Given an action$G\times X\to X$ of a (discrete) group$G$ on a set $X$, any set of the form $G x = \{g x|g\in G\}$ for a fixed $x\in X$ is called an orbit of the action, or the $G$-orbit through the point $x$. The set $X$ is a disjoint union of its orbits.

If the original space is paracompact Hausdorff, then every orbit $G x$ as a topological $G$-space is isomorphic to $G/H$, where $H$ is the stabilizer subgroup of $x$.

The orbit method is a method in representation theory introduced by Kirillov, Kostant and Souriau; it is a special case of geometric quantization. The orbit method is based on the study of the representations constructed from the coadjoint orbits with Kirillov symplectic structure. The terminology ‘geometric quantization’ allows for more general underlying spaces.

Given a compactLie group$K$ with complexification$G$ and a unitary representation$\rho$ of $K$ on a finite-dimensional complex space $V$, the real orbits of the highest weight vector agrees with the complex orbits, i.e. the orbits of the extension of this representation to the representation of the complexification. These are the coherent state orbits; there is also an infinite-dimensional version for reductive groups and representations which allows them (so-called coherent state representations).

Revised on November 22, 2015 18:33:51
by Urs Schreiber
(193.55.36.81)