Fundamental groups of split real Kac-Moody groups and generalized real flag manifolds. With appendices by Tobias Hartnick and Ralf K\"ohl and by Julius Gr\"uning and Ralf K\"ohl

TL;DR

This paper computes the fundamental groups of split real Kac-Moody groups with the Kac-Peterson topology by analyzing their flag varieties, extending known results from finite-dimensional cases to infinite-dimensional Kac-Moody groups.

Contribution

It determines the fundamental groups of symmetrizable split real Kac-Moody groups and their flag varieties, generalizing finite-dimensional results to infinite-dimensional Kac-Moody settings.

Findings

Fundamental groups of Kac-Moody groups are determined via flag varieties.

Results apply to all symmetrizable cases with CW decompositions.

The structure of fundamental groups extends to non-symmetrizable two-spherical cases.

Abstract

We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, the Iwasawa decomposition $G = KAU_+$ provides a weak homotopy equivalence $K \hookrightarrow G$, implying $\pi_1(G) = \pi_1(K)$. It thus suffices to determine $\pi_1(K)$ which we achieve by investigating the fundamental groups of generalized flag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized flag variety is a CW decomposition $-$ in particular, we cover the complete symmetrizable situation; the result concerning the structure of $\pi_1(K)$ more generally also holds in the non-symmetrizable two-spherical situation.

Equations

$${\pi }_1(G(\Pi ))\cong {\mathbb{Z}}^{n(g)}\times C_2^{n(b)}.$$

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{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{33.63199pt}{-113.81104pt}\pgfsys@lineto{51.72629pt}{-113.81104pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}} {{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-118.99026pt}\pgfsys@lineto{0.0pt}{-137.08456pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\pi_{1}(G(X))\cong\mathbb{Z}^{2}\times C_{2}^{2}\\ \hline\cr\end{array}$$

$$\mathrm{Spin}(3)\to \mathrm{Spin}(\Pi )\to \mathrm{Spin}(\Pi )/\mathrm{Spin}(3)$$

$$\{1\}={\pi }_1(\mathrm{Spin}(3))\to {\pi }_1(\mathrm{Spin}(\Pi ))\to {\pi }_1(\mathrm{Spin}(\Pi )/\mathrm{Spin}(3))$$

$$⟨x_i;i\in I\mid x_i{x}_j^{\epsilon (i,j)}=x_j{x}_i,x_k=1;i,j\in I,k\in J⟩.$$

$${\pi }_1(\mathrm{Spin}(\Pi ,\kappa ))\cong {\mathbb{Z}}^{n(g)}\times C_2^{n(b,\kappa )}.$$

$${\mathfrak{g}}_{\alpha }^k:=\{X\in {\mathfrak{g}}_k(\mathbf{A})\mid \forall H\in {\mathfrak{h}}_k(\mathbf{A}{)}^{\ast }:[H,X]=\alpha (H)X\}.$$

$${\mathfrak{g}}_k(\mathbf{A})={\mathfrak{h}}_k(\mathbf{A})\oplus \underset{\alpha \in \Delta }{⨁}{\mathfrak{g}}_{\alpha }^k.$$

$${\mathfrak{u}}^{\pm }:=\underset{\alpha \in {\Delta }_{\pm }}{⨁}{\mathfrak{g}}_{\alpha }^k$$

$${\sigma }_i(\lambda ):=\lambda -\lambda ({\check{\alpha }}_i){\alpha }_i.$$

$$\mathrm{Ad}:G_{\mathbf{A}}(\mathbb{R})\to \mathrm{Aut}({\mathfrak{g}}_{\mathbb{R}}(\mathbf{A})),$$

$$\mathrm{Ad}:G(\Pi )\to \mathrm{Aut}({\mathfrak{g}}_{\mathbb{R}}^{\prime }(\mathbf{A}))$$

$$\mathrm{ad}(X)(e_i)=[X,e_i]={\alpha }_i(X)e_i,$$

$$\mathrm{ad}(X)(f_i)=[X,f_i]=-{\alpha }_i(X)f_i,$$

$$\delta :G/B\times G/B$$

$$\delta (gB,hB)=w$$

$$C_w(gB):=\{hB\in G/B\mid \delta (gB,hB)=w\},$$

$$C_{\le w}(gB):=\bigcup _{v\le w}{C}_v(gB)$$

$$C_{<w}(gB):=C_{\le w}(gB)\setminus C_w(gB).$$

$$\mathrm{Spin}(3)\to \mathrm{Spin}(\Pi )\to \mathrm{Spin}(\Pi )/\mathrm{Spin}(3)$$

$$\{1\}={\pi }_1(\mathrm{Spin}(3))\to {\pi }_1(\mathrm{Spin}(\Pi ))\to {\pi }_1(\mathrm{Spin}(\Pi )/\mathrm{Spin}(3))=1.$$

$$\mathrm{cl}Bw{P}_J/P_J=\bigcup _{x\le w}Bx{P}_J/P_J=\bigcup _{\begin{array}{c}x\le w\\ x\in W^J\end{array}}Bx{P}_J/P_J,$$

$${\psi }^{-1}(A)\cap {\bar{e}}_w^{\prime }={\psi }^{-1}(A)\cap {\psi }^{-1}({\bar{e}}_w)={\psi }^{-1}(A\cap {\bar{e}}_w)={\psi }_{\bar{w}}^{-1}(A\cap {\bar{e}}_w).$$

$$⟨[{\chi }_{\mu }],\mu \in {\Lambda }_1\mid [{\gamma }_{\lambda }],\lambda \in {\Lambda }_2⟩$$

$${\gamma }_i(R(s)){\gamma }_j(R(t))B$$

$$⟺({\gamma }_i(R(\tilde{s})){)}^{-1}{\gamma }_i(R(s)){\gamma }_j(R(t))B$$

$${\chi }_{i,j}(s,t)$$

$$=x_i{b}^{-1}\cdot b{x}_{j}B$$

$$=x_i{x}_{j}B.$$

$$⟨x_i;i\in I\mid x_i{x}_j^{\epsilon (i,j)}=x_j{x}_i,x_k=1;i,j\in I,k\in J⟩.$$

$${\phi }_1(t)$$

$${\phi }_2(t)$$

Full text

1 Introduction

The structure of maximal compact subgroups in semisimple Lie groups was investigated by Cartan and, later, Mostow: In [Mos49], Mostow gives a new proof of a Cartan’s theorem stating that a connected semisimple Lie group $G$ is a topological product of a maximal compact subgroup $K$ and a Euclidean space, implying in particular that $G$ and $K$ have isomorphic fundamental groups. Subsequent case-by-case analysis provided the isomorphism types of these maximal compact subgroups – which in the split real situation turn out to be all classical – and their fundamental groups; tables of the maximal compact subgroups can be found in [Hel78, p 518], their fundamental groups in [SBG*+*95, 94.33].

Starting in the 1940’s, Dynkin diagrams, introduced in [Dyn46], have been used to describe the structure of simple Lie groups. In this article, we present a uniform result which makes it possible to determine the fundamental group of any algebraically simply connected split real simple Lie group – and, more generally, any algebraically simply-connected semisimple split real topological Kac–Moody group – directly from its Dynkin diagram.

In [Tit87, Theorem 1], Tits for every generalized Cartan matrix $\mathbf{A}$ provides a functor $G_{\mathbf{A}}:\mathfrak{cRings}\to\mathfrak{Groups}$ from commutative rings into groups. Let $\Pi$ be the Dynkin diagram of $\mathbf{A}$.

Definition 1.1.

We set $G(\Pi):=[G_{\mathbf{A}}(\mathbb{R}),G_{\mathbf{A}}(\mathbb{R})]$ and refer to this group as the algebraically simply-connected semisimple split real Kac–Moody group of type $\Pi$.

Kac–Moody groups endowed with the Kac–Peterson topology have been studied extensively by the second author together with Glöckner and Hartnick in [GGH10] and with Hartnick and Mars in [HKM13]. Our result is applicable to those Kac–Moody groups whose Bruhat decompositions are CW decompositions and for which the embedding $K\hookrightarrow G$ is a weak homotopy equivalence.

In order to fix notations, let $G=G(\Pi)$ be the algebraically simply-connected split real semisimple Kac–Moody group associated to an irreducible diagram $\Pi=(V,E)$ endowed with the Kac–Peterson topology (for definitions, see Section 2). Let $K=K(\Pi)$ be the so-called maximal compact subgroup of the topological group $G(\Pi)$, i.e., the subgroup fixed by the Cartan–Chevalley involution $\theta$ of $G(\Pi)$. We stress that in the infinite-dimensional non-Lie case this topological group $K$ is not a compact group, only a $k$-group, in fact a $k_{\omega}$-group.

Given the Dynkin diagram $\Pi=(V,E)$ with a fixed labelling $\lambda:\{1,\dots,n\}=:I\to V$, we define a modified diagram $\Pi^{\mathrm{adm}}$ with vertex set $V$ and $\{i^{\lambda},j^{\lambda}\}\in V\times V$ edge if and only if $\varepsilon(i,j)=\varepsilon(j,i)=-1$, where $\varepsilon(i,j)$ denotes the parity of the corresponding Cartan matrix entry. To each connected component $\bar{\Pi}^{\mathrm{adm}}$ of $\Pi^{\mathrm{adm}}$ we then assign a colour as follows: Let $\bar{\Pi}^{\mathrm{adm}}$ be coloured red (denoted by $r$) if it contains a vertex $i^{\lambda}$ such that there exists a vertex $j^{\lambda}\in V$ satisfying $\varepsilon(i,j)=1$ and $\varepsilon(j,i)=-1$; let $\bar{\Pi}^{\mathrm{adm}}$ be coloured green ($g$) if it is not red and consists only of an isolated vertex; and blue ($b$) else.

One can then read off the isomorphism type of $\pi_{1}(G(\Pi))$ from the coloured diagram $\Pi^{\mathrm{adm}}$ as specified in the following theorem.

Theorem**.**

Let $\Pi$ be an irreducible Dynkin diagram such that the Bruhat decomposition of $G(\Pi)$ provides a CW decomposition (i.e., such that the conclusion of Proposition 3.7 holds) and such that the embeding $K\hookrightarrow G(\Pi)$ is a weak homotopy equivalence (i.e., such that the conclusion of Theorem A.15 holds). Let $n(g)$ and $n(b)$ be the number of connected components of $\Pi^{\mathrm{adm}}$ of colour $g$ and $b$, respectively. Then

[TABLE]

In particular, this statement holds in the symmetrizable case.

Example: 0 Isomorphism types of $\pi_{1}(G(\Pi))$ for the spherical Dynkin diagrams111Dynkin diagram LaTeX styles kindly provided by Max Horn at [Hor].

[TABLE]

Example: 0 Isomorphism types of $\pi_{1}(G(\Pi))$ for selected indefinite Dynkin diagrams222Dynkin diagram LaTeX styles kindly provided by Max Horn at [Hor].

[TABLE]

While in the classical finite-dimensional Lie case, one has a topological Iwasawa decomposition $G=K\times A\times U^{+}$ with $A$ and $U^{+}$ contractible, implying $\pi_{1}(K)\cong\pi_{1}(G)$, it is currently unknown whether the corresponding Iwasawa decomposition in the general Kac–Moody case is also topological. However, using a fibration result by Palais (see Proposition A.13), in the appendix Hartnick and the second author prove that the isomorphism between the fundamental groups still exists in the general symmetrizable case, therefore reducing the problem to the computation of $\pi_{1}(K)$.

In [GHKW17, Section 16], the group ${\mathrm{Spin}(\Pi,\kappa)}$ – where $\kappa$ denotes a so-called admissible colouring of the vertices of $\Pi$ – is defined as the canonical universal enveloping group of a $\mathrm{Spin}(2)$-amalgam $\mathcal{A}(\Pi,\mathrm{Spin}(2))=\{\tilde{G}_{ij},\tilde{\phi}_{ij}^{i}\mid i\neq j\in I\}$ where the isomorphism type of $\tilde{G}_{ij}$ depends on the $(i,j)$- and $(j,i)$-entries of the Cartan matrix of $\Pi$ as well as the values of $\kappa$ on the corresponding vertices.

It is shown in [GHKW17, Section 17] that there exists a finite central extension $\mathrm{Spin}(\Pi,\kappa)\to K(\Pi)$ which implies that the subspace topology on $K(\Pi)$ inherited from the Kac–Peterson topology on $G(\Pi)$ defines a unique topology on $\mathrm{Spin}(\Pi,\kappa)$ that turns the central extension into a covering map. The resulting group topology on $\mathrm{Spin}(\Pi,\kappa)$ is called the Kac–Peterson topology on $\mathrm{Spin}(\Pi,\kappa)$.

In the simply-laced case, there is a unique non-trivial admissible colouring $\kappa$ and the corresponding group $\mathrm{Spin}(\Pi):=\mathrm{Spin}(\Pi,\kappa)$ double-covers $K$ as shown in [GHKW17]. We prove here that in the simply-laced case $\mathrm{Spin}(\Pi)$ is simply connected which then implies that $\pi_{1}(K)\cong C_{2}$.

The strategy of proof in the simply-laced case is to study fibre bundles of the form

[TABLE]

arising from embeddings of $\mathrm{Spin}(3)$ along subdiagrams of type $A_{2}$, which yield exact sequences of the form

[TABLE]

and establishes the equivalence of simple-connectedness of $\mathrm{Spin}(\Pi)$ with the simple-connectedness of $\mathrm{Spin}(\Pi)/\mathrm{Spin}(3)$.

A key to the proof both in the simply-laced and in the general case is the computation of the fundamental groups of generalized flag varieties – that is, spaces of the form $G/P_{J}$ for a parabolic subgroup $P_{J}$ of $G$ corresponding to an index subset $J\subseteq I$. It turns out that the aforementioned space $\mathrm{Spin}(\Pi)/\mathrm{Spin}(3)$ is a universal covering space of an appropriately chosen generalized flag variety. In general, we prove the following theorem:

Theorem**.**

Let $\Pi$ be an irreducible Dynkin diagram such that the Bruhat decomposition of $G(\Pi)$ provides a CW decomposition (i.e., such that the conclusion of Proposition 3.7 holds), let $I$ be the index set of the Dynkin diagram, let $J\subseteq I$, and let $P_{J}$ be a parabolic of type $J$. Then a presentation of $\pi_{1}(G/P_{J})$ is given by

[TABLE]

In particular, this statement holds in the $2$-spherical and in the symmetrizable case.

We refer to [Wig98] for the analog result in the finite-dimensional situation.

In order to determine $\pi_{1}(K)$ in the general case, we compute subgroups of $\pi_{1}(K)$ corresponding to the index sets of connected components of $\Pi^{\mathrm{adm}}$ using the above theorem and covering maps of the type $K/K_{J}\to K/(K\cap T)K_{J}$ where $T$ is a maximal split torus of $G(\Pi)$ and $K_{J}$ is the subgroup of fixed points of a Levi factor of $P_{J}$ with both $T$ and $P_{J}$ invariant under the Cartan–Chevalley involution. We then show that $\pi_{1}(K)$ is a direct product of appropriately chosen such subgroups.

In a very similar way, the fundamental group of $\mathrm{Spin}(\Pi,\kappa)$ is determined, establishing the following theorem:

Theorem**.**

Let $\Pi$ be an irreducible Dynkin diagram such that the Bruhat decomposition of $G(\Pi)$ provides a CW decomposition (i.e., such that the conclusion of Proposition 3.5 holds). Let $n(g)$ be the number of connected components of $\Pi^{\mathrm{adm}}$ of colour $g$. Let $n(b,\kappa)$ be the number of connected components of $\Pi^{\mathrm{adm}}$ on which $\kappa$ takes the value 1 and which have colour $b$. Then

[TABLE]

In particular, this statement holds in the $2$-spherical and in the symmetrizable case.

