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\begin{document}

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\section*{orbit}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{discrete_case}{Discrete case}\dotfill \pageref*{discrete_case} \linebreak
\noindent\hyperlink{category_of_orbits}{Category of orbits}\dotfill \pageref*{category_of_orbits} \linebreak
\noindent\hyperlink{topological_case}{Topological case}\dotfill \pageref*{topological_case} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{orbit_method}{Orbit method}\dotfill \pageref*{orbit_method} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{discrete_case}{}\subsubsection*{{Discrete case}}\label{discrete_case}

Given an [[action]] $G\times X\to X$ of a ([[discrete group|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 \textbf{orbit} of the action, or the \textbf{$G$-orbit through the point $x$}. The set $X$ is a [[disjoint union]] of its orbits.

\hypertarget{category_of_orbits}{}\subsubsection*{{Category of orbits}}\label{category_of_orbits}

The \emph{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 $G$.

\hypertarget{topological_case}{}\subsubsection*{{Topological case}}\label{topological_case}

If $G$ is a [[topological group]], $X$ a [[topological space]] and the action [[continuous map|continuous]], then one can distinguish [[closed subset|closed]] orbits from those which are not. Even when one starts with $G,X$ [[Hausdorff space|Hausdorff]], the space of orbits is typically non-Hausdorff. (This problem is one of the motivations of the [[noncommutative geometry]] of Connes' school.)

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

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[coadjoint orbit]]


\item [[orbit category]]


\item [[coinvariant]]


\item [[coset space]]



\end{itemize}
\hypertarget{orbit_method}{}\subsubsection*{{Orbit method}}\label{orbit_method}

The \textbf{[[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 [[compact space|compact]] [[Lie 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).

[[!redirects orbits]]

[[!redirects orbit space]] [[!redirects orbit spaces]]



\end{document}