Imaginary cones and limit roots of infinite Coxeter groups

Abstract

Let (W, S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropic cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots. In this article we study the close relations of the imaginary cone with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.

Appendix: Relation of limit roots to Benoist’s limit sets

Let $\Phi$ be a based root system, associated to a Coxeter group W. We assume here that $\Phi$ is irreducible, of indefinite type and of rank at least three, and that $\Phi$ spans V linearly. When $\Phi$ is non-degenerate (i.e., the associated bilinear form is non-degenerate), we explain in this appendix how the set of limit roots $E(\Phi )$ can be identified with one of the projected limit sets of Benoist [5], which are limit sets associated to a Zariski dense subgroup of a connected reductive algebraic group. Benoist’s framework is described in “ A.1”. Constructing the identification involves generalizing first a result by Benoist-De la Harpe [4] on the Zariski closure of a Coxeter group (“ A.2”). In “ A.5”, we prove the identification of E with Benoist’s limit set (Theorem A.3) and we obtain this way a new characterization of the set of limit roots in the non-degenerate case (Corollary A.4).

These results do not extend directly to the case where $\Phi$ is degenerate, because the natural ambient algebraic group is not reductive (“ A.3”). However, in this case $E(\Phi )$ will project onto some $E(\Psi )$ with $\Psi$ non-degenerate, as explained in “ A.6”.

1.1 A.1

This subsection describes, somewhat informally and imprecisely, a part of the results of [5], referring to [7] and [35, Chapter1] for the necessary background. Let $\Gamma$ be a Zariski dense subsemigroup of the group of k-points G(k) of a connected, reductive algebraic group G defined over a local field k. Benoist attaches to $\Gamma$ certain (equivalent) notions of “limit set” for $\Gamma$ in G. We discuss the realization of the limit set as a subset $\Lambda$ of a suitable flag variety Y. Our applications involve only the special case in which $k=\mathbb {R}$, G is semisimple and $\Gamma$ is a group, and we assume this henceforward for simplicity. Below, the set of k-points X(k) of a complex algebraic variety X defined over k is always considered as an analytic k-variety (in particular, it is taken to have the standard Hausdorff topology induced from that of k).

The standard parabolic k-subgroups of G may be naturally indexed as $P_{\theta }$ for subsets $\theta$ of the set $\Pi$ of restricted simple roots, so that $P_{\theta }\supseteq P_{\theta '}$ if $\theta \subseteq \theta '$. Attached to $P_\theta$, one has a “flag variety” $Y_{\theta }\,{:}{=}\,G(k)/P_\theta (k)$ which is a compact analytic k-manifold on which a maximal compact subgroup K of G(k) acts transitively. There is a K-invariant probability measure $\mu _{\theta }$ on $Y_{\theta }$. Define $\Lambda _{\theta }$ to be the set of all points x in $Y_{\theta }$ such that there is a sequence $(\gamma _{n})_{n\in \mathbb {N}}$ in $\Gamma$ such that the sequence $\gamma _{n}^{*}(\mu _{\theta })$ of pullback measures converges to a Dirac mass concentrated at x.

The following facts are from [5, 3] (see especially the first paragraph of 3 and 3.5–3.6). The set $\Lambda _{\theta }$ is a closed, $\Gamma$-invariant subset of $Y_\theta$. Let us denote by $Y\,{:}{=}\,Y_ \Pi$ the flag variety associated to the minimal parabolic k-subgroup, and $\Lambda \,{:}{=}\,\Lambda _ \Pi$ the set of associated limit points in Y for $\Gamma$. Because of our assumption $k=\mathbb {R}$, one has $\Lambda \ne \varnothing$ and any non-empty $\Gamma$-invariant closed subset of Y contains $\Lambda$. For $\theta \subseteq \Pi$, $\Lambda _{\theta }$ is the image of $\Lambda$ under the natural projection $Y\rightarrow Y_{\theta }$.