Acknowledgements. The research leading to this article has been partially funded by DFG via the project KO 4323/11. The authors thank Julius Grüning and two anonymous referees for various helpful remarks on earlier versions of this article.

2 Split-real Kac–Moody groups

In [Kac90, §1.3], Kac associates with every generalized Cartan matrix $\mathbf{A}=(a_{ij})_{1\leq i,j\leq n}\in\mathbb{Z}^{n\times n}$ a quadruple $(\mathfrak{g}_{\mathbb{C}}(\mathbf{A}),\mathfrak{h}_{\mathbb{C}}(\mathbf{A}),\Psi,\check{\Psi})$ of a complex Lie algebra $\mathfrak{g}_{\mathbb{C}}(\mathbf{A})$, an abelian subalgebra $\mathfrak{h}_{\mathbb{C}}(\mathbf{A})$ and linearly independent finite subsets $\Psi=\{\alpha_{1},\dots,\alpha_{n}\}\subseteq\mathfrak{h}_{\mathbb{C}}(\mathbf{A})^{*}$ and $\check{\Psi}=\{\check{\alpha}_{1},\dots,\check{\alpha}_{n}\}\subseteq\mathfrak{h}_{\mathbb{C}}(\mathbf{A})$ called simple roots and simple coroots, respectively, such that $\alpha_{j}(\check{\alpha}_{i})=a_{ij}$. Associated with such a quadruple is a Lie algebra generating set $\{e_{1},\dots,e_{n},f_{1},\dots,f_{n}\}\cup\mathfrak{h}_{\mathbb{C}}(\mathbf{A})$. The complex Lie algebra $\mathfrak{g}_{\mathbb{C}}(\mathbf{A})$ is called the complex Kac–Moody algebra associated with $\mathbf{A}$, and $\mathfrak{h}_{\mathbb{C}}(\mathbf{A})$ its standard Cartan subalgebra.

Since $a_{ij}\in\mathbb{R}$, one can analogously define a quadruple $(\mathfrak{g}_{\mathbb{R}}(\mathbf{A}),\mathfrak{h}_{\mathbb{R}}(\mathbf{A}),\Psi,\check{\Psi})$ where $\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$ is a real Lie algebra that embeds naturally into $\mathfrak{g}_{\mathbb{C}}(\mathbf{A})$ as the real form given by the involution induced by complex conjugation. One refers to $\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$ as the split real Kac–Moody algebra associated with $\mathbb{R}$ and to $\mathfrak{h}_{\mathbb{R}}(\mathbf{A})$ as its standard split Cartan subalgebra.

Let $Q\subseteq\mathfrak{h}_{\mathbb{R}}(\mathbf{A})^{*}$ be the group generated by $\Psi$ and $Q_{\pm}$ the subsemigroups generated by $\pm\Psi$, respectively. For $k\in\{\mathbb{C},\mathbb{R}\}$ and $\alpha\in\mathfrak{h}_{k}(\mathbf{A})^{*}$ define the root space

[TABLE]

The set $\Delta$ of $\mathfrak{h}_{k}(\mathbf{A})$ roots in $\mathfrak{g}_{k}(\mathbf{A})$ is defined as $\Delta:=\{\alpha\in Q\setminus\{0\}\mid\mathfrak{g}_{\alpha}^{k}\neq\{0\}\}$. One has the root space decomposition

[TABLE]

The set $\Delta$ decomposes as a disjoint union into the subsets $\Delta_{\pm}:=\Delta\cap Q_{\pm}$ called positive, respectively negative roots. The restriction of the Lie bracket on $\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$ to

[TABLE]

turns $\mathfrak{u}^{+}$ and $\mathfrak{u}^{-}$ into Lie subalgebras of $\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$.

For $i=\{1,\dots,n\}$ define the fundamental root reflection $\sigma_{i}\in\mathrm{GL}(\mathfrak{h}_{\mathbb{R}}(\mathbf{A})^{*})$ by

[TABLE]

Then the Weyl group of $\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$ is defined as $W:=\langle\sigma_{1},\dots,\sigma_{n}\rangle\leq\mathrm{GL}(\mathfrak{h}_{\mathbb{R}}(\mathbf{A})^{*})$ and forms a Coxeter system together with the set of fundamental root reflections. Finally, define the set of real roots $\Phi:=W.\Psi\subseteq\Delta$ and $\Phi^{\pm}:=\Delta_{\pm}\cap\Phi$, the positive, respectively negative real roots.

The construction in [Tit87] of $G_{\mathbf{A}}(\mathbb{R})$ (see Definition 1.1) provides a representation of $G_{\mathbf{A}}(\mathbb{R})$ on $\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$ by Lie algebra automorphisms, which is denoted by

[TABLE]

and referred to as the adjoint representation of $G_{\mathbf{A}}(\mathbb{R})$. Since the subgroup $\operatorname{Ad}(G(\Pi))$ of $G(\Pi)$ under this representation preserves the commutator subalgebra $\mathfrak{g}_{\mathbb{R}}^{\prime}(\mathbf{A})$, one obtains an adjoint representation

[TABLE]

for $G(\Pi)$. The kernels of the adjoint representations of $G_{\mathbf{A}}(\mathbb{R})$ and $G(\Pi)$ are given by the respective centres.

An element $X\in\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$ is $\operatorname{ad}$-locally-finite if for every element $Y\in\mathfrak{g}_{\mathbb{R}}(\mathbf{A})$ there exists an $\operatorname{ad}(X)$-invariant finite-dimensional subspace $W$ with $Y\in W$. As pointed out in [Mar18, p. 64], this implies that $\left.\operatorname{ad}(X)\right|_{W}$ is a (finite) matrix in some basis of $W$, so the exponential $\exp(\operatorname{ad}(X))$ can be defined in the ususal way. By [Tit87, (KMG5), p. 545] and the uniqueness properties of $G_{\mathbf{A}}(\mathbb{R})$ established in [Tit87, Theorem 1], $\exp(\operatorname{ad}(X))\in\operatorname{Ad}(G_{\mathbf{A}}(\mathbb{R}))$. Let $F_{\mathfrak{g}_{\mathbb{R}}(\mathbf{A})}$ and $F_{\mathfrak{g}_{\mathbb{R}}^{\prime}(\mathbf{A})}$ be the subsets of $\operatorname{ad}$-locally-finite elements of the respective algebras. The maps $\exp:F_{\mathfrak{g}_{\mathbb{R}}(\mathbf{A})}\to\operatorname{Ad}(G_{\mathbf{A}}(\mathbb{R}))$ and $\exp:F_{\mathfrak{g}_{\mathbb{R}}^{\prime}(\mathbf{A})}\to\operatorname{Ad}(G(\Pi))$ given by $X\mapsto\exp(\operatorname{ad}(X))$ can be lifted to exponential functions $\exp:F_{\mathfrak{g}_{\mathbb{R}}(\mathbf{A})}\to G_{\mathbf{A}}(\mathbb{R})$ and $\exp:F_{\mathfrak{g}_{\mathbb{R}}^{\prime}(\mathbf{A})}\to G(\Pi)$.

For $X\in\mathfrak{h}_{\mathbb{R}}(\mathbf{A})\subseteq F_{\mathfrak{g}_{\mathbb{R}}(\mathbf{A})}$, one has

[TABLE]

cf. [Kum02, Section 6.1.6], [Tit87, (KMG5), p. 545].

The same constructions apply also to $\mathbb{C}$ instead of $\mathbb{R}$. Since $\mathfrak{h}_{\mathbb{C}}(\mathbf{A})\subseteq F_{\mathfrak{g}_{\mathbb{C}}(\mathbf{A})}$, one can define $T_{\mathbb{C}}:=\exp(\mathfrak{h}_{\mathbb{C}}(\mathbf{A}))$. Note that $\exp(\mathfrak{h}_{\mathbb{R}}(\mathbf{A}))=:A_{\mathbb{R}}\subsetneq T_{\mathbb{R}}:=T_{\mathbb{C}}\cap G_{\mathbf{A}}(\mathbb{R})$. There is a unique Lie group topology on $T_{\mathbb{R}}$ in which $T_{\mathbb{R}}\cong(\mathbb{R}^{\times})^{n}$ and $A_{\mathbb{R}}=T_{\mathbb{R}}^{\circ}\cong(\mathbb{R}_{>0})^{n}$. The centre of $G_{\mathbf{A}}(\mathbb{R})$ is contained in $T_{\mathbb{R}}$.

The intersection $T:=G(\Pi)\cap T_{\mathbb{C}}$ is called the standard split maximal torus of $G(\Pi)$; again, $A_{\mathbb{R}}\cap T$ is of finite index in $T$ and $T$ contains the centre of $G(\Pi)$.

The Lie algebra $\mathfrak{g}_{\mathbb{R}}({\mathbf{A}})$ admits a unique involution $\theta$ which maps $e_{j}$ to $f_{j}$ for all $j=1,\dots,r$ and acts as $-1$ on $\mathfrak{h}_{\mathbb{R}}({\mathbf{A}})$. There exists a unique involutive automorphism $\theta:G_{\mathbf{A}}(\mathbb{R})\to G_{\mathbf{A}}(\mathbb{R})$ such that $\theta(\exp(X))=\exp(\theta(X))$ for all $X\in F_{\mathfrak{g}_{\mathbb{R}}({\mathbf{A}})}$, and this involutive automorphism is called the Cartan–Chevalley involution of $G_{\mathbf{A}}(\mathbb{R})$. We denote by $K_{\mathbf{A}}(\mathbb{R}):=G_{\mathbf{A}}(\mathbb{R})^{\theta}\subset G_{\mathbf{A}}(\mathbb{R})$ the fixed point subgroup of this involution and define $K(\Pi):=K_{\mathbf{A}}(\mathbb{R})\cap G(\Pi)$.

Let $\alpha\in\Phi$ be a real root. Then $\mathfrak{g}_{\alpha}^{\mathbb{R}}$ is one-dimensional and consists of ad-locally-finite elements. One can therefore define the root group $U_{\alpha}:=\exp(\mathfrak{g}_{\alpha}^{\mathbb{R}})\subseteq G_{\mathbf{A}}(\mathbb{R})$. Each root group $U_{\alpha}$ carries a unique Lie group topology such that $U_{\alpha}\cong\mathbb{R}$ as topological groups. Root groups corresponding to positive real roots are called positive root groups, root groups corresponding to negative real roots are called negative root groups.

Define the positive, respectively negative maximal unipotent subgroup $U^{\pm}$ of $G_{\mathbf{A}}(\mathbb{R})$ as the group generated respectively by the positive and negative root groups. One has $U^{\pm}\subseteq G(\Pi)$. The groups $U^{\pm}$ are normalized by $T_{\mathbb{R}}$ and intersect $T_{\mathbb{R}}$ trivially. In particular, they intersect the centres of $G_{\mathbf{A}}(\mathbb{R})$ and $G(\Pi)$ trivially and hence embed into both $\operatorname{Ad}(G_{\mathbf{A}}(\mathbb{R})$ and $\operatorname{Ad}(G(\Pi))$.

If $\alpha\in\Phi^{+}$, then $-\alpha\in\Phi^{-}$ and the group $G_{\alpha}:=\langle U_{\alpha},U_{-\alpha}\rangle\leq G(\Pi)$ is isomorphic to $\mathrm{SL}_{2}(\mathbb{R})$. The groups $G_{\alpha}$ with $\alpha\in\Phi^{+}$ are called the rank-$1$ subgroups and the groups $G_{1}:=G_{\alpha_{1}},\dots,G_{n}:=G_{\alpha_{n}}$ are called the fundamental rank-$1$ subgroups of $G(\Pi)$.

One can show that the pair $((U_{\alpha})_{\alpha\in\Phi},T)$ defines an RGD system for $G(\Pi)$. For details concerning RGD systems, we refer the reader to [AB08, Chapter 8].

Recall that the generalized Cartan matrix $\mathbf{A}$ is called $2$-spherical, if $a_{ij}a_{ji}\leq 3$ for all $i\neq j\in I$; in other words, if the orders of the products $\sigma_{i}\sigma_{j}$ are always finite. The generalized Cartan matrix $\mathbf{A}$ is symmetrizable, if it is the product of a symmetric and a diagonal matrix. These notions are also applied to any and all objects that are derived from $\mathbf{A}$ such as the (extended) Weyl group, the Kac–Moody group, their buildings, etc.

Definition and Remark 2.1.

The Kac–Peterson topology on $G_{\mathbf{A}}(\mathbb{R})$ equals the finest group topology on $G_{\mathbf{A}}(\mathbb{R})$ such that the natural embeddings $(U_{\alpha}\hookrightarrow G_{\mathbf{A}}(\mathbb{R}))_{\alpha\in\Phi}$ and $T_{\mathbb{R}}\hookrightarrow G_{\mathbf{A}}(\mathbb{R})$ are continuous when $T_{\mathbb{R}}$ and the root groups $U_{\alpha}$ are endowed with their Lie group topologies.

The Kac–Peterson topology is $k_{\omega}$ by [HKM13, Proposition 7.10] and, in particular, Hausdorff. Moreover, for every $\alpha\in\Phi^{+}$, it induces the unique connected Lie group topology on $G_{\alpha}$ and on $T_{\mathbb{R}}$ by [HKM13, Corollary 7.16]

For more details on the Kac–Peterson topology, see [HKM13, Chapter 7].

Notation 2.2.

Throughout this paper, let $G:=G(\Pi):=[G_{\mathbf{A}}(\mathbb{R}),G_{\mathbf{A}}(\mathbb{R})]$ be the algebraically simply connected semisimple split real Kac–Moody group associated to an irreducible generalized Dynkin diagram $\Pi=(V,E)$ with (bijective) labelling $\lambda:\{1,\dots,n\}=:I\to V$. Let $K:=K(\Pi)$ be the maximal compact subgroup of $G$, i.e., the subgroup fixed by the Cartan–Chevalley involution $\theta$.

Denote by $B:=B_{+}$ the positive Borel subgroup of the twin $BN$-pair of $G$, by $T$ the standard split maximal torus and by $W$ the Weyl group of $G$ with generating set $S=\{\sigma_{i}\}_{i\in I}$. For each $\sigma_{i}\in S$, take $s_{i}\in G$ to be a fixed representative of order $4$ for $\sigma_{i}$. The group $\tilde{W}:=\langle s_{i}\mid i\in I\rangle\leq G$ is called the extended Weyl group. By [DMGH09, Corollary 1.7], one has an Iwasawa decomposition $G=KB$.

The groups $G$ and $K$ are always endowed with the subspace topologies induced by the Kac–Peterson topology on $G_{\mathbf{A}}(\mathbb{R})$ and $G/B$ with the quotient topology.

Unless specified more explicitly, the symbol $J$ will always denote an arbitrary subset of the index set $I$, the symbol $\Pi_{J}$ the subdiagram of $\Pi$ corresponding to $J$, the symbol $G_{J}$ the subgroup $G(\Pi_{J})$ of $G$, and the symbols $K_{J}$ and $B_{J}$ the intersections $G_{J}\cap K$ and $G_{J}\cap B$, respectively. This is consistent with the notation for the fundamental rank one subgroups: One has $G(\Pi_{i})=G_{i}=G_{\alpha_{i}}$.

Remark 2.3.

Due to the structure theory of RGD systems (cf. [AB08, Chapter 8], most notably the fact that restricting an RGD system to a subdiagram again yields an RGD system), for each fundamental rank one subgroup $G_{i}$ there exists an (abstract) isomorphism $\gamma_{i}:\mathrm{SL}(2,\mathbb{R})\to G_{i}$ with the following properties: Let $B_{\mathrm{SL}(2,\mathbb{R})}$ be the group of upper triangular matrices in $\mathrm{SL}(2,\mathbb{R})$ and let $U_{\pm\beta}$ denote the canonical root subgroups of $\mathrm{SL}(2,\mathbb{R})$. Then

• •

$\gamma_{i}(U_{\pm\beta})=U_{\pm\alpha_{i}}$.

