The Orbit Method

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 $G$

  1. Describe the unitary dual $\widehat{G}$ as a topological space - Take the space $\mathcal{O}(G)$ of coadjoint orbits with the quotient topology.
  2. 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 $\rho_{F,H}(\exp X) = e^{2\pi i \langle F,X\rangle}$.
  3. 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.
  4. Describe the spectrum of $\operatorname{Ind}^G_H\pi_\omega$ - Take the $G$-saturation of $p^{-1}(\omega)$ and split it into $G$-orbits.
  5. 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.
  6. 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 $\langle \chi_\Omega,\phi\rangle = \int_\Omega\tilde{\phi}(F)e^\sigma$.
  7. 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.