramanujan conjecture: any irreducible representation appearing in automorphic forms for $GL(n)$ (real or complex) must correspond to an orbit. Also the orbit method is the only map found for representations of general Lie groups, similar to the Langlands philosophy.

## “Dictionary”

Given a connected, simply connected Lie group $G$

- Describe the unitary dual $\widehat{G}$ as a topological space - Take the space $\mathcal{O}(G)$ of coadjoint orbits with the quotient topology.
- Construct the irreducible unitary representation $\pi_\Omega$ associated to the orbit $\Omega\subset\mathfrak{g}^\ast$ - Choose a point $F\in\Omega$ and take a subalgebra $\mathfrak{h}$ of maximal dimension subordinate to $F$. Put $\pi_\Omega = \operatorname{Ind}^G_H\rho_{F,H}$, where $\rho_{F,H}$ is the 1-dim unirrep of $H$ given by $\rho_{F,H}(\exp X) = e^{2\pi i \langle F,X\rangle}$.
- Describe the spectrum of $\operatorname{Res}^G_H\pi_\Omega$ - Take the projection $p(\Omega)$, where $p:\mathfrak{g}^\ast\to \mathfrak{h}^\ast$ is the canonical projection, and split it into $H$-orbits.
- Describe the spectrum of $\operatorname{Ind}^G_H\pi_\omega$ - Take the $G$-saturation of $p^{-1}(\omega)$ and split it into $G$-orbits.
- Describe the spectrum of the tensor product $\pi_{\Omega_1}\otimes \pi_{\Omega_2}$ - take the arithmetic sum $\Omega_1 + \Omega_2$ and split it into orbits.
- Compute the generalized character of $\pi_\Omega$ - $\operatorname{tr}\ \pi_\Omega (\exp X) = \int_\Omega e^{2\pi i\langle F,X\rangle + \sigma}$ where $\sigma$ is the canonical symplectic structure on a coadjoint orbit, or $\langle \chi_\Omega,\phi\rangle = \int_\Omega\tilde{\phi}(F)e^\sigma$.
- Compute the Plancherel measure $\mu$ on $\widehat{G}$ - The measure on $\mathcal{O}(G)$ arising when the Lebesgue measure on $\mathfrak{g}^\ast$ is decomposed into canonical measures on coadjoint orbits.