• •

$\gamma_{i}(B_{\mathrm{SL}(2,\mathbb{R})})=B_{i}$.

• •

For each $x\in\mathrm{SL}(2,\mathbb{R})$, $\gamma_{i}((x^{t})^{-1})=(\gamma_{i}(x))^{\theta}$, and hence

• •

$\gamma_{i}(\mathrm{SO}(2,\mathbb{R}))=K_{i}$.

By [HKM13, Corollary 7.16], the restriction of the Kac–Peterson topology to any spherical subgroup $H$ of $G$ coincides with its Lie topology. That is, the groups $G_{i}$ inherit their Lie group topology from the topological Kac–Moody group $G$. By the classical theory of Lie groups this yields the existence of a diffeomorphism $\gamma_{i}$ with the desired properties; in particular, $\gamma_{i}$ is an open map.

Definition 2.4.

Using the Bruhat decomposition $G=\bigsqcup_{w\in W}BwB$ ([AB08, Theorem 6.56, Remark (1)]), let

[TABLE]

be the Weyl distance function on $G/B$, and let $l_{S}$ be the length function that associates to each element the (unique) length of a corresponding reduced expression in $S$. Let $\leq$ be the strong Bruhat order on $W$: Recall that for $w_{1},w_{2}\in W$ one has $w_{1}\leq w_{2}$ if there exist reduced expressions $s_{i_{1}}\dots s_{i_{l_{S}(w_{1})}}$ of $w_{1}$ and $s_{j_{1}}\dots s_{j_{l_{S}(w_{2})}}$ such that the former is a (not necessarily consecutive) substring of the latter.

For $w\in W$ and a chamber $gB\in G/B$ define

[TABLE]

[TABLE]

and

[TABLE]

In particular, one has $C_{w}(B)=BwB/B$ and $C_{\leq\sigma}(B)=B\langle s\rangle B/B$ for $\sigma\in S$ with representative $s\in\tilde{W}\subseteq G$ in the extended Weyl group $\tilde{W}$. A set $C_{\leq\sigma}(gB)$ is called a $\sigma$-panel.

Moreover, for a subset $\{\sigma_{i}\}_{i\in J}\subseteq S$ with representatives $\{s_{i}\}_{i\in J}\subseteq\tilde{W}$ define $P_{J}$ to be the standard parabolic subgroup corresponding to the index set $J$, that is, $P_{J}:=B\langle\{s_{i}\}_{i\in J}\rangle B$.

Throughout this paper, $C_{w}(gB)$ and $C_{\leq w}(gB)$ will always be endowed with the subspace topologies induced by $G/B$.

Lemma 2.5.

Let $\sigma_{i}\neq\sigma_{j}\in S$. Then the following hold:

1. (a)

$P_{i}=G_{i}B$* $=K_{i}B$. In particular, $C_{\leq\sigma_{i}}(B)=K_{i}B/B$.* 2. (b)

$Bs_{i}s_{j}B=Bs_{i}BBs_{j}B$. In particular, $C_{\leq\sigma_{i}\sigma_{j}}(B)=K_{i}BK_{j}B/B$

Proof.

Assertions (a) and (b) follow from [AB08, Remark 8.51] and [AB08, Remark (2) after Theorem 6.56], respectively, and the Iwasawa decomposition $G_{i}=K_{i}B_{i}$. ∎

3 The fundamental group of the generalized flag variety $G/P_{J}$

For a moment, let $\Pi$ be an irreducible simply-laced diagram distinct from $A_{1}$, and let $G=G(\Pi)$ and $K=K(\Pi)$ be as in the preceding section. Moreover, let $\mathrm{Spin}(\Pi)$ be the double cover of $K(\Pi)$ constructed in [GHKW17, Lemma 16.18] (see Definition 4.6 below). By construction, any $A_{2}$-subdiagram of $\Pi$ yields an embedding $\mathrm{Spin}(3)\hookrightarrow\mathrm{Spin}(\Pi)$ and, since $\mathrm{Spin}(3)$ inherits the Lie topology from the Kac–Peterson topology on $\mathrm{Spin}(\Pi)$ by [HKM13, Corollary 7.16], one obtains a locally trivial fibre bundle

[TABLE]

by [Pal61] (see Proposition A.13). It will turn out in Section 4 below that $\mathrm{Spin}(\Pi)/\mathrm{Spin}(3)$ is a universal covering space of the generalized flag variety $G/P_{J}$ where $J\subset I$ equals the set consisting of the two types involved in the chosen $A_{2}$-subdiagram. The fundamental group of $\mathrm{Spin}(\Pi)$ then follows from the homotopy exact sequence

[TABLE]

This motivates our interest in the fundamental group and covering theory of generalized flag varieties $G/P_{J}$.

Throughout this section, let $J\subseteq I$, let $W_{J}$ be the subgroup of $W$ generated by $\{\sigma_{i}\}_{i\in J}$, and let $W^{J}\subseteq W$ be a set of representatives of the cosets in $W/W_{J}$ that have minimal length in the coset they define.

Lemma 3.1 (Bruhat decomposition).

One has $G/P_{J}=\bigsqcup_{w\in W^{J}}BwP_{J}/P_{J}$.

Proof.

This follows immediately from [AB08, Theorem 6.56, Remark (1)]. ∎

Lemma 3.2.

Let $G$ be a topological group and $H_{1}\leq H_{2}$ subgroups of $G$ and endow $G/H_{i}$ with the quotient topology. Then the following hold:

1. (a)

The projection map $\pi:G\to G/H_{1}$ is continuous and open. 2. (b)

The canonical map $\psi:G/H_{1}\to G/H_{2}$ is continuous and open.

Proof.

This is a standard exercise for topological groups. ∎

Definition and Remark 3.3.

For $w\in W$, define the following restrictions of the canonical map $\psi:G/B\to G/P_{J}$:

• •

$\psi_{w}:BwB/B\to BwP_{J}/P_{J}$,

• •

$\psi_{\bar{w}}:\bigcup_{x\leq w}BxB/B\to\bigcup_{x\leq w}BxP_{J}/P_{J}$.

Since $\psi$ is continuous, the same holds for the two restrictions. The space $\bigcup_{x\leq w}BxB/B$ is compact by [HKM13, Corollary 3.10] and so $\psi_{\bar{w}}$ is a quotient map.

Lemma 3.4.

Let $G$ be $2$-spherical or symmetrizable and let $w\in W^{J}$. Then the canonical map $\psi_{w}$ is a homeomorphism.

Proof.

By Remark 3.3, $\psi_{\bar{w}}$ is a quotient map. One has $\psi_{\bar{w}}^{-1}(BwP_{J}/P_{J})=BwB/B$: Let $x\leq w$ such that $BxP_{J}/P_{J}=BwP_{J}/P_{J}$. Then $x\in BwP_{J}=BwW_{J}B$ where the equality holds since by definition of $W^{J}$ one has $l(ww^{\prime})=l(w)+l(w^{\prime})$ for all $w^{\prime}\in W_{J}$ which implies $Bww^{\prime}B=BwBw^{\prime}B$. The Bruhat decomposition of $G$ yields $x\in wW_{J}$ and hence, $l(x)\geq l(w)$. This implies $x=w$.

Now, since $BwB/B$ is open in its closure $\bigcup_{x\leq w}BxB/B$ in $G/B$ (see [HKM13, Proposition 5.9] plus Corollary B.8), the preceding observations yield that $\psi_{w}$ is an injective quotient map and therefore a homeomorphism. ∎

Lemma 3.5.

Let $\sigma_{i}\in S$. Then each panel $C_{\leq\sigma_{i}}(B)$ is homeomorphic to the $1$-sphere $\mathbb{S}^{1}$.

Proof.

The panel $C_{\leq\sigma_{i}}(B)$ is a subbuilding of $G/B$ corresponding to the RGD system $\{G_{i},U_{\alpha_{i}},U_{-\alpha_{i}},T\cap G_{i}\}$. By Remark 2.3 one has $G_{i}\cong\mathrm{SL}(2,\mathbb{R})$, $T\cap G_{i}\cong T_{\mathrm{SL}(2,\mathbb{R})}$ and $U_{\pm\alpha_{i}}\cong U_{\pm\alpha}$ where $T_{\mathrm{SL}(2,\mathbb{R})}$ denotes the subgroup of diagonal matrices and $U_{\pm\alpha}$ denote the canonical root subgroups of $\mathrm{SL}(2,\mathbb{R})$. This implies that $C_{\leq\sigma_{i}}(B)$ is homeomorphic to the building $\mathrm{SL}(2,\mathbb{R})/B_{\mathrm{SL}(2,\mathbb{R})}\simeq\mathbb{P}_{1}(\mathbb{R})\simeq\mathbb{S}^{1}$. ∎

Definition 3.6.

Following [Rot88, Chapter 8], a CW complex is an ordered triple $(X,E,\chi)$, where $X$ is a Hausdorff space, $E$ is a family of cells in $X$, and $\chi=\{\chi_{e}\mid e\in E\}$ is a family of maps, such that

1. (a)

$X=\bigsqcup_{e\in E}{E}$. 2. (b)

For $k\in\mathbb{N}$, let $X^{(k)}\subseteq X$ be the union of all cells of dimension $\leq k$. Then for each $(k+1)$-cell $e\in E$, the map $\chi_{e}:(D^{k+1},S^{k})\to(e\cup X^{(k)},X^{(k)})$, is a relative homeomorphism, i.e., it is a continuous map and its restriction $D^{k+1}\setminus S^{k}\to e$ is a homeomorphism. 3. (c)

If $e\in E$, then its closure $\operatorname{cl}{e}$ is contained in a finite union of cells in $E$. 4. (d)

$X$ has the weak topology determined by $\{\operatorname{cl}{e}\mid e\in E\}$, i.e., a subset $A$ of $X$ is closed if and only if $A\cap\operatorname{cl}{e}$ is closed in $\operatorname{cl}{e}$ for each $e\in E$.

For $k\in\mathbb{N}$, let $\Lambda_{k}$ be an index set for the $k$-dimensional cells, so that $X^{(k)}\setminus X^{(k-1)}=\bigsqcup_{\lambda\in\Lambda_{k}}e_{\lambda}$ and set $\chi_{\lambda}:=\chi_{e_{\lambda}}$. This map is called the characteristic map of $e_{\lambda}$.

Proposition 3.7.

Let $G$ be $2$-spherical or symmetrizable. Then for each $w\in W$, the set $C_{w}(B)=BwB/B$ is a cell of dimension $l(w)$ that is open in its compact closure $C_{\leq w}(B)$ in $G/B$. For each subset $J\subseteq I$, the Bruhat decomposition $G/P_{J}=\bigsqcup_{w\in W^{J}}BwP_{J}/P_{J}$ is a CW decomposition.

Proof.

The first statement is immediate by [HKM13, Corollary 3.10 and Proposition 5.9] plus Corollary B.8, see also [Kra01, p. 170, 171]. Furthermore, [HKM13, Proposition 5.9] combined with Corollary B.8 states that the Bruhat decomposition of $G/B$ is a CW decomposition. By Lemma 3.4, $G/P_{J}$ is composed of cells that are homeomorphic to cells in $G/B$, so composing the characteristic maps of the latter cells with the canonical map $\psi:G/B\to G/P_{J}$ yields characteristic maps for the cells in $G/P_{J}$.

For the closure-finiteness, let $BwP_{J}/P_{J}$ be a cell in $G/P_{J}$. Since $\psi$ is continuous and restricts to a homeomorphism $BwB/B\to BwP_{J}/P_{J}$, it maps $\operatorname{cl}{BwB/B}$ surjectively onto $\operatorname{cl}{BwP_{J}/P_{J}}$. Now, $\operatorname{cl}BwB/B=\bigcup_{x\leq w}BxB/B$, which implies that

[TABLE]

where the last equality holds since $W_{J}\subseteq P_{J}$. This proves that $\operatorname{cl}{BwP_{J}/P_{J}}$ is contained in a finite union of cells.

It remains to show that $G/P_{J}$ has the weak topology determined by the cell closures.

For $w\in W$ and a minimal-length representative $\tilde{w}\in W^{J}$ of $wW_{J}$, one has $BwP_{J}/P_{J}=B\tilde{w}P_{J}/P_{J}$. Let $e_{w}:=BwP_{J}/P_{J}=B\tilde{w}P_{J}/P_{J}$ and $e_{w}^{\prime}:=BwB/B$. Let $\bar{e}_{w}=\operatorname{cl}{e_{w}}=\bigcup_{x\leq\tilde{w}}BxP_{J}/P_{J}$ and $\bar{e}_{w}^{\prime}:=\operatorname{cl}{e^{\prime}_{w}}=\bigcup_{x\leq w}BxB/B$.

Let $A$ be a closed subset of $G/P_{J}$ and let $e_{w}$, $w\in W^{J}$, be an arbitrary cell. Then $\psi^{-1}(A)$ is closed in $G/B$ since $\psi$ is continuous, so $\psi^{-1}(A)\cap\bar{e}_{w}^{\prime}$ is closed in $\bar{e}_{w}^{\prime}$ since $G/B$ is a CW complex. Now,

[TABLE]

Since $\psi_{\bar{w}}$ is a quotient map by Remark 3.3, this implies that $A\cap\bar{e}_{w}$ is closed in $\bar{e}_{w}$.

Now, let $A$ be a subset of $G/P_{J}$ such that $A\cap\bar{e}_{w}$ is closed in $\bar{e}_{w}$ for all $w\in W^{J}$. Since for each $w\in W$ one has $e_{w}=e_{\tilde{w}}$ for any minimal-length representative $\tilde{w}\in W^{J}$ of $wW_{J}$, in fact $A\cap\bar{e}_{w}$ is closed in $\bar{e}_{w}$ for all $w\in W$. Therefore $\psi_{\bar{w}}^{-1}(A\cap\bar{e}_{w})$ is closed in $\bar{e}_{w}^{\prime}$ for all $w\in W$. Since $\psi_{\bar{w}}^{-1}(A\cap\bar{e}_{w})=\psi^{-1}(A)\cap\bar{e}_{w}^{\prime}$, the fact that $G/B$ is a CW complex implies that $\psi^{-1}(A)$ is closed in $G/B$. Since $\psi$ is open by Lemma 3.2, it follows that $A$ is closed in $G/P_{J}$. This proves that $G/P_{J}$ is a CW complex. ∎

The preceding result combined with the following lemma (which is a consequence of [Mas77, Ch. 7, Thm 2.1]) will allow us to efficiently compute the fundamental group of a generalized flag variety in Theorem 3.15 below.

Lemma 3.8.

Let $X$ be a $CW$ complex with only one [math]-cell $x_{0}$. For each $\lambda\in\Lambda_{2}$, let $f_{\lambda}:[0,1]\to\mathbb{S}^{1}$ be a loop whose homotopy class generates $\pi_{1}(\mathbb{S}^{1})$ and whose image $\gamma_{\lambda}:=\chi_{\lambda}\circ f_{\lambda}$ under $\chi_{\lambda}$ is a loop in $X^{(1)}$ starting at $x_{0}$. Then

[TABLE]

is a presentation of $\pi_{1}(X,x_{0})$, where the brackets denote the respective homotopy classes in $X^{(1)}$.

Next, we study the characteristic maps of the CW decomposition of a generalized flag variety explicitly.

Notation 3.9.

Define $R:[0,1]\to\mathrm{SO}(2,\mathbb{R}),s\mapsto\begin{pmatrix}\cos(s\pi)&-\sin(s\pi)\\ \sin(s\pi)&\cos(s\pi)\end{pmatrix}.$

Lemma 3.10.

$R$* induces a continuous, surjective map $\tilde{R}:[0,1]\to\mathrm{SL}(2,\mathbb{R})/B_{\mathrm{SL}(2,\mathbb{R})}$ which maps the interior $(0,1)$ homeomorphically onto its image and maps the boundary $\{0,1\}$ surjectively onto its image.*

Proof.

Let $\{x_{0}\}:=\left\langle\begin{pmatrix}1&0\end{pmatrix}^{\intercal}\right\rangle\in\mathbb{P}^{1}$ where $\mathbb{P}^{1}$ denotes the real projective line, modelled as the subset of one-dimensional subspaces of $\mathbb{R}^{2}$. Since each one-dimensional subspace in $\mathbb{P}^{1}\setminus\{x_{0}\}$ contains exactly one element in the upper half circle $R([0,1])\cdot\begin{pmatrix}1&0\end{pmatrix}^{\intercal}$ while $x_{0}$ contains the two boundary points corresponding to $R(0)$ and $R(1)$, one has a surjection from $[0,1]$ onto $\mathbb{P}^{1}$ given by $t\mapsto\left\langle R(t)\cdot\begin{pmatrix}1&0\end{pmatrix}^{\intercal}\right\rangle$ which maps $(0,1)$ bijectively onto $\mathbb{P}^{1}\setminus\{x_{0}\}$. Since $\mathrm{SL}(2,\mathbb{R})$ acts transitively on the real projective line $\mathbb{P}^{1}$ with $B_{\mathrm{SL}(2,\mathbb{R})}$ being the stabilizer of $x_{0}:=\left\langle\begin{pmatrix}1&0\end{pmatrix}^{\intercal}\right\rangle$, one has a bijective correspondence $gB\mapsto gx_{0}$ between $\mathrm{SL}(2,\mathbb{R})/B_{\mathrm{SL}(2,\mathbb{R})}$ and $\mathbb{P}^{1}$. This yields the desired surjectivity and bijectivity properties of $\tilde{R}$. Continuity is clear, as well as the fact that the restriction to the interior is a homeomorphism. ∎

Definition 3.11.

Let $D^{1}=[0,1]$ be the $1$-dimensional unit disc and note that $D^{2}\simeq D^{1}\times D^{1}$. For $i,j\in I$ let $\gamma_{i},\gamma_{j}$ be as in Remark 2.3. Let $p:G\to G/B$ be the canonical projection. Define $\chi_{i}:D^{1}\to G/B$ and $\chi_{(i,j)}:D^{1}\times D^{1}\to G/B$ by

• •

$\chi_{i}(s):=p(\gamma_{i}(R(s)))=\gamma_{i}(R(s))\cdot B$,

• •

$\chi_{(i,j)}(s,t):=p(\gamma_{i}(R(s))\gamma_{j}(R(t)))=\gamma_{i}(R(s))\gamma_{j}(R(t))\cdot B$.

The following lemma was inspired by [Pro07, Ch. 10, second Proposition of 6.8], see also [Kac85, §2.6, p. 198].

Lemma 3.12.

Let $G$ be $2$-spherical or symmetrizable. Then the maps defined above are characteristic maps for the following cells:

1. (a)

$\chi_{i}$* for $C_{{\sigma}_{i}}(B)=B{s}_{i}B/B$,* 2. (b)

$\chi_{(i,j)}$* for $C_{{\sigma}_{i}{\sigma}_{j}}(B)=B{s}_{i}{s}_{j}B/B$.*

Proof.

(a): One has to show that $\chi_{i}([0,1])\subseteq C_{\leq\sigma_{i}}(B)$ and that $\chi_{i}$ is a continuous map which maps $(0,1)$ homeomorphically to $C_{\sigma_{i}}(B)$. The first assertion is clear, since by Lemma 2.5 one has $C_{\leq\sigma_{i}}=G_{i}B/B$.

By Lemma 2.5, one has $C_{\sigma_{i}}(B)=\{kB\mid k\in K_{i}\setminus(K_{i}\cap B)\}$. Let $k\in K_{i}\setminus(K_{i}\cap B)$. Then $\gamma_{i}^{-1}(p^{-1}(kB))=\gamma_{i}^{-1}(k)\cdot B_{\mathrm{SL}(2,\mathbb{R})}\in\mathrm{SL}(2,\mathbb{R})/B_{\mathrm{SL}(2,\mathbb{R})}\setminus B_{\mathrm{SL}(2,\mathbb{R})}$. By Lemma 3.10, there exists a unique $s\in(0,1)$ satisfying $R(s)B_{\mathrm{SL}(2,\mathbb{R})}=\gamma_{i}^{-1}(k)B_{\mathrm{SL}(2,\mathbb{R})}$. Hence, $s$ is the unique preimage of $kB$ under $\chi_{i}$. This yields the desired bijectivity property. The continuity properties are clear.

(b): Since by Lemma 2.5 (b) one has $C_{\leq\sigma_{i}\sigma_{j}}(B)=K_{i}BK_{j}B/B$, it is clear that $\chi_{(i,j)}([0,1]\times[0,1])\subseteq C_{\leq{\sigma}_{i}{\sigma}_{j}}(B)$. For the injectivity of the restriction, let $(s,t),(\tilde{s},\tilde{t})\in(0,1)^{2}$ such that $\chi_{(i,j)}(s,t)=\chi_{(i,j)}(\tilde{s},\tilde{t})$. Then

[TABLE]

This implies $R(\tilde{s})^{-1}R(s)\in B_{\mathrm{SO}(2,\mathbb{R})}$, since otherwise the left expression is in $C_{\sigma_{i}\sigma_{j}}(B)$, contradicting $C_{\sigma_{i}\sigma_{j}}(B)\cap C_{\sigma_{j}}(B)=\emptyset$. Since $s,\tilde{s}\in(0,1)$, one obtains $\tilde{s}=s$. It follows that $\chi_{j}(t)=\chi_{j}(\tilde{t})$, hence $t=\tilde{t}$ by (a).

For the surjectivity, note that by Lemma 2.5 (b), one has $C_{\sigma_{i}\sigma_{j}}(B)=Bs_{i}s_{j}B/B=Bs_{i}BBs_{j}B/B$. Let $x_{i}x_{j}B$ be an arbitrary element of $C_{\sigma_{i}\sigma_{j}}(B)$ with $x_{i}=b_{1}s_{i}b_{2}\in Bs_{i}B$ and $x_{j}\in Bs_{j}B$. By (a), there exists an $s\in(0,1)$ with $\gamma_{i}(R(s))B=b_{1}s_{i}B\in C_{\sigma_{i}}(B)$. Hence, there exists a $b\in B$ with $(\gamma_{i}(R(s))b=b_{1}s_{i}b_{2}=x_{i}$. Again by (a), there exists a $t\in(0,1)$ with $\gamma_{j}(R(t))B=bx_{j}B\in C_{\sigma_{j}}(B)$. This yields

[TABLE]

This proves that $\chi_{i,j}$ maps $(0,1)\times(0,1)$ bijectively to $C_{\sigma_{i}\sigma_{j}}(B)$ The continuity properties are clear. ∎

Notation 3.13.

For $i,j\in I$, let $\varepsilon(i,j):=(-1)^{\alpha_{j}(\check{\alpha}_{i})},$ where $\alpha_{j}(\check{\alpha}_{i})=a_{ij}$ is the $(i,j)$-entry of the Cartan matrix $\mathbf{A}$ of $\Pi$.

Lemma 3.14 [GHKW17, Remark 15.4(1)].

Let $e_{i}:=\gamma_{i}(-I)\in G_{i}$ with $\gamma_{i}$ as in Remark 2.3 and $k_{j}\in K_{j}$. Then $e_{i}k_{j}e_{i}=k_{j}^{\varepsilon(i,j)}$.

Theorem 3.15**.**

If the Bruhat decomposition satisfies the conclusion of Proposition 3.7, then a presentation of $\pi_{1}(G/P_{J})$ is given by

[TABLE]

In particular, this statement holds in the $2$-spherical and the symmetrizable case.

Proof.

By Lemma 3.4 and Proposition 3.7, the Bruhat decomposition $G/P_{J}=\bigsqcup_{w\in W^{J}}BwP_{J}/P_{J}$ is a CW decomposition where each cell $BwP_{J}/P_{J}$ has dimension $l(w)$. The characteristic maps of the 1-cells $Bs_{i}P_{J}/P_{J}$ and 2-cells $Bs_{i}s_{j}P_{J}/P_{J}$ are given by the compositions $\tilde{\chi}_{i}:=\psi_{s_{i}}\circ\chi_{i}$, respectively $\tilde{\chi}_{(i,j)}:=\psi_{s_{i}s_{j}}\circ\chi_{(i,j)}$ ($\psi_{s_{i}}$ and $\psi_{s_{i}s_{j}}$ denoting the canonical homeomorphisms from Lemma 3.4).

Lemma 3.8 gives a presentation of $\pi_{1}(G/P_{J})$. The generating elements are given by the homotopy classes $x_{i}:=[\tilde{\chi}_{i}]$ of the characteristic maps of the 1-cells – namely, the cells $Bs_{i}P_{J}/P_{J}$ where $i\in I\setminus J$. For the homotopy classes $x_{k}$ with $k\in J$, note that $\gamma_{k}(R(t))\in G_{k}\subseteq P_{J}$, and so $\tilde{\chi}_{k}(t)=\gamma_{k}(r(t))\cdot P_{J}=P_{J}$ which implies $x_{k}=[\tilde{\chi}_{k}]=1_{\pi_{1}(G/P_{J})}$. This yields the desired generating set as well as the trivial relation $x_{k}=1$ for $i\in J$.

To obtain the set of relators, for $k=1,\dots,4$ let $\varphi_{k}:[0,1]\to[0,1]\times[0,1]$ where

[TABLE]

Then the concatenation $\varphi:=\varphi_{1}*\varphi_{2}*\varphi_{3}*\varphi_{4}$ is a loop in the relative boundary $\partial([0,1]\times[0,1])\simeq\mathbb{S}^{1}$ which generates its fundamental group. Moreover, for each characteristic map $\tilde{\chi}_{(i,j)}$ of a 2-cell, one has $\tilde{\chi}_{(i,j)}(\varphi(0))=\tilde{\chi}_{(i,j)}((0,0))=\psi_{s_{i}s_{j}}({\chi}_{(i,j)}(0,0))=\psi_{s_{i}s_{j}}(B)=P_{J}$ where $P_{J}$ is the unique 0-cell of the CW complex. Therefore, Lemma 3.8 implies that the set of relators is given by $\{[\tilde{\chi}_{(i,j)}\circ\varphi]\mid\sigma_{i}\sigma_{j}\in W^{J},l(\sigma_{i}\sigma_{j})=2\}$. Now,

[TABLE]

where

$\tilde{\chi}_{(i,j)}(s,t)=\alpha_{i}(R(s))\alpha_{j}(R(t))\cdot P_{J}$ with $R(0)=I_{\mathrm{SO}(2,\mathbb{R})},R(1)=-I_{\mathrm{SO}(2,\mathbb{R})}\in B_{\mathrm{SO}(2,\mathbb{R})}$ which implies

[TABLE]

Moreover,

[TABLE]

Since $R(t)^{-1}=R(1-t)$, this yields $[\tilde{\chi}_{(i,j)}\circ\varphi_{2}]=x_{j}^{\varepsilon(i,j)}$. One therefore obtains $[\tilde{\chi}_{(i,j)}\circ\varphi]=x_{i}\cdot x_{j}^{\varepsilon(i,j)}\cdot x_{i}^{-1}\cdot x_{j}^{-1}$. This proves the assertion.

∎

Lemma 3.16.

Let $\Pi$ be irreducible simply-laced distinct from $A_{1}$ and $\emptyset\neq J\subset I=\{1,...,n\}$. Then $\pi_{1}(G/P_{J})\cong{C_{2}}^{n-|J|}$.

Proof.

For each generator $x_{h}$ in the presentation of Theorem 3.15, one has $x_{h}^{2}=1$: Recall that $\lambda$ denotes the labelling map $I\to V$ of the vertex set of $\Pi$. Since $\Pi$ is connected, one has a minimal path $(i_{1},\dots,i_{m}=h)^{\lambda}$ in $\Pi$ such that $i_{1}\in J$. If $m=1$, one has $x_{h}=1$ by the presentation above. Let $x_{i_{1}},\dots,x_{i_{m-1}}$ have order $\leq 2$. Since $\Pi$ is simply-laced, $\varepsilon(m-1,h)=-1=\varepsilon(h,m-1)$ which implies $x_{h}x_{i_{m-1}}^{-1}x_{h}^{-1}x_{i_{m-1}}^{-1}=1$ and $x_{i_{m-1}}x_{h}^{-1}x_{i_{m-1}}^{-1}x_{h}^{-1}=1$. Multiplying these expressions yields $x_{h}^{2}=1$.

Since each generator has order $\leq 2$, the relations show that the group is abelian. One concludes that $\pi_{1}(G/P_{J})\cong{C_{2}}^{n-|J|}$. ∎

4 The fundamental groups of $G(\Pi)$ and $\mathrm{Spin}(\Pi,\kappa)$

The Iwasawa decomposition $G=KAU_{+}$ implies that $K$ acts transitively on the generalized flag varieties $G/P_{J}$. In this section, we describe the generalized flag varieties and suitable covering spaces as coset spaces of $K$ and its various spin covers defined in [GHKW17]. This will then allow us to compute the fundamental group of $K$ and its various spin covers via locally trivial fibre bundles and homotopy exact sequences.

Lemma 4.1.

The canonical map $\psi:K/(K\cap P_{J})\to G/P_{J}$ is a homeomorphism. In particular, there exists a homeomorphism $G/P_{J}\to K/(K\cap T)K_{J}$.

Proof.

Bijectivity follows from the product formula for subgroups since $G=KP_{J}$. By Lemma 3.2, the map $\tilde{\psi}:G/(K\cap P_{J})\to G/P_{J}$ is continuous, so the same holds for its bijective restriction $\psi:K/(K\cap P_{J})\to G/P_{J}$.

In order to show that $\psi$ is closed, let $P:=P_{J}$ and let $\tilde{P}:=P_{J}\cap K$. Consider the commutative diagram

[TABLE]

where $\iota$ denotes the canonical embedding and $\varphi$ denotes the canonical map from $G/\tilde{P}$ to $G/P$. Since $K$ is closed in $G$ by [FHHK20, Section 3F], the map $\iota$ is closed. By Lemma 3.2, $\varphi$ is open.

Let $X\tilde{P}\subseteq K/\tilde{P}$ be a closed subset of $K/\tilde{P}$ and suppose that $\psi(X\tilde{P})=XP$ is not closed in $G/P$. Then the complement $\mathsf{C}_{G/P}(XP)$ is not open in $G/P$, hence the complement $\mathsf{C}_{G/\tilde{P}}(\varphi^{-1}(XP))=\varphi^{-1}(\mathsf{C}_{G/P}(XP))$ is not open in $G/\tilde{P}$. Therefore, $\varphi^{-1}(XP)$ is not closed in $G/\tilde{P}$. This yields that $X\tilde{P}=\psi^{-1}(XP)=\iota^{-1}(\varphi^{-1}(XP))$ is not closed in $K/\tilde{P}$, a contradiction.

For the second claim, since $P_{J}=G_{J}B$ and $\theta(P_{J})\cap P_{J}=G_{J}T$, one has $P_{J}\cap K=K_{J}(K\cap T)$. Furthermore, $G_{J}$ is normal in $G_{J}T$ which implies $K_{J}(K\cap T)=(K\cap T)K_{J}$. The claim follows. ∎

The key advantage of the description of a generalized flag variety as a $K$-coset space lies in the fact that $K\cap T$ is a finite group. It is therefore straightforward to write down covering spaces of generalized flag varieties via the following well-known basic observation from covering theory.

