Structure of compact groups

Denote by $\widehat{G}$ the set of classes of irreducible continuous unitary representations of G on finite dimensional complex Hilbert spaces (Spectral Theory, in preparation). For all $u\in \widehat{G}$, let $Hu$ be the space of $u$ and $\rho_u: G\rightarrow \mathbf{U}(H_u)$ the homomorphism associated to $u$. By the Peter-Weyl theorem (Spectral Theory, in preparation), the continuous homomorphism $\rho = (\rho_u)_{u\in\widehat{G}}$ from G to $\prod_{u\in\widehat{G}}\mathbf{U}(H_u)$ is injective; since G is compact, $\rho$ induces an isomorphism

from G onto a closed subgroup of the group $\prod_{u\in\widehat{G}}\mathbf{U}(H_u)$.

Let $(K_{\lambda})_{\lambda\in L}$ be a family of compact Lie groups such that G can be identified with a closed subgroup of $\prod_{\lambda\in L}K_{\lambda}$; for $\lambda \in L$, demote by $p_{\lambda}: G\rightarrow K_{\lambda}$ the restriction to G of the canonical projection. There exists a finite subset $J\subset L$, and for each $\lambda \in J$ a neighbourhood $V_{\lambda}$ of the origin in $K_{\lambda}$, such that V contains $\bigcap_{\lambda\in J}p^{-1}_{\lambda}(V_{\lambda})$. It now suffices to put $H =\bigcap_{\lambda\in J}$ Ker($p_{\lambda}$).

Denote by $(H_{\alpha})_{\alpha\in I}$ the decreasing filtered family of closed normal subgroups of G, such that the quotient $G/H_{\alpha}$ is a Lie group. Consider the projective system of compact Lie groups $G/H_{\alpha}$ (cf. General Topology, Chap. III, §7, no. 2, Prop. 2).

Indeed, Cor. 1 implies that condition (PA) of General Topology, Chap. III, §7, no. 2, is satisfied; the assertion now follows from Prop. 2 of loc. cit.

The necessity of this condition has already been proved (Chap. III, §4, no. 2, Cor. 1 of Th. 2), and the sufficiency is an immediate consequence of Cor. 1.

a) The subgroups $D(G_{\alpha})$ (resp. $C(G_{\alpha})$, resp. $C(G_{\alpha})_0$) form a projective system of subsets of $G_{\alpha}$.

b) We have D(G) = lim$_{_{\leftarrow -}\alpha}\overline{D(G_{\alpha})}$ and C(G) = lim$_{_{\leftarrow -}\alpha}C(G_{\alpha})$.

c) If $G_{\alpha}$ is compact for all $\alpha \in I$, then $C(G)_0=$ lim$_{_{\leftarrow -}\alpha}C(G_{\alpha})_0$.

Let $\alpha , \beta$ be two elements of I, with $\alpha \leq \beta$. Then $f_{\alpha \beta}(D(G_{\beta}))\subset D(G_{\alpha})$, and $f_{\alpha \beta}(C(G_{\beta}))\subset C(G_{\alpha})$ since $f_{\alpha \beta}$ is surjective; since $f_{\alpha \beta}$ is continuous, it follows that $f_{\alpha \beta}(\overline{D(G_{\beta})})\subset \overline{D(G_{\alpha})}$ and $f_{\alpha \beta}(C(G_{\beta})_0)\subset C(G_{\alpha})_0$, hence a). Since $f_{\alpha}$ is surjective, $f_{\alpha}(D(G)) = D(G_{\alpha})$ (Algebra, Chap. I, §6, no. 2, Prop. 6), so $\overline{D(G)}=$ lim$_{\leftarrow -}\overline{D(G_{\alpha})}$ (General Topology, Chap. I, §4, no. 4, Cor. of Prop. 9). The surjectivity of $f_{\alpha}$ also implies the inclusion $f_{\alpha}(C(G))\subset$ $C(G_{\alpha})$ and hence $C(G)\subset$ lim$_{\leftarrow -}C(G_{\alpha})$; the opposite inclusion is immediate. Finally, assertion c) follows from b) and General Topology, Chap. III, §7, no. 2, Prop. 4).

Denote by $\mathfrak{s}_a$ (resp. $\mathfrak{t}_b$) the Lie algebra of $S_a$ (resp. $T_b$) for $a\in A$ (resp. $b\in B$), and consider the homomorphism $L(u) :\prod_{a\in A}\mathfrak{s}_a\rightarrow \prod_{b\in B}\mathfrak{t}_b$. Its kernel

is an ideal of the semi-simple Lie algebra $\prod_{a\in A}\mathfrak{s}_a$, and hence is of the form $\prod_{a\in A''}\mathfrak{s}_a$, with $A''\subset A$ (Chap. I, §6, no. 2, Cor. 1). Put $A'= A$-- $A''$. By

restriction, $L(u)$ induces an isomorphism $f:\prod_{a\in A'}\mathfrak{s}_a\rightarrow \prod_{b\in B}\mathfrak{t}_b$. By loc. cit., for all $a\in A'$ the ideal $f(\mathfrak{s}_a)$ is equal to one of the $\mathfrak{t}_b$; hence, there exists a bijection $l: B\rightarrow A'$ such that $f(\mathfrak{s}_{l(b)}) =\mathfrak{t}_b$ for $b\in B$, and $f$ induces an isomorphism $f_b:\mathfrak{s}_{l(b)}\rightarrow \mathfrak{t}_b$. Since the groups $S_a$ and $T_b$ are simply-connected, there exist isomorphisms $u_b: S_{l(b)}\rightarrow T_b$ such that $L(u_b) =f_b$ for $b\in B$ (Chap. III, §6, no. 3, Th. 3).

Denote by $\widetilde{u}:\prod_{a\in A}S_a\rightarrow \prod_{b\in B}T_b$ the morphism defined by $\widetilde{u}((s_a)_{a\in A}) =$

$(u_b(s_{l(b)}))_{b\in B}$. By construction, $L(\widetilde{u}) =f= L(u)$, so $\widetilde{u}=u$, which proves the lemma.

For all $\alpha \in I$, the group $G_{\alpha}$ is the direct product of a finite family of almost simple, simply-connected subgroups $(S^{\lambda}_{\alpha})_{\lambda\in L_{\alpha}}$ (Chap. III, §9, no. 8, Prop. 28). Let $\beta \in I,\beta \geq \alpha$. By Lemma 2, there exists a map $l_{\beta \alpha}: L_{\alpha}\rightarrow L_{\beta}$ such that $f_{\alpha \beta}(S^{l_{\beta \alpha}(\lambda)}_{\beta}) = S^{\lambda}_{\alpha}$ for $\lambda \in L_{\alpha}$. We have $l_{\gamma \beta}\circ l_{\beta \alpha}=l_{\gamma \alpha}$ for $\alpha \leq \beta \leq \gamma$, so $(L_{\alpha}, l_{\beta \alpha})$ is an inductive system of sets relative to I. Let L be its limit; the maps $l_{\beta \alpha}$ being injective, $L_{\alpha}$ can be identified with a subset of L, so that $L =\bigcup_{\alpha\in I}L_{\alpha}$.

Let $\lambda \in L$. Put $S^{\lambda}_{\alpha}=\{1\}$ when $\lambda \notin L_{\alpha}$, and denote by $\varphi^{\lambda}_{\alpha \beta}: S^{\lambda}_{\beta}\rightarrow S^{\lambda}_{\alpha}$ the morphism induced by $f_{\alpha \beta}$; this gives a projective system of topological groups $(S^{\lambda}_{\alpha}, \varphi^{\lambda}_{\alpha \beta})$, whose limit is isomorphic to $S_{\lambda}$. The canonical homomorphism of topological groups

lim$_{_{\leftarrow -\alpha\in I}}((\prod_{\lambda\in L}S^{\lambda}_{\alpha})\rightarrow \prod_{\lambda\in L}($lim$_{_{\alpha\leftarrow -\in I}}S^{\lambda}_{\alpha})$

is bijective (Theory of Sets, Chap. III, §7, no. 3, Cor. 2); it is thus an isomorphism since the groups in question are compact. But the first of these groups can be identified with G and the second with the product of the $S_{\lambda}$, hence the lemma.

a$) C(G)_0$ is a commutative connected compact group; D(G) is a connected compact group, equal to its derived group.

b) The continuous homomorphism $(x, y) \rightarrow xy$ from $C(G)_0\times D(G)$ to G is surjective and its kernel is a central subgroup of $C(G)_0\times D(G)$ that is compact and totally discontinuous.

c) There exists a family $(S_{\lambda})_{\lambda\in L}$ of almost simple compact Lie groups and a surjective continuous homomorphism $\prod_{\lambda\in L}S_{\lambda}\rightarrow D(G)$, whose kernel is a

totally discontinuous, compact, central subgroup.

Let $(G_{\alpha}, f_{\alpha \beta})$ be a projective system of compact Lie groups, relative to a filtered set I, such that G is isomorphic to lim$_{\leftarrow -}G_{\alpha}$ and such that the canonical maps $f_{\alpha}: G\rightarrow G_{\alpha}$ are surjective (Cor. 2 of Prop. 1). For $\alpha \in I$, let $\pi_{\alpha}: \widetilde{D}(G_{\alpha})\rightarrow D(G_{\alpha})$ be a universal covering of the group $D(G_{\alpha})$. The $f_{\alpha \beta}$ induce morphisms $\widetilde{f}_{\alpha \beta}: \widetilde{D}(G_{\beta})\rightarrow \widetilde{D}(G_{\alpha})$, and $( \widetilde{D}(G_{\alpha}),\widetilde{f}_{\alpha \beta})$ is a projective system of topological groups satisfying the hypotheses of Lemma 3.

It follows from this lemma that the topological group lim$_{\leftarrow -}\widetilde{D}(G_{\alpha})$ is isomorphic to the product of a family $(S_{\lambda})_{\lambda\in L}$ of almost simple compact Lie groups. By Lemma 1, the limit of the projective system of homomorphisms $(\pi_{\alpha})$ can be identified with a continuous homomorphism $\pi :\prod_{\lambda\in L}S_{\lambda}\rightarrow D(G)$,

which is surjective (General Topology, Chap. I, §9, no. 6, Cor. 2 of Prop. 8).

Now observe that the group $\prod_{\lambda\in L}S_{\lambda}$ is equal to its derived group: this

follows from §4, no. 5, Cor. of Prop. 10. The same is true for D(G), since $\pi$ is surjective. Consequently, $D(G)\supset D(D(G)) = D(G)$. Thus, the group D(G) is compact and equal to its derived group; this proves a), since the assertions concerning $C(G)_0$ are trivial.

On the other hand, the kernel of $\pi :\prod_{\lambda\in L}S_{\lambda}\rightarrow D(G)$ can be identified

with lim$_{\leftarrow -}$ Ker($\pi_{\alpha}$) $($Algebra, Chap. II, §6, no. 1, Remark 1), and thus with a

compact, totally discontinuous, central subgroup, hence c).

We prove b). For all $\alpha$ in I, the morphism $s_{\alpha}: C(G_{\alpha})_0\times D(G_{\alpha})\rightarrow G_{\alpha}$ such that $s_{\alpha}(x, y) =xy$ for $x\in C(G_{\alpha})_0, y\in D(G_{\alpha})$, is surjective and its kernel is a finite central subgroup (§1, no. 4, Cor. 1 of Prop. 4). The $s_{\alpha}$ form a projective system of maps, whose limit can, by the preceding, be identified with the homomorphism $(x, y) \rightarrow xy$ from $C(G)_0\times D(G)$ to G. We now see as before that this map is surjective and that its kernel is central and totally discontinuous, hence b).

Indeed, the derived group is then solvable and equal to its derived group (Prop. $2a$)$)$, hence reduced to the identity element.