The results of the preceding paragraph imply that for all $\theta \subseteq \Pi$, $\Lambda _{\theta }$ is a non-empty, closed, $\Gamma$-invariant subset of $Y_{\theta }$, and that any non-empty ${\Gamma }$-invariant closed subset of $Y_{\theta }$ contains $\Lambda _{\theta }$. These properties uniquely characterize $\Lambda _{\theta }$ and may be taken as definitions for our purposes below. They imply in particular that the $\Gamma$-action on each $\Lambda _{\theta }$ is minimal. We call $\Lambda$ the limit set of $\Gamma$. By a projected limit set, we mean a set $\Lambda _{\theta }$ for some $\theta \subseteq \Pi$.

1.2 A.2

For non-degenerate, spanning based root systems (associated to a Coxeter group W), the set of limit roots will be identified below with a suitable projected limit set. First we need to understand what is the right algebraic group to consider. The Zariski closure of W in its standard reflection representation is described by Benoist-De la Harpe in [4]. We extend below their result to the more general class of reflection representations considered in our paper. Although this result is used here only for non-degenerate forms, we state it in natural generality corresponding to that in op. cit.

Fix a based root system $(\Phi ,\Delta )$ in (V, B), together with its Coxeter group W. We assume from now on that $\Phi$ is irreducible of indefinite type and of rank at least three, and that $\Phi$ spans V linearly. Let $O(V)=O(V,B)$ denote the orthogonal group of (V, B); that is,

$$\begin{aligned} O(V)\,{:}{=}\,\{g\in \text { GL}(V)\,|\,B(gv,gv')=B(v,v')\text {for all }v,v'\in V\}. \end{aligned}$$

Let $V^\perp =\{v\in V\,|\,B(v,v')=0 \text { for all }v'\in V\}$ denote the radical of (V, B), and define the following subgroup of O(V):

$$\begin{aligned} H(B)\,{:}{=}\,\{g\in O(V)\,|\,g(v)=v \text {for all }v\in V^\perp \}. \end{aligned}$$

Let $(V_{\mathbb {C}},B_{\mathbb {C}})$ denote the quadratic space arising as the complexification of (V, B) (i.e., $V_{\mathbb {C}}\,{:}{=}\,V\otimes _{\mathbb {R}}\mathbb {C}$ and $B_{\mathbb {C}}$ is the symmetric bilinear form on $V_{\mathbb {C}}$ arising by extension of scalars to $\mathbb {C}$ from B on V). Similarly as above, we define the orthogonal group $O(V_{\mathbb {C}})=O(V_{\mathbb {C}},B_{\mathbb {C}})$ and its subgroup $H(B_{\mathbb {C}})=H(V_{\mathbb {C}},B_{\mathbb {C}})$. Regarding the natural map $\mathrm{GL}(V,\mathbb {R})\rightarrow \mathrm{GL}(V_{\mathbb {C}},\mathbb {C})$ as an inclusion, we regard H(B) as a subgroup of $H(B_{\mathbb {C}})$. Note that $H(B_{\mathbb {C}})$ is a linear algebraic group, since it is closed in the Zariski topology of $\mathrm{GL}(V_{\mathbb {C}},\mathbb {C})$. More precisely, we view $H\,{:}{=}\,H(B_{\mathbb {C}})$ as a complex linear algebraic group defined over k with $H(k)\,{:}{=}\,H(B)$ as its (Zariski dense) group of k-points.

The main result of [4] extends to this setting as follows:

Proposition A.1

The Zariski closure of W is H (i.e., W is Zariski dense in H(k)).