Lemma 4.2.

Let $\varphi:X\to Y$ be a continuous, open, surjective map between Hausdorff topological spaces. If all fibers are finite and of constant cardinality, then $\varphi$ is a covering map.

This readily applies in our setting:

Lemma 4.3.

The canonical map $\psi:K/K_{J}\to K/(K\cap T)K_{J}$ is a covering map of degree $2^{n-|J|}$.

Proof.

By Lemma 3.2, $\psi$ is continuous, open and surjective.

By [FHHK20, Lemma 3.20 and the discussion after Prop 3.8], the group $\tilde{T}:=(K\cap T)$ has order $2^{n}$. Note that one has $T_{J}\cap T_{I\setminus J}=\{1\}$, since the Kac–Moody group $G$ being algebraically simply connected implies $T\cong T_{J}\times T_{I\setminus J}$. Now, for $k\in K$ one has $\psi^{-1}(k\tilde{T}K_{J})=\{ktK_{J}\mid t\in\tilde{T}\}$, and since $T_{J}\cap T_{I\setminus J}=\{1\}$, one has $kt_{i}K_{J}\neq kt_{j}K_{J}$ for $t_{i}\neq t_{j}\in T\cap K_{I\setminus J}$. This yields $|\psi^{-1}(k\tilde{T}K_{J})|=|\{ktK_{J}\mid t\in\tilde{T}\}|=|\{ktK_{J}\mid t\in T\cap K_{I\setminus J}\}|=|T\cap K_{I\setminus J}|=|T_{I\setminus J}\cap K_{I\setminus J}|=2^{n-|J|}$. Lemma 4.2 now shows that $\psi$ is a covering map.∎

Definition 4.4 [GHKW17, Definition 16.2].

Let $\Pi^{\mathrm{adm}}$ be the graph on the vertex set $V$ with edge set

[TABLE]

where $\varepsilon(i,j)$ denotes the parity of the corresponding Cartan matrix entry, as defined in Notation 3.13.

An admissible colouring of $\Pi$ is a map $\kappa:V\to\{1,2\}$ such that

1. (a)

$\kappa(i^{\lambda})=1$ whenever there exists $j\in I\setminus\{i\}$ with $\varepsilon(i,j)=1$ and $\varepsilon(j,i)=-1$. 2. (b)

the restriction of $\kappa$ to any connected component of the graph $\Pi^{\mathrm{adm}}$ is a constant map.

Define $c(\Pi,\kappa)$ to be the number of connected components of $\Pi^{\mathrm{adm}}$ on which $\kappa$ takes the value 2. For a subgraph $\Pi^{\mathrm{adm}}_{J}$ of $\Pi^{\mathrm{adm}}$ that is a union of connected components of $\Pi^{\mathrm{adm}}$ let $\kappa_{J}$ be the corresponding restriction of $\kappa$.

Definition 4.5.

Let be the colouring $\gamma:V\to\{r,g,b\}$ of $\Pi^{\mathrm{adm}}$ that to each connected component $\bar{\Pi}^{\mathrm{adm}}$ of $\Pi^{\mathrm{adm}}$ assigns a colour as follows: Let $\bar{\Pi}^{\mathrm{adm}}$ be coloured red (denoted by $r$) if it contains a vertex $i^{\lambda}$ such that there exists a vertex $j^{\lambda}\in V$ satisfying $\varepsilon(i,j)=1$ and $\varepsilon(j,i)=-1$; let $\bar{\Pi}^{\mathrm{adm}}$ be coloured green ($g$) if it is not red and consists only of an isolated vertex; and blue ($b$) else.

We refer to the introduction for a discussion of various examples.

Definition and Remark 4.6.

Recall from the introduction that in [GHKW17, Definition 16.16], the spin group $\mathrm{Spin}(\Pi,\kappa)$ with respect to $\Pi$ and $\kappa$ is defined as the universal enveloping group of a particular $\mathrm{Spin}(2)$-amalgam $\{\tilde{G}_{ij},\tilde{\phi}_{ij}^{i}\mid i\neq j\in I\}$ where the isomorphism type of $\tilde{G}_{ij}$ depends on the $(i,j)$- and $(j,i)$-entries of the Cartan matrix of $\Pi$ as well as the values of $\kappa$ on the corresponding vertices. The group $K(\Pi)$ can be regarded as (being uniquely isomorphic to) the universal enveloping group of an $\mathrm{SO}(2,\mathbb{R})$-amalgam $\{G_{ij},\phi_{ij}^{i}\mid i\neq j\in I\}$ where each $\tilde{G}_{ij}$ covers $G_{ij}$ via an epimorphism $\alpha_{ij}$. By [GHKW17, Lemma 16.18] there exists a canonical central extension $\rho_{\Pi,\kappa}:\mathrm{Spin}(\Pi,\kappa)\to K(\Pi)$ that makes the following diagram commute for all $i\neq j\in I$:

[TABLE]

Here, $\tilde{\tau}_{ij}$ and ${\tau}_{ij}$ denote the respective canonical maps into the universal enveloping groups.

By [GHKW17, Proposition 3.9], one has

[TABLE]

Each connected component of $\Pi^{\mathrm{adm}}$ that admits a vertex $i^{\lambda}$ with $\kappa(i^{\lambda})=2$ contributes a factor $2$ to the order of $\ker(\rho_{\Pi,\kappa})$ so that $\mathrm{Spin}(\Pi,\kappa)$ is a $2^{c(\Pi,\kappa)}$-fold central extension of $K(\Pi)$.

In particular, this implies that the subspace topology on $K(\Pi)$ defines a unique topology on $\mathrm{Spin}(\Pi)$ that turns the extension into a covering map. The resulting group topology on $\mathrm{Spin}(\Pi,\kappa)$ is called the Kac–Peterson topology on $\mathrm{Spin}(\Pi,\kappa)$.

In the case of an irreducible simply-laced diagram $\Pi$, the only admissible colourings are the (trivial) constant colouring $V\to\{1\}$ (that every diagram admits) and the constant colouring $\kappa:V\to\{2\}$; we define the spin group $\mathrm{Spin}(\Pi)$ with respect to $\Pi$ as $\mathrm{Spin}(\Pi):=\mathrm{Spin}(\Pi,\kappa)$.

Before turning to the general case, we will first consider the simply-laced case and formulate and prove the corresponding simplified versions of the main theorems.

Lemma 4.7.

Let $\Pi$ be irreducible simply-laced distinct from $A_{1}$ and let $\{i,j\}\subseteq I$ be the index set of an $A_{2}$-subdiagram of $\Pi$. Then the spaces $\mathrm{Spin}(\Pi)/\mathrm{Spin}(\Pi_{ij})$ and $K/K_{ij}$ are homeomorphic.

Proof.

From [GHKW17] (exact references below) it follows that the kernel of the covering map $\mathrm{Spin}(\Pi)\to K$ coincides with the kernel of the covering map $\mathrm{Spin}(\Pi_{ij})\to K_{ij}$ and is equal to the group $Z:=\{\pm 1_{\mathrm{Spin}(\Pi)}\}$ (for the definition of $-1_{\mathrm{Spin}(\Pi)}$, see below). This is a consequence of the following facts regarding an irreducible simply-laced diagram $\Pi$ (all referring to [GHKW17]):

• •

There is an epimorphism $\mathrm{Spin}(2)\to\mathrm{SO}(2,\mathbb{R})$ with kernel $\{\pm 1_{\mathrm{Spin}(2)}\}$ (see [Theorem 6.8]).

• •

In $\mathrm{Spin}(\Pi)$, all elements $\tilde{\tau}_{ij}(\tilde{\phi}_{ij}^{i}(-1_{\mathrm{Spin}(2)}))$ coincide (see [Lemma 11.7]).

• •

Let $-1_{\mathrm{Spin}(\Pi)}:=\tilde{\tau}_{ij}(\tilde{\phi}_{ij}^{i}(-1_{\mathrm{Spin}(2)}))$ for an arbitrary pair $i\neq j\in I$. Then $1_{\mathrm{Spin}(\Pi)}\neq-1_{\mathrm{Spin}(\Pi)}$ (see [Corollary 11.16]).

• •

$\mathrm{Spin}(\Pi)$ is a 2-fold central extension of $K(\Pi)$ (see [Theorem 11.17]).

Hence, the 2-fold covering map $\tilde{\varphi}:\mathrm{Spin}(\Pi)\to K(\Pi)$ induces a continuous bijective map $\varphi:\mathrm{Spin}(\Pi)/\mathrm{Spin}(\Pi_{ij})\to(\mathrm{Spin}(\Pi)/Z)/(\mathrm{Spin}(\Pi_{ij})/Z)\to K/K_{ij}$. One has a commutative diagram

[TABLE]

Since $\tilde{\varphi}$ is open as a covering map and $\pi_{2}$ is open by Lemma 3.2, it follows that $\varphi$ is a homeomorphism. ∎

Lemma 4.8.

Let $\Pi$ be irreducible simply-laced distinct from $A_{1}$ and $\emptyset\neq J\subset I=\{1,...,n\}$. Then $K/K_{J}$ is simply connected.

Proof.

$K/K_{J}$ is connected since $K$ is generated by connected groups isomorphic to $\mathrm{SO}(2,\mathbb{R})$. Hence by Lemma 4.3 it is a non-trivial cover of $K/(K\cap T)K_{J}$ of degree $2^{n-|J|}$. The claim now follows from Corollary 3.16 and Corollary 4.1. ∎

The following proposition provides our main result in the simply laced case.

Proposition 4.9.

Let $\Pi$ be irreducible simply-laced distinct from $A_{1}$. Then $\mathrm{Spin}(\Pi)$ is simply connected with respect to the Kac–Peterson topology. In particular, $\pi_{1}(G)\cong C_{2}$.

Proof.

By [Hus94, 4.2.4], for a closed subgroup $H$ of a topological group $G$, the projection $p:G\to G/H$ is a principal $H$-bundle. By Lemma A.13, this bundle is locally trivial if $H$ is a (closed) Lie group (note that, by [HR63, Theorem 5.11], every locally compact subgroup of a topological group is closed). Since locally trivial bundles admit local cross sections, [Ste99, Corollary in Section 7.4] implies that, if $H$ is a closed Lie group, then $p:G\to G/H$ is a fibre bundle with fibre $H$. This yields a locally trivial fibre bundle

[TABLE]

By [Hat02, Chapter 4], this yields the homotopy long exact sequence

[TABLE]

from which one extracts the exact sequence

[TABLE]

By Lemmas  4.7 and 4.8 one has $\pi_{1}(\mathrm{Spin}(\Pi)/\mathrm{Spin}(\Pi_{ij}))\cong\pi_{1}(K/K_{ij})=\{1\}$ and so by exactness $\pi_{1}(\mathrm{Spin}(\Pi))=\{1\}$.

The second assertion follows from the fact that $\pi_{1}(G)\cong\pi_{1}(K)$ by Corollary A.15 and the fact that $\mathrm{Spin}(\Pi)$ is a 2-fold central extension of $K$ by [GHKW17, Theorem 11.17]. ∎

We will now return to the case of a general irreducible Dynkin diagram $\Pi$.

Notation 4.10.

For a subset $J\subseteq I$ let

[TABLE]

Lemma 4.11.

Let $J\subseteq I$ be the index set of a connected component $\Pi^{\mathrm{adm}}_{J}$ of $\Pi^{\mathrm{adm}}$. Then the following hold:

1. (a)

If $\Pi^{\mathrm{adm}}_{J}$ has colour $r$, then $H_{J}\cong C_{2}^{|J|}$. 2. (b)

If $\Pi^{\mathrm{adm}}_{J}$ has colour $g$, then $|J|=1$ and $H_{J}\cong\mathbb{Z}$. 3. (c)

If $\Pi^{\mathrm{adm}}_{J}$ has colour $b$, then $|H_{J}|=2^{|J|+1}$.

Proof.

(a): If $\Pi^{\mathrm{adm}}_{J}$ has colour $r$, then there exist $i\in J,j\in I\setminus\{i\}$ with $\varepsilon(i,j)=1$ and $\varepsilon(j,i)=-1$. This implies $x_{i}x_{j}=x_{j}x_{i}$ and $x_{j}x_{i}^{-1}=x_{i}x_{j}$ which yields $x_{i}^{2}=1$. Now, if $\{i^{\lambda},k^{\lambda}\}$ is an edge in $\Pi^{\mathrm{adm}}$, then $x_{i}x_{k}^{-1}x_{i}^{-1}x_{k}^{-1}=1=x_{k}x_{i}^{-1}x_{k}^{-1}x_{i}^{-1}$. Multiplying these expressions shows that $x_{i}^{2}=1$ implies $x_{k}^{2}=1$. Since $\Pi^{\mathrm{adm}}_{J}$ is connected, this yields $x_{k}^{2}=1$ for each $k\in J$. Commutativity then follows from the relations of $H_{J}$.

(b): By definition, nodes of colour $g$ are isolated in $\Pi^{\mathrm{adm}}$.

(c): Let $\Pi_{J}^{\mathrm{sl}}$ be the simply laced Dynkin diagram with vertex set $J^{\lambda}$ and edge set $\{\{i,j\}\in J\times J\mid\{i,j\}\text{ edge in }\Pi^{\mathrm{adm}}\}$. Let $\tilde{T}:=K(\Pi_{J}^{\mathrm{sl}})\cap T(\Pi_{J}^{\mathrm{sl}})$ where $T(\Pi_{J}^{\mathrm{sl}})$ denotes the standard maximal torus of $G(\Pi_{J}^{\mathrm{sl}})$. Then by Lemma 4.3 and Proposition 4.9, $\mathrm{Spin}(\Pi_{J}^{\mathrm{sl}})\to K(\Pi_{J}^{\mathrm{sl}})\to K(\Pi_{J}^{\mathrm{sl}})/\tilde{T}$ is a universal covering map where $K(\Pi_{J}^{\mathrm{sl}})\to K(\Pi_{J}^{\mathrm{sl}})/\tilde{T}$ has degree $2^{|J|}$ and $\mathrm{Spin}(\Pi_{J}^{\mathrm{sl}})\to K(\Pi_{J}^{\mathrm{sl}})$ has degree $2$ according to [GHKW17, Theorem 11.17]. Since $\pi_{1}(K(\Pi_{J}^{\mathrm{sl}})/\tilde{T})\cong H_{J}$ by Theorem 3.15 and Lemma 4.1, this implies $|H_{J}|=2^{|J|+1}$. ∎

Proposition 4.12.

Let $J_{1}\sqcup\dots\sqcup J_{k}=I$ be the index sets of the connected components of $\Pi^{\mathrm{adm}}$. If the Bruhat decomposition satisfies the conclusion of Proposition 3.7, then

[TABLE]

Proof.

By Theorem 3.15, $\pi_{1}(G/B)\cong H_{I}$ where

[TABLE]

as defined in 4.10. For $J\subseteq I$, let

[TABLE]

the set of relators of $H_{J}$. Let

[TABLE]

the set of commutators of pairs of generators from different connected components of $\Pi^{\mathrm{adm}}$. Then

[TABLE]

Let $\pi_{H_{I}}$ and $\pi_{H}$ be the canonical homomorphisms from the free group $\langle x_{i};\quad i\in I\rangle$ to $H_{I}$ and $H$, respectively. It suffices to show that $\bigcup_{l=1}^{k}R_{J_{l}}\cup R^{c}\subseteq\ker\pi_{H_{I}}$ and $R_{I}\subseteq\ker\pi_{H}$. It is clear that a relator $x_{i}x_{j}^{\varepsilon(i,j)}x_{i}^{-1}x_{j}^{-1}\in R_{I}$ with $i^{\lambda}$ and $j^{\lambda}$ in a common connected component is contained in $\bigcup_{l=1}^{k}R_{J_{l}}\subseteq\ker\pi_{H}$, so let $x_{i}x_{j}^{\varepsilon(i,j)}x_{i}^{-1}x_{j}^{-1}\in R_{I}$ with $i^{\lambda}$ and $j^{\lambda}$ in different connected components of $\Pi^{\mathrm{adm}}$. Then one has $(\varepsilon(i,j),\varepsilon(j,i))\in\{(1,1),(1,-1),(-1,1)\}$. If $\varepsilon(i,j)=1$, then $x_{i}x_{j}^{\varepsilon(i,j)}x_{i}^{-1}x_{j}^{-1}\in R^{c}\subseteq\ker\pi_{H}$, so let $\varepsilon(i,j)=-1$ and $\varepsilon(j,i)=1$. Then $j^{\lambda}$ is contained in a connected component $\Pi^{\mathrm{adm}}_{J_{m}}$ of colour $r$, and by Lemma 4.11, $\langle x_{l};\quad l\in J_{m}\mid R_{J_{m}}\rangle=H_{J_{m}}\cong C_{2}^{|J_{m}|}$.

This implies that $x_{j}$ has order $2$ in ${H_{J_{m}}}$, hence $x_{j}^{2}\in\langle\langle R_{J_{m}}\rangle\rangle_{\langle x_{i};i\in I\rangle}$, the normal closure of $R_{J_{m}}$ in the free group.

Since $\langle\langle R_{J_{m}}\rangle\rangle_{\langle x_{i};i\in I\rangle}\subseteq\ker\pi_{H}$, one obtains $x_{j}^{2}\in\ker\pi_{H}$. Since $x_{j}x_{i}x_{j}^{-1}x_{i}^{-1}\in R^{c}\subseteq\ker\pi_{H}$ and $\varepsilon(i,j)=-1$, one therefore has

[TABLE]

Conversely, it is clear that $\bigcup_{l=1}^{k}R_{J_{l}}\subseteq R_{I}\subseteq\ker\pi_{H_{I}}$, so let $x_{i}x_{j}x_{i}^{-1}x_{j}^{-1}\in R^{c}$ with $i^{\lambda}$ and $j^{\lambda}$ in different connected components. As above, we can assume that $\varepsilon(i,j)=-1$ and $\varepsilon(j,i)=1$. Since $x_{j}x_{i}^{\varepsilon(j,i)}x_{j}^{-1}x_{i}^{-1}\in\ker\pi_{H_{I}}$, this implies

[TABLE]

This proves the assertion. ∎

Theorem 4.13**.**

Let $\Pi$ be an irreducible Dynkin diagram such that $G(\Pi)$ satisfies the conclusions of Proposition 3.7 and of Theorem A.15. Let $n(g)$ and $n(b)$ be the number of connected components of $\Pi^{\mathrm{adm}}$ of colour $g$ and $b$, respectively. Then

[TABLE]

In particular, this statement holds in the symmetrizable case.

Proof.

By Theorem A.15, $\pi_{1}(G)\cong\pi_{1}(K)$, so it suffices to prove that $\pi_{1}(K)$ is of the given isomorphism type; note that Theorem A.15 has only been established in the symmetrizable case. Let $J\subseteq I$. The diagram

[TABLE]

with all maps being the respective canonical maps, commutes. Since the maps are continuous by Lemma 3.2, one obtains a commutative diagram of induced homomorphisms

[TABLE]

where $p_{*}$ and $q_{*}$ are injective, because $p$ and $q$ are covering maps (see Lemma 4.3). By Theorem 3.15 and Lemma 4.1, $\pi_{1}(K/(K\cap T))$ and $\pi_{1}(K/(K\cap T)K_{J})$ can be identified with $H_{I}=\langle x_{i};\quad i\in I\mid R_{I}\rangle$ and $\langle x_{i};\quad i\in I\mid R_{I}\cup\{x_{j}\mid j\in J\}\rangle$, respectively ($R_{I}$ as in (4) in the above proof), where $\psi_{*}$ corresponds to the canonical homomorphism between these groups as the proof of Theorem 3.15 shows.

For the index set $J_{m}$ of a connected component of $\Pi^{\mathrm{adm}}$, let $\bar{J}_{m}:=I\setminus J_{m}$. Then by Proposition 4.12,

[TABLE]

Summing up, one obtains a commutative diagram

[TABLE]

having replaced $p_{*}$ and $q_{*}$ from above with the corresponding monomorphisms.

By Lemma 4.3, the covering $K/K_{\bar{J}_{m}}\to K/K_{\bar{J}_{m}}(K\cap T)$ has degree $2^{n-|\bar{J}_{m}|}=2^{|J_{m}|}.$ This implies that $\tilde{H}_{m}:=q_{*}(\pi_{1}(K/K_{\bar{J}_{m}}))$ is a subgroup of $H_{J_{m}}$ of index $2^{|J_{m}|}$. The isomorphism type of $\tilde{H}_{m}$ is uniquely determined by this index and Lemma 4.11: One has

[TABLE]

Again by Lemma 4.3, the covering $K\to K/(K\cap T)$ has degree $2^{n}$, so $p_{*}(\pi_{1}(K))$ is a subgroup of index $2^{n}$ of $\prod_{i=1}^{k}{H_{J_{i}}}$. The commutative diagram above implies that $\pi_{1}(K)\cong p_{*}(\pi_{1}(K))\subseteq\pi_{m}^{-1}(\tilde{H}_{m})$. Since this holds for the index set of every connected component of $\Pi^{\mathrm{adm}}$, one has $p_{*}(\pi_{1}(K))\subseteq\tilde{H}_{1}\times\dots\times\tilde{H}_{m}$. But the latter is a subgroup of index $2^{|J_{1}|}\cdot\dots\cdot 2^{|J_{m}|}=2^{n}$ of $\prod_{i=1}^{k}{H_{J_{i}}}$, so equality holds. This proves the assertion. ∎

Theorem 4.14**.**

Let $\Pi$ be an irreducible Dynkin diagram such that $G(\Pi)$ satisfies the conclusion of Proposition 3.7. Let $n(g)$ be the number of connected components of $\Pi^{\mathrm{adm}}$ of colour $g$. Let $n(b,\kappa)$ be the number of connected components of $\Pi^{\mathrm{adm}}$ on which $\kappa$ takes the value 1 and which have colour $b$. Then

[TABLE]

In particular, this statement holds in the $2$-spherical and the symmetrizable case.

Proof.

By [GHKW17, Theorem 17.1], the map $\rho_{\Pi,\kappa}:\mathrm{Spin}(\Pi,\kappa)\to K$ is a $2^{c(\Pi,\kappa)}$-fold central extension. Let $J$ be the index set of a connected component of $\Pi^{\mathrm{adm}}$ and let $\bar{J}:=I\setminus\bar{J}$. Let $U_{\bar{J}}:=\langle\tilde{G}_{ij}\mid i\neq j\in J\rangle_{\mathrm{Spin}(\Pi,\kappa)}$.

Since $\rho_{\Pi,\kappa}(U_{\bar{J}})\subseteq K_{\bar{J}}$, one has a continuous induced map $\rho_{\Pi,\kappa}^{J}:\mathrm{Spin}(\Pi,\kappa)/\mathrm{Spin}(\Pi_{\bar{J}},\kappa_{\bar{J}})\to K/K_{\bar{J}}$ making the following diagram commute, where $\tilde{\varphi}$ and $\varphi$ denote the respective canonical maps:

[TABLE]

Each fiber of $\rho_{\Pi,\kappa}^{J}$ has cardinality

[TABLE]

Since $\rho_{\Pi,\kappa}$ is open as a covering map and $\varphi$ is open by Lemma 3.2, it follows from Lemma 4.2 that $\rho_{\Pi,\kappa}^{J}$ is a covering map.

From here the proof is analogous to the proof of Theorem 4.13, after extending the commutative diagram at the beginning of the latter proof:

[TABLE]

One obtains that $\pi_{1}(\mathrm{Spin}(\Pi,\kappa))\cong\prod_{i=1}^{k}H^{\prime}_{J_{i}}$ where each $H^{\prime}_{J_{m}}$ is a subgroup of index $2^{c(\Pi,\kappa)-c(\Pi_{\bar{J}_{m}},\kappa_{\bar{J}_{m}})}$ of

[TABLE]

Since $\Pi^{\mathrm{adm}}_{\bar{J}_{m}}$ is the union of all connected components except $\Pi^{\mathrm{adm}}_{J_{m}}$, one has $c(\Pi,\kappa)-c(\Pi_{\bar{J}_{m}},\kappa_{\bar{J}_{m}})\in\{0,1\}$, depending on whether $\kappa$ is constant $1$ or $2$ on $\Pi^{\mathrm{adm}}_{J_{m}}$. This implies

[TABLE]

This proves the assertion. ∎

Now all theorems from the introduction have been proved.

Appendix A Maximal unipotent subgroups of Kac–Moody groups and applications to Kac–Moody symmetric spaces (by Tobias Hartnick and Ralf Köhl)

Throughout this appendix we fix a symmetrizable generalized Cartan matrix ${\bf A}$ with underlying diagram $\Pi$. We consider the corresponding algebraically simply-connected semisimple split real Kac–Moody group $G:=G(\Pi)=[G_{\mathbf{A}}(\mathbb{R}),G_{\mathbf{A}}(\mathbb{R})]$ as given by Definition 1.1. As in Section 2 we also denote by $K_{\mathbf{A}}(\mathbb{R})\leq G_{\mathbf{A}}(\mathbb{R})$ the fixed point subgroup of the Cartan–Chevalley involution $\theta$ and set $K:=K(\Pi)=K_{\mathbf{A}}(\mathbb{R})\cap G$. We equip all of these groups with the restrictions of the Kac–Peterson topology.

The goal of this appendix is to relate the topology of $G$ to the topology of $K$. Our main result (see Theorem A.15 below) asserts that the inclusion $K\hookrightarrow G$ is a weak homotopy equivalence. This implies in particular that $\pi_{1}(G)\cong\pi_{1}(K)$ and thus allows the computation of $\pi_{1}(G)$ by the methods presented in the main part of the article.

In the spherical case the subgroup $K<G$ is even a deformation retract and hence the inclusion $K\hookrightarrow G$ is a homotopy equivalence, as a consequence of the topological Iwasawa decomposition of $G$. This decomposition also implies that the associated Riemannian symmetric space $G/K$ is contractible.

While real Kac–Moody groups also possess an Iwasawa decomposition, it is currently unknown whether this decomposition is topological. To establish our main result we thus have to work with a certain central quotient $\overline{G}$ of $G$, for which the topological Iwasawa decomposition was established in [FHHK20]. We will show that the image $\overline{K}$ of $K$ in $\overline{G}$ is a strong deformation retract and that the reduced Kac–Moody symmetric space $\overline{G}/\overline{K}$ is contractible. Since the finite-dimensional central extension $G\to\overline{G}$ is a Serre fibration by a classical result of Palais [Pal61], this will allow us to deduce the desired result about $G$ and $K$.

A.1 The topological Iwasawa decomposition

Let us denote by $\operatorname{Ad}:G_{\mathbf{A}}(\mathbb{R})\to\operatorname{Aut}(\mathfrak{g}_{\mathbb{R}}(\mathbf{A}))$ and $\operatorname{Ad}:G(\Pi)\to\operatorname{Aut}(\mathfrak{g}_{\mathbb{R}}^{\prime}(\mathbf{A}))$ the adjoint representations of $G_{\mathbf{A}}(\mathbb{R})$ and $G=G(\Pi)$ respectively. We recall from [FHHK20] that the quotient map $G\to{\rm Ad}(G)$ factors as

[TABLE]

where $\overline{G}$ is uniquely determined by the fact that $\overline{T}:=p_{1}(T)\cong(\mathbb{R}^{\times})^{{\rm rk}(\mathbf{A})}$ is a torus and $p_{2}$ has finite kernel. The group $\overline{G}$ is referred to as the semisimple adjoint quotient of $G$, and we equip it with the quotient topology with respect to the Kac–Peterson topology on $G$. We will denote by $U^{\pm}$ the positive, respectively negative maximal unipotent subgroup of $G(\Pi)$ as introduced in Section 2. Also recall from Section 2 that $A_{\mathbb{R}}:=\exp(\mathfrak{h}_{\mathbb{R}}(\mathbf{A}))\leq G_{\mathbf{A}}(\mathbb{R})$ and set $A:=A_{\mathbb{R}}\cap G$.

Lemma A.1 (Iwasawa decomposition).

Multiplication induces continuous bijections

[TABLE]

Proof.

This follows from [KP85, Proposition 5.1(a)]. ∎

A more refined statement has been established in [FHHK20] for the semisimple adjoint quotient $\overline{G}$ of $G$. To state this result, denote by

[TABLE]

the canonical quotient maps from (5) and set $\overline{K}:=p_{1}(K)$, $\overline{T}:=p_{1}(T)\cong(\mathbb{R}^{\times})^{{\rm rk}(\mathbf{A})}$, $\overline{A}:=p_{1}(A)=\overline{T}^{o}$ and $\overline{U^{+}}:=p_{1}(U^{+})$. Equip these groups with their respective quotient topologies and note that $p_{1}$ restricts to a bijection between $U^{+}$ and $\overline{U^{+}}$.

Theorem A.2** (Topological Iwasawa decomposition, [FHHK20, Theorem 3.23]).**

Multiplication induces homeomorphisms

[TABLE]

Since $\overline{A}$ is contractible, in order to show that $\overline{K}$ is a deformation retract of $\overline{G}$ it will suffice to show that $\overline{U^{+}}$ is contracible. We thus need to understand the topology induced by the Kac–Peterson topology on the standard unipotent subgroups.

A.2 The Kac–Peterson topology on $U^{\pm}$

We now turn to the study of the restriction of the Kac–Peterson topology to the standard maximal unipotent subgroups $U^{-}$ and $U^{+}$. Recall from Section 2 that the Weyl group $W$ is a Coxeter group, so elements of $W$ can be represented by reduced words in the generators $s_{1},\dots,s_{r}$. Given such a reduced word $w=(s_{i_{1}},\dots,s_{i_{r}})$ in $W$ with corresponding simple roots $\alpha_{i_{1}},\dots\alpha_{i_{r}}$ we define positive roots $\beta_{1},\dots,\beta_{r}$ by

[TABLE]

We then set $U_{w}:=U_{\beta_{1}}\cdots U_{\beta r}\subset U^{+}$ and define a map

[TABLE]

It is established in [CR09, Section 5.5, Lemma] that the map $\mu_{w}$ is a bijection for every reduced word $w$, and that its image $U_{w}$ depends only on the Weyl group element represented by $w$, but not on the chosen reduced expression. Since $G_{\mathbf{A}}(\mathbb{R})$ is a topological group, the bijection $\mu_{w}$ is continuous. In fact, one can show that $\mu_{w}$ is a homeomorphism. A proof of this fact was sketched in [HKM13, Lemma 7.25]; since openness of the maps $\mu_{w}$ is crucial for everything that follows, we fill in the details of this sketch here:

Lemma A.3.

For every reduced word $w$ the map $\mu_{w}$ is a homeomorphism onto its image.

Proof.

We argue by induction on the length $m$ of $w$ and observe that the case $m=1$ holds by definition. Since the linear functionals $\alpha_{1},\dots,\alpha_{r}$ are linearly independent, there exists an element $X\in\mathfrak{h}_{\mathbb{R}}(\mathbf{A})$ (see Section 2) such that $\alpha_{i_{1}}(X)=0$ and $\alpha_{j}(X)<0$ for all $j\in\{1,\dots,\widehat{i_{1}},\dots,r\}$. It follows that $\beta_{1}(X)=\alpha_{i_{1}}(X)=0$ and $\beta_{k}(X)<0$ for all $k=2,\dots,m$. Indeed, since the word $w$ is reduced, none of the positive real roots $\beta_{2},\dots,\beta_{m}$ equals $\alpha_{i_{1}}$, and since $n\alpha_{i_{1}}$ is not a root for any $n\geq 2$ (cf. [Kac90, Proposition 5.1]), each of them contains at least one other positive simple root as a summand. Now for $j\in\{1,\dots,m\}$ and $Y\in\mathfrak{g}_{\beta_{j}}$ we have ${\rm ad}(X)(Y)=\beta_{j}(X)(Y)$, and thus

[TABLE]

We conclude that if $x_{j}\in U_{\beta_{j}}$, then

[TABLE]

where the convergence is uniform on compacta. This shows that the map

[TABLE]

is continuous, and hence the map

[TABLE]

is continuous. Now let $w^{\prime}=(r_{i_{2}},\dots,r_{i_{m}})$ and let $\beta_{2}^{\prime}=s_{i_{1}}(\beta_{2})$, …, $\beta_{m}^{\prime}:=s_{i_{1}}(\beta_{m})$. Now by Axiom (RGD2) of an RGD system (see [AB08, Chapter 8]) there exists an element $g\in G_{\mathbf{A}}(\mathbb{R})$ such that $gU_{\beta_{j}}g^{-1}=U_{\beta_{j}^{\prime}}$ for all $j=2,\cdots,m$, and by induction hypothesis we have a homeomorphism

[TABLE]

Conjugating the inverse of this homeomorphism by $g^{-1}$ we obtain a homeomorphism

[TABLE]

Composing this homeomorphism with the map (7) now provides the desired continuous inverse to $\mu_{w}$. ∎

To describe the topology on $U^{+}$ we recall that there exist several distinct but related partial orders on $W$ which in different places in the literature are referred to as the Bruhat order on $W$. In the sequel we will consider the following version; here $\ell$ denotes the length function with respect to the generating set $\{s_{1},\dots,s_{r}\}$.

Definition A.4.

The weak right Bruhat order on $W$ is the partial order $\leq_{w}$ defined as

[TABLE]

According to [CR09, p. 44] we have $w_{1}\leq_{w}w_{2}$ if and only if there exists a reduced word $(r_{i_{1}},\dots,r_{i_{\ell(w_{2})}})$ for $w_{2}$ such that $w_{1}=r_{i_{1}}\cdots r_{i_{\ell(w_{1})}}$.

Recall that for the strong Bruhat order $\leq$ one has $w_{1}\leq w_{2}$ if there exists a reduced word $(r_{i_{1}},\dots,r_{i_{m}})$ for $w_{2}$ and a reduced word $(r_{j_{1}},\dots,r_{j_{l}})$ for $w_{1}$ such that $(r_{j_{1}},\dots,r_{j_{l}})$ is a substring of $(r_{i_{1}},\dots,r_{i_{m}})$ (not necessarily consecutive). By definition,

[TABLE]

but the converse is not true. An important difference between the weak right Bruhat order and the strong Bruhat order is that $(W,\leq)$ contains a cofinal chain, i.e., a totally ordered subset $T\subset W$ such that for every $w\in W$ there exists $t\in T$ such that $w\leq t$, whereas for the weak right Bruhat order, such a cofinal chain does not exist. In fact, given $w_{1},w_{2}\in W$ there will in general not exist an element $w_{3}\in W$ with $w_{1}\leq_{w}w_{3}$ and $w_{2}\leq_{w}w_{3}$.

Note that if $w_{1}\leq_{w}w_{2}$, then we can choose a reduced word $(r_{i_{1}},\dots,r_{i_{\ell(w_{2})}})$ for $w_{2}$ such that $w_{1}=r_{i_{1}}\cdots r_{i_{\ell(w_{1})}}$. Thus if we define $\beta_{1},\dots,\beta_{\ell(w_{2})}$ as above then we have a commuting diagram

[TABLE]

where the horizontal maps are inclusions, and the vertical maps are homeomorphisms. In particular, we have a continuous inclusion $\iota_{w_{1}}^{w_{2}}:U_{w_{1}}\hookrightarrow U_{w_{2}}$, hence we may form the colimit

[TABLE]

in the category of topological spaces. We emphasize that in view of the previous remark the system $((U_{w})_{w\in W},(\iota_{w_{1}}^{w_{2}})_{w_{1}\leq_{w}w_{2}})$ is not directed, hence this colimit is not a direct limit.

Proposition A.5.

The $k_{\omega}$-space $U^{+}$ is given by the colimit

[TABLE]

both in the category of topological spaces and in the category of $k_{\omega}$-topological spaces.

Proof.

The corresponding statement in the category of sets is established in [CR09, Theorem 5.3]. For the topological statement see [HKM13, Proposition 7.27]. ∎

In view of the applications to Kac–Moody symmetric spaces that we have in mind we recall that $U^{\pm}$ are subgroups of the commutator subgroup $G$ of $G_{\mathbb{R}}(\mathbf{A})$, in particular we can consider their images $\overline{U}^{\pm}:=p_{1}(U^{\pm})$ under the map $p_{1}:G\to\overline{G}$ from (5). In this context we will need the following fact:

Proposition A.6.

The map $p_{1}$ induces homeomorphisms $U^{\pm}\to\overline{U}^{\pm}$.

Proof.

By [HKM13, Proposition 7.27] the map $T\times U^{+}\to TU^{+}$ is a homeomorphism and the kernel of $p_{1}$ is contained in $T$. The latter implies that $p_{1}$ restricts to a continuous bijection $U^{+}\to\overline{U^{+}}$, and the former implies that this bijection is open. ∎

A.3 Dilation structures on $U^{\pm}$

Definition A.7.

Let $U$ be a topological group. By a dilation structure on $U$ we mean a family of maps $(\Phi_{t}:U\to U)_{t\in\mathbb{R}}$ with the following properties:

1. (a)

Each $\Phi_{t}$ is a continuous automorphisms of the topological group $U$. 2. (b)

$(\Phi_{t})_{t\in\mathbb{R}}$ is a $1$-parameter group, i.e. $\Phi_{0}={\rm Id}$ and $\Phi_{s+t}=\Phi_{s}\circ\Phi_{t}$ for all $s,t\in\mathbb{R}$. 3. (c)

If we define $\Phi_{-\infty}:U\to U$ by $\Phi_{-\infty}(u):=e$, then the map

[TABLE]

is continuous.

Remark A.8.

Note that if a topological group $U$ admits a dilation structure, then it is in particular contractible. Indeed, if we define $\Psi_{t}:=\Phi_{\frac{t}{t-1}}$, then

[TABLE]

is continuous with $\Psi_{0}=\Phi_{0}={\rm Id}$ and $\Psi_{1}=\Phi_{-\infty}$, hence a contraction to the identity.

Dilation structures on finite-dimensional simply-connected nilpotent Lie groups play a major role in conducting analysis on such groups, see e.g. [Goo76]. Not every finite-dimensional simply-connected nilpotent Lie group admits a dilation structure, but if $U$ is the unipotent radical of a minimal parabolic subgroup of a semisimple Lie group, then such a dilation structure always exists. The methods of [Kum02] allow one to extend this result to the Kac–Moody setting.

Following [Kac90, §3.12], we define the fundamental chamber of $\mathfrak{h}_{\mathbb{R}}(\mathbf{A})$ as

[TABLE]

Since the family $(\alpha_{i})_{1\leq i\leq n}$ is linearly independent, there exists

[TABLE]

Indeed, by the linear independence of $(\alpha_{i})_{1\leq i\leq n}$ the solution space for the system of $n-1$ linear equations $\forall 2\leq i\leq n:\alpha_{1}(x)-\alpha_{i}(x)=0$ has strictly larger dimension than the solution space for the system of $n$ linear equations $\forall 1\leq i\leq n:\alpha_{i}(x)=0$.

We now define a $1$-parameter subgroup of $A_{\mathbb{R}}$ by $a_{t}:=\exp(tX_{0})$ and denote by

[TABLE]

the associated automorphism of the Lie algebra $\mathfrak{u}^{+}=\bigoplus_{\alpha\in\Delta_{+}}\mathfrak{g}_{\alpha}^{k}$. Similarly we denote by

[TABLE]

the restriction of the conjugation-action of $a_{t}$ on $G_{\mathbf{A}}(\mathbb{R})$ to $U^{+}$. Note that if $X\in\mathfrak{u}^{+}$ is ad-locally finite then

[TABLE]

From (1) and the defining property of $X_{0}$ one deduces that for every positive root $\alpha$ with height $|\alpha|$

[TABLE]

It follows that for all positive roots $\alpha$ one has

[TABLE]

(see [Tit87, (4), p. 549]), where $\{x_{\alpha}(s)\mid s\in\mathbb{R}\}\cong(\mathbb{R},+)$ is the root subgroup of $G_{\mathbf{A}}(\mathbb{R})$ corresponding to the root space $\mathfrak{g}_{\alpha}$. As a consequence, if one endows each of the root subgroups $\{x_{\alpha}(s)\mid s\in\mathbb{R}\}$ with the natural topology of $\mathbb{R}$, then $\Phi_{t}$ contracts each of them. We are now in a position to reproduce the following result and proof by Kumar:

Theorem A.9** ([Kum02, Proposition 7.4.17]).**

The family $(\Phi_{t})_{t\in\mathbb{R}}$ defines a dilation structure on $U^{+}$.

Proof.

Let $w$ be a reduced word and write $w=s_{i_{1}}\cdots s_{i_{r}}\in W$ with corresponding simple roots $\alpha_{i_{1}},\dots\alpha_{i_{r}}$. Recall that multiplication induces a homeomorphism

[TABLE]

where the roots $\beta_{1},\dots,\beta_{r}$ are given by

[TABLE]

Given an element $x_{\beta_{1}}(y_{1})x_{\beta_{2}}(y_{2})\cdots x_{\beta_{r}}(y_{r})\in U_{w}$ by (8) one has

[TABLE]

Setting $\Phi_{-\infty}(u):=e$ for all $u\in U^{+}$, we deduce that the map

[TABLE]

is continuous and that $\Phi_{0}={\rm Id}_{U_{w}}$. Combining this with Proposition A.5 one deduces that the map

[TABLE]

is continuous, hence a dilation structure. ∎

Recall that $U^{+}$ is isomorphic to $U^{-}$ under the Cartan–Chevalley involution of $G_{\mathbf{A}}(\mathbb{R})$, which maps $a_{t}$ to $a_{-t}$. Thus if we define $\Phi^{-}_{t}:=c_{a_{-t}}|_{U^{-}}$ then we obtain:

Corollary A.10.

The family $(\Phi^{-}_{t})_{t\in\mathbb{R}}$ defines a dilation structure on $U^{-}$.∎

Combining this with Remark A.8 and Proposition A.6 we can record:

Corollary A.11.

The topological groups $U^{+}$ and $U^{-}$ are contractible. Consequently, the groups $\overline{U}^{+}$ and $\overline{U}^{-}$ are contractible.∎

A.4 Homotopy groups of real-split semisimple Kac–Moody groups

Corollary A.12.

The subgroup $\overline{K}<\overline{G}$ is a deformation retract. In particular the inclusion $i_{K}:K_{\mathbf{A}}(\mathbb{R})\hookrightarrow G_{\mathbf{A}}(\mathbb{R})$ is a homotopy equivalence and thus induces isomorphisms $(i_{K})_{*}:\pi_{n}(\overline{K})\to\pi_{n}(\overline{G})$ for all $n\geq 0$.

Proof.

We have established in Corollary A.11 that $\overline{U^{+}}$ is contractible, and $\overline{A}$ is contractible since it is homeomorphic to $\mathbb{R}^{{\rm rk}(\mathbf{A})}$. The assertion now follows from Theorem A.2. ∎

Since it is currently unknown whether the Iwasawa decomposition of $G$ is also a topological decomposition, the strategy of the above proof can not be applied to $G$. However, using the following result of Palais [Pal61, Section 4.1, Corollary], one can still obtain an isomorphism between the fundamental groups of $G$ and $K$.

Proposition A.13 (Palais).

Let $G$ be a topological group and let $H<G$ be a subgroup which is homeomorphic to a Lie group. Then the fibration $H\hookrightarrow G\to G/H$ is locally trivial, in particular a Hurewicz fibration, hence there is a long exact sequence of homotopy groups

[TABLE]

Recall that the kernel of the quotient map $G\to\overline{G}$ is homeomorphic to $(\mathbb{R}^{\times})^{{\rm cork}(\mathbf{A})}$. In particular it has $2^{{\rm cork}(\mathbf{A})}$ connected components, whereas its higher homotopy groups vanish. Applying Proposition A.13 to the diagram of fibrations

[TABLE]

we thus obtain:

Corollary A.14.

There is a commutative diagram with exact rows

[TABLE]

Moreover, for $n\geq 2$ there are isomorphisms $\pi_{n}(G_{\mathbf{A}}(\mathbb{R}))\cong\pi_{n}(G)$ and $\pi_{n}(K_{\mathbf{A}}(\mathbb{R}))\cong\pi_{n}(K)$.

Combining this with Corollary A.12 we deduce:

Theorem A.15**.**

For every $n\geq 0$ the inclusion $K\hookrightarrow G$ induces isomorphisms

[TABLE]

hence is a weak homotopy equivalence. In particular, $\pi_{1}(G)\cong\pi_{1}(K)$. ∎

A.5 Kac–Moody symmetric spaces and causal contractions

We conclude this appendix with an application to the results obtained so far to Kac–Moody symmetric spaces. It was established in [FHHK20] that the homogeneous spaces $G_{\mathbf{A}}(\mathbb{R})/K_{\mathbf{A}}(\mathbb{R})$ and $G/K$ carry the natural structure of topological reflection spaces, and the same holds for their quotients ${\rm Ad}(G_{\mathbf{A}}(\mathbb{R}))/{\rm Ad}(K_{\mathbf{A}}(\mathbb{R}))$ and ${\rm Ad}(G)/{\rm Ad}(K)$. The topological reflection space $\mathcal{X}=G/K$ is called the unreduced Kac–Moody symmetric space of type ${\mathbf{A}}$, and the topological reflection space $\overline{\mathcal{X}}={\rm Ad}(G)/{\rm Ad}(K)=\overline{G}/\overline{K}$ is called the reduced Kac–Moody symmetric space of type ${\mathbf{A}}$.

Corollary A.16.

The reduced symmetric space $\overline{\mathcal{X}}$ is contractible.

Proof.

In view of the topological Iwasawa decomposition the orbit map at the basepoint $o=e\overline{K}$

[TABLE]

is a homeomorphism. Since $\overline{U^{+}}$ and $\overline{A}$ are contractible, this implies contractability of $\overline{\mathcal{X}}$. ∎

The proof of Theorem A.9 can be used to provide an explicit contraction for $\overline{\mathcal{X}}\simeq\overline{U^{+}}\times\overline{A}$, using the contraction by conjugation with suitable elements of the torus $T_{\mathbb{R}}$ on the group $\overline{U^{+}}$ and the standard contraction on the finite-dimensional real vector space $A$. It turns out that this contraction has interesting additional properties. Recall from [FHHK20, Section 7] that the symmetric space $\overline{\mathcal{X}}$ admits future and past boundaries $\Delta^{+}_{\|}$ and $\Delta^{-}_{\|}$ that both carry a simplicial structure which turns them in the geometric realizations of the positive and negative halves of the twin building of $G_{\mathbf{A}}(\mathbb{R})$. Following [FHHK20, Section 7], a causal ray is a geodesic ray of $\overline{\mathcal{X}}$ whose parallelity class equals a point in $\Delta^{+}_{\|}$ and a piecewise geodesic causal curve is the concatenation of a finite set of segments of causal rays that can be parametrized in such a way that the walking direction always points towards the future boundary. Given $x,y\in\overline{\mathcal{X}}$ we say that $x$ causally preceeds $y$ (in symbols $x\preceq y$) if there exists a piecewise geodesic causal curve from $x$ to $y$.

Since both conjugation by elements of $T_{\mathbb{R}}$ and the standard contraction of the vector space $A$ preserve geodesic rays and the future and past boundaries (cf. [FHHK20, Section 7]), the set of piecewise geodesic causal curves of $\overline{\mathcal{X}}$, and hence the causal pre-order $\preceq$, are invariant under the given contraction.

Corollary A.17.

The reduced symmetric space $\overline{\mathcal{X}}$ is causally contractible, i.e., it admits a contraction that preserves $\preceq$.

Appendix B The Bruhat decomposition is a CW decomposition (by Julius Grüning and Ralf Köhl)

Let $G$ be a Kac–Moody group endowed with the Kac–Peterson topology and let $T$ be the standard maximal torus and $U^{+}$, $U^{-}$ the standard unipotent subgroups. [KP83, Theorem 4(a)] asserts without proof that the multiplication map

[TABLE]

is a homeomorphism with respect to the Kac–Peterson topology. In this note we provide a proof in the symmetrizable case that makes use of this fact in the two-spherical case ([HKM13, Proposition 7.31]), of the embedding of Kac–Moody groups constructed in [Mar19, Theorem 3.15(2)], and of the fact that the Kac–Peterson topology is $k_{\omega}$. Among the various consequences of this result is that the Bruhat decomposition of a symmetrizable topological Kac–Moody group is a CW decomposition.

Recall that a $k$-space (alternatively: compactly generated space) is a topological space $X$ in which a set $C\subset X$ is closed if and only if its intersection $C\cap K$ with any compact subset $K$ of $X$ is compact. That is, a $k$-space is a topological space $X$ whose topology is coherent with the family of all compact subspaces of $X$. A $k_{\omega}$-space is a topological space $X$ whose topology is coherent with respect to a countable ascending family of compact subspaces. By (3) of [FST77] any $k_{\omega}$-space is a $k$-space.

Proposition B.1 ([Pal70, Corollary]).

A continuous proper map $f:X\to Y$ from a topological space $X$ to a $k$-space $Y$ is closed. In particular, a continuous injection $\iota:X\to Y$ into a $k_{\omega}$-space $Y=\bigcup_{n\in\mathbb{N}}Y_{m}$ with compact $Y_{m}$ such that for each $m\in\mathbb{N}$ the pre-image $\iota^{-1}(Y_{m})$ is also compact is a topological embedding, i.e., it is a homeomorphism onto its image.

Proof.

The first statement is exactly [Pal70, Corollary]. The second statement is an immediate consequence of the first, since a $k_{\omega}$-space is a $k$-space in which any compact subset $K$ of $Y$ is contained in some $Y_{m}$ of the ascending family $(Y_{m})_{m\in\mathbb{N}}$ of compact subsets (statement (3) of [FST77]). ∎

Remark B.2.

The authors thank Tobias Hartnick and Stefan Witzel for various lively discussions concerning the correct formulation and application of Proposition B.1. Moreover, they thank Stefan Witzel for suggesting to make use of the concept of proper maps.

A subgroup of a Kac–Moody group is bounded if it lies in the intersection of two spherical parabolic subgroups of opposite signs. In other words, it is bounded if and only if it stabilises a point the Davis CAT(0) realization of each half of its twin building. The maximal bounded subgroups of a Kac–Moody group have been determined in [CM06, Theorem 4.1].

Proposition B.3.

Let $G$ be a split real Kac–Moody group. Then the Kac–Peterson topology $\tau_{\mathrm{KP}}$ on $G$ equals the finest group topology $\tau_{\mathrm{MB}}$ on $G$ such that the embeddings of the maximal bounded subgroups, each endowed with its Lie group topology, are continuous.

Proof.

By [Mar15, Lemma 4.3], the Kac–Peterson topology $\tau_{\mathrm{KP}}$ on $G$ induces the Lie group topology on its maximal bounded subgroups. A fundamental $\mathrm{SL}_{2}(\mathbb{R})$ is bounded and, in particular, embeds as a closed subgroup into a maximal bounded subgroup. Therefore its subspace topology equals its Lie group topology; by [HKM13, Proposition 7.21] the topology $\tau_{\mathrm{KP}}$ equals the finest group topology on $G$ such that the embeddings of the fundamental $\mathrm{SL}_{2}(\mathbb{R})$ Lie subgroups is continuous, whence $\tau_{\mathrm{KP}}$ is finer than or equal to the final group topology $\tau_{\mathrm{MB}}$ with respect to the embedded maximal bounded subgroups. Again, since by [Mar15, Lemma 4.3] the Kac–Peterson topology on $G$ induces the Lie group topology on its maximal bounded subgroups, the two described topologies actually coincide. ∎

Corollary B.4.

Let $G$ be a split real Kac–Moody group endowed with the Kac–Peterson topology and let $(G_{i})_{i\in I}$ be a finite family of Lie-subgroups of $G$ such that each fundamental $\mathrm{SL}_{2}(\mathbb{R})$ is contained in at least one of the $G_{i}$. Then the Kac–Peterson topology on $G$ equals the finest group topology on $G$ such that the embeddings of the $(G_{i})_{i}$, each endowed with its Lie group topology, are continuous.

Proposition B.5 (cf. [HKL15, 1.5, 1.10], [Mar19, Theorem 3.15(2)]).

Any symmetrizable topological Kac–Moody group endowed with the Kac–Peterson topology admits a continuous injective group homomorphism into a simply laced topological Kac–Moody group with closed image with respect to the Kac–Peterson topology.

Proof.

By [Mar19, Theorem 3.15(2)] for any symmetrizable Kac–Moody group $G$ there is an injective group homomorphism $\iota:G\to H$ into a simply laced Kac–Moody group $H$ embedding each fundamental rank-$1$ subgroup $G_{\alpha_{i}}\cong\mathrm{SL}_{2}(\mathbb{R})$ diagonally into the direct product

[TABLE]

of a suitable (finite) family of fundamental rank-$1$ subgroups $H_{\alpha_{i,j}}$ of $H$.

The restriction of this map to any fundamental rank-$1$ subgroup $G_{\alpha}$ of $G$ is continuous with respect to the Lie group topology on $G_{\alpha}$ and the Kac–Peterson topology on $H$. Hence, by universality (see [HKM13, Proposition 7.21]), the map $\iota:G\to H$ is continuous with respect to the Kac–Peterson topology on both $G$ and $H$.

One has

[TABLE]

where $\varphi_{\sigma}$ is the automorphism of $H$ given by

[TABLE]

for some $\sigma=(\sigma_{1},\dots,\sigma_{N})$, where $\sigma_{i}\in\mathrm{Sym}(n_{i})$ acting by permuting the factors of the direct product $\prod_{j=1}^{n_{i}}H_{\alpha_{i,j}}\cong\mathrm{SL}_{2}(\mathbb{R})^{n_{i}}$. Since the automorphisms $\varphi_{\sigma}$ are continuous with respect to the Kac–Peterson topology on $H$, the group $\iota(G)$ is a closed subgroup of $H$. ∎

The embedding $\iota:G\to H$ corresponds to an embedding of the twin building $\Delta_{G}$ of $G$ into the twin building $\Delta_{H}$ of $H$ such that $\Delta_{G}=\bigcap_{\sigma}\operatorname{Fix}(\varphi_{\sigma})$ (with the $\varphi_{\sigma}$ now considered as twin building automorphisms) and the additional property that two chambers of $\Delta_{G}$ are opposite in $\Delta_{G}$ if and only if they are opposite in $\Delta_{H}$.

Indeed, this is immediate from an argument along the lines of descent in buildings (cf. [MPW15]). The automorphisms $\varphi_{\sigma}$ act on the twin apartment defined by the fundamental chambers $c_{+}$, $c_{-}$ of $\Delta_{H}$ and, by definition, the fixed substructure is isometric to a twin apartment of $\Delta_{G}$. The claim then follows from the fact that $G$ acts transitively on the twin apartments of $\Delta_{G}$.

In particular, this embedding

[TABLE]

of twin buildings induces an embedding of opposite geometries

[TABLE]

Specialising to the embedding of a fundamental rank-$1$ subgroup $G_{\alpha_{i}}\cong\mathrm{SL}_{2}(\mathbb{R})$ of $G$ diagonally into the direct product

[TABLE]

of a suitable (finite) family of fundamental rank-$1$ subgroups $H_{\alpha_{i,j}}$ of $H$, one obtains an embedding of the real projective line $\mathbb{S}^{1}$ (the building of type $A_{1}$) diagonally into a suitable product $\left(\mathbb{S}^{1}\right)^{n_{i}}$ of real projective lines (the building of type ${A_{1}}^{n_{i}}=\underbrace{A_{1}\oplus A_{1}\oplus\cdots\oplus A_{1}}_{n_{i}}$).

This in turn yields an embedding of the corresponding opposites geometries of pairs of distinct points of $\mathbb{S}^{1}$ with adjacency relation given by the complete relation (the opposite geometry of type $A_{1}$ of diameter $1$), respectively of $n_{i}$-tuples of pairs of distinct points of $\mathbb{S}^{1}$ with adjacency relation given by equality in all up to at most one component (the opposite geometry of type ${A_{1}}^{n_{i}}$ of diameter $n_{i}$).

Refer to [Gra09, Section 4.3] for more details, some examples, and applications of the opposite geometry. The most striking application of the opposite geometry is a proof of [Tit74, Theorem 13.32] via its simple connectedness and Mühlherr’s generalization to Kac–Moody groups333A manuscript that has never been published and unfortunately seems to be lost. To the second author’s dismay he has lost his copy that he once owned.; see also [AM97].

The following result follows immediately from the preceding discussion:

Proposition B.6.

Let $\iota:G\to H$ be the injective group homomorphism from Proposition B.5, let $\Delta_{G}=(\Delta^{+}_{G},\Delta^{-}_{G},\delta^{*}_{G})\to\Delta_{H}=(\Delta^{+}_{H},\Delta^{-}_{H},\delta^{*}_{H})$ be the induced embedding of twin buildings, and $\mathrm{Opp}(\Delta_{G})=\{(c,d)\in\Delta^{+}_{G}\times\Delta^{-}_{G}\mid\delta^{*}_{G}(c,d)=1\}\to\mathrm{Opp}(\Delta_{H})=\{(c,d)\in\Delta^{+}_{H}\times\Delta^{-}_{H}\mid\delta^{*}_{H}(c,d)=1\}$ the resulting embedding of opposite geometries. Given $(c_{+},c_{-})\in\mathrm{Opp}(\Delta_{G})$, for all $n\in\mathbb{N}$ exists $m\in\mathbb{N}$ such that the intersection of $\mathrm{Opp}(\Delta_{G})$ with the ball of radius $n$ in $\mathrm{Opp}(\Delta_{H})$ around $(c_{+},c_{-})$ is contained in the ball of radius $m$ in $\mathrm{Opp}(\Delta_{H})$ around $(c_{+},c_{-})$.

Corollary B.7.

Let $G$ be a topological Kac–Moody group endowed with the Kac–Peterson topology. If it is two-spherical or symmetrizable, then the multiplication map $\varphi:U^{+}\times T\times U^{-}\to G$ is a homeomorphism onto its image.

Proof.

The two-spherical case is [HKM13, Proposition 7.31]. In the symmetrizable case note that Proposition B.1 is applicable since the Kac–Peterson topology is $k_{\omega}$ by [HKM13, Proposition 7.10]. Consequently, the injection from Proposition B.5 yields a topological embedding $\iota:G\to H$, provided one can find $k_{\omega}$-decompositions $G=\bigcup_{n}G_{n}$ and $H=\bigcup_{m}H_{m}$ such that each intersection $H_{m}\cap\iota(G)$ lies in some $\iota(G_{n})$. (Indeed, $\iota^{-1}(H_{m})$ is closed by continuity of $\iota$, so it is compact once it lies inside some compact set $G_{n}$, which is equivalent to $H_{m}\cap\iota(G)\subset\iota(G_{n})$.)

For $G$ and $H$ choose $k_{\omega}$-decompositions making use of Corollary B.4 and $k_{\omega}$-decompositions of the fundamental subgroups $G_{\alpha_{i}}\cong\mathrm{SL}_{2}(\mathbb{R})$ of $G$ and the corresponding subgroups $\prod_{j=1}^{n_{i}}H_{\alpha_{i,j}}\cong\mathrm{SL}_{2}(\mathbb{R})^{n_{i}}$ of $H$ into which the $G_{\alpha_{i}}$ embed diagonally, endowed with their Lie group topology. That is,

[TABLE]

where each of the $X_{t}^{t}$ is the ball of radius $t$ around $1$ of the maximal bounded subgroup $X_{t}$ endowed with some suitable metric inducing its Lie group topology, with $X\in\{G,H\}$ and lower index $t$ taken modulo the total number of maximal bounded subgroups.

By construction, each $H_{j}^{j}$ intersects $\iota(G)$ in some compact subset of a fundamental subgroup $G_{\alpha_{i}}$ of $G$ with respect to the Lie group topology. In other words, each $H_{j}^{j}\cap\iota(G)$ lies in some $\iota(G_{k}^{k})$. Forming finite products of such sets and using Proposition B.6 one concludes that $H_{t}\cap\iota(G)=(H_{1}^{1}\cdots H_{t}^{t})\cap\iota(G)$ lies in some suitable product $G_{t}=G_{1}^{1}\cdots G_{t}^{t}$; that is, the injective homomorphism $\iota:G\to H$ indeed is a topological embedding.

Since $\iota$ restricts to maps {\left.\kern-1.2pt\iota\vphantom{\big{|}}\right|_{U_{+}^{G}}}:U_{+}^{G}\to U_{+}^{H}, {\left.\kern-1.2pt\iota\vphantom{\big{|}}\right|_{U_{-}^{G}}}:U_{-}^{G}\to U_{-}^{H}, {\left.\kern-1.2pt\iota\vphantom{\big{|}}\right|_{T^{G}}}:T^{G}\to T^{H}, one can conclude that the diagram

[TABLE]

commutes, which proves that the map $\varphi^{G}$ is a homeomorphism onto its image, since $\varphi^{H}$ is a homeomorphism onto its image by [HKM13, Proposition 7.31]. ∎

Corollary B.8.

Let $G$ be a topological Kac–Moody group endowed with the Kac–Peterson topology. If it is two-spherical or symmetrizable, then the associated twin building with the quotient topology is a strong topological twin building.

Proof.

The two-spherical case is [HKM13, Theorem 1]. In the symmetrizable case it follows by replacing [HKM13, Proposition 7.31] with Corollary B.7; cf. the discussion after [HKM13, Theorem 1]. ∎

Corollary B.9.

Let $G$ be a topological Kac–Moody group endowed with the Kac–Peterson topology. If it is two-spherical or symmetrizable, then the Bruhat decomposition of a symmetrizable Kac–Moody group is a CW decomposition.

Proof.

This is a restatement of Proposition 3.7 from the main text. Its proof heavily relies on Corollary B.8. ∎

Corollary B.10.

Let $G$ be a topological Kac–Moody group endowed with the Kac–Peterson topology. If it is two-spherical or symmetrizable, then the coset model, the group model, and the involution model of the reduced Kac–Moody symmetric space are pairwise homeomorphic with respect to their internal topologies.

Proof.

The two-spherical case is [FHHK20, Proposition 4.19]. In the symmetrizable case it follows from [FHHK20, Proposition 4.19] and Corollary B.7. ∎