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.

Category of orbits

The category of orbits of a group$G$ is the full subcategory of the category of sets with an action of $G$.

Since any orbit of $G$ is isomorphic to the orbit $G/H$ for some group $H$, the category of $G$-orbits admits the following alternative description: its objects are subgroups $H$ of $G$ and morphisms $H_1\to H_2$ are elements $[g]\in G/H_2$ such that $H_1\subset gH_2g^{-1}$.

In particular, the group of automorphisms of a $G$-orbit $G/H$ is $N_G(H)/H$, where $N_G(H)$ is the normalizer of $H$ in $G$.

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).

Last revised on June 16, 2019 at 09:48:54.
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