Proof

In case $\Delta$ is linearly independent the argument is the same, mutatis mutandis, as that in [4]. The general case can be reduced to that case as follows. Choose a subset $\Delta '$ of $\Delta$ which is inclusion maximal subject to the requirements that $\Delta '$ is linearly independent and the corresponding standard parabolic subgroup $W_{\Delta '}=\langle s_\alpha \,|\, \alpha \in \Delta '\rangle$ is irreducible. We claim that $\Delta '$ spans V. Otherwise, there is some $\alpha \in \Delta {\setminus } \mathrm{span}(\Delta ')$. By irreducibility of W, one may suppose without loss of generality that $B(\alpha ,\beta )\ne 0$ for some $\beta \in \Delta \cap \mathrm{span}(\Delta ')$. This implies $B(\alpha ,\gamma )\ne 0$ for some $\gamma \in \Delta '$. Then $\Delta ''\,{:}{=}\,\Delta '\cup \{\alpha \}$ is linearly independent and $W_{\Delta ''}$ is irreducible, contrary to maximality of $\Delta '$. Note $W_{\Delta '}$ is of rank at least three and is of indefinite type, since its type is determined by the signature of B. By the case of linearly independent simple roots, $W_{\Delta '}$ is Zariski dense in H(k) and hence so is $W\supseteq W_{\Delta '}$. $\square$

1.3 A.3

To apply Benoist’s results, the ambient algebraic group should be connected and reductive. Since H is not connected, we will first need to replace H with $H^0$, its connected component of the identity, and W with $W\cap H^0(k)$. We therefore need the following simple fact.

Proposition A.2

The algebraic group $H^0$ is reductive if and only if $V^\perp =\{0\}$.

Proof

Choose a complementary subspace U to $V^\perp$ in V and let $B_{U}$ be the restriction of B to a symmetric bilinear form on U. Let r denote the dimension of $V^\perp$ and m that of U . Let A denote the $m\times m$ matrix (with respect to some basis) of $B_{U}$ on U . Then (with respect to a basis obtained by extending that basis by a basis of $V^\perp$) the matrix of B on V is a diagonal block matrix $\mathrm{diag}(A,0_r)$ , where $0_r$ is the $r\times r$ zero matrix. Then H(k) (resp., H) identifies with the group of all real (resp., complex) block matrices of the form

$$\begin{aligned} \begin{bmatrix}X&\quad 0\\ Y&\quad \mathrm{Id}_{r}\end{bmatrix} \end{aligned}$$

(A.1)

where Y, of size $r\times m$, is arbitrary and X satisfies $X^{t}AX=A$. The subgroup of such (complex) matrices with $Y= 0$ identifies with the complex semisimple algebraic group $(O(U )_{\mathbb {C}},(B_{U} )_{\mathbb {C}})\cong O(m,\mathbb {C})$. On the other hand, the set of complex matrices (A.1) with $X=\mathrm{Id}_{m}$ is a unipotent normal (abelian) subgroup of H; it is the unipotent radical $R_{u}H^{0}$. It follows that $H^{0}$ is reductive if and only if $V^\perp =\{0\}$, in which case $H^{0}$ is semisimple.

1.4 A.4

The following notation will prove convenient below. For any finite-dimensional real vector space $U'$, let $\mathbb {P}(U')$ denote the projective space with points the real lines in $U'$, in the usual (Hausdorff) topology. For $X\subseteq U'$, let $[X]\subseteq \mathbb {P}(U')$ denote the set of lines spanned by non-zero points of X.

1.5 A.5

We assume in this subsection that $(\Phi ,\Delta )$ is (spanning and) non-degenerate, i.e. $V^\perp =\{0\}$. Let us denote as usual by Q the isotropic cone of B, $Q\,{:}{=}\,\{\,v\in V\,|\,B(v,v)=0\,\}$. In the following we explain how to identify the set of isotropic lines [Q] with some partial flag variety $Y_\theta$, for some $\theta \subseteq \Pi$ as in ” A.1”.

The assumed non-degeneracy of B implies that $H= O(V_{\mathbb {C}}, B_{\mathbb {C}})$, so the connected component $G\,{:}{=}\,H^0= SO(V_{\mathbb {C}}, B_{\mathbb {C}})$ of the identity of H is a semