Theorems Structure of Disconnected Compact Lie Groups Attributions

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Statement

Let $G$ be a compact Lie group with identity element $e$, and let $G^\circ$ denote the [connected component](/page/Connected%20Component) of $e$ in the underlying [topological space](/page/Topological%20Space) of $G$. Define the component group by

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\begin{align*} \pi_0(G):=G/G^\circ. \end{align*}

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Then $G^\circ$ is a compact connected normal embedded Lie subgroup of $G$, and $\pi_0(G)$ is finite.

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For each $a\in G^\circ$, let

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\begin{align*} c_a:G^\circ\to G^\circ \end{align*}

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denote the inner automorphism $c_a(x)=axa^{-1}$. Define

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\begin{align*} \operatorname{Inn}(G^\circ):=\{c_a:a\in G^\circ\}\le \operatorname{Aut}(G^\circ) \end{align*}

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and

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\begin{align*} \operatorname{Out}(G^\circ):=\operatorname{Aut}(G^\circ)/\operatorname{Inn}(G^\circ). \end{align*}

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Conjugation in $G$ induces a canonical [group homomorphism](/page/Group%20Homomorphism)

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\begin{align*} \Theta:\pi_0(G)\to \operatorname{Out}(G^\circ). \end{align*}

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Explicitly, if $g\in G$, then $\Theta(gG^\circ)$ is the class in $\operatorname{Out}(G^\circ)$ of the Lie group automorphism

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\begin{align*} c_g:G^\circ\to G^\circ \end{align*}

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given by $c_g(x)=gxg^{-1}$.

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Assume in addition that $G^\circ$ is semisimple. Let $T\le G^\circ$ be a maximal torus, let $\mathfrak g^\circ$ be the [Lie algebra](/page/Lie%20Algebra) of $G^\circ$, and define

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\begin{align*} \mathfrak g^\circ_{\mathbb C}:=\mathfrak g^\circ\otimes_{\mathbb R}\mathbb C. \end{align*}

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Define the character lattice and cocharacter lattice by

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\begin{align*} X^*(T):=\operatorname{Hom}_{\mathrm{cts}}(T,S^1) \end{align*}

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and

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\begin{align*} X_*(T):=\operatorname{Hom}_{\mathrm{cts}}(S^1,T). \end{align*}

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Let $R\subset X^*(T)$ be the root system of $\mathfrak g^\circ_{\mathbb C}$ with respect to $T$, and let $R^\vee\subset X_*(T)$ be the corresponding coroot system. The root datum of $G^\circ$ with respect to $T$ is

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\begin{align*} \mathcal R(G^\circ,T):=(X^*(T),R,X_*(T),R^\vee). \end{align*}

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Let

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\begin{align*} N_{G^\circ}(T):=\{a\in G^\circ:aTa^{-1}=T\} \end{align*}

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and define the Weyl group

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\begin{align*} W(G^\circ,T):=N_{G^\circ}(T)/T. \end{align*}

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For each component $gG^\circ\in\pi_0(G)$, the automorphism $c_g$ sends $T$ to another maximal torus. If $a\in G^\circ$ satisfies $a c_g(T)a^{-1}=T$, then $c_a\circ c_g$ preserves $T$ and induces an automorphism of the root datum $\mathcal R(G^\circ,T)$. The class of this root-datum automorphism modulo the Weyl-[group action](/page/Group%20Action) of $W(G^\circ,T)$ is independent of the representative $g$ and of the choice of $a$. This gives a homomorphism from $\pi_0(G)$ to the group of automorphisms of $\mathcal R(G^\circ,T)$ modulo the Weyl-group action.

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Equivalently, choose a positive root system $R^+\subset R$, let $\Delta\subset R^+$ be the corresponding simple roots, and let $\Delta^\vee\subset R^\vee$ be the corresponding simple coroots. The based root datum is

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\begin{align*} \mathcal R_b(G^\circ,T,R^+):=(X^*(T),\Delta,X_*(T),\Delta^\vee). \end{align*}

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For compact connected semisimple $G^\circ$, the standard automorphism theorem for root data identifies $\operatorname{Out}(G^\circ)$ with $\operatorname{Aut}(\mathcal R_b(G^\circ,T,R^+))$, equivalently with automorphisms of $\mathcal R(G^\circ,T)$ modulo the Weyl-group action. Under this identification, $\Theta$ is recorded by a homomorphism

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\begin{align*} \Theta_b:\pi_0(G)\to \operatorname{Aut}(\mathcal R_b(G^\circ,T,R^+)). \end{align*}

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In Dynkin-diagram language, these are precisely the diagram automorphisms that preserve the character and cocharacter lattices of $G^\circ$, including the finite central quotient data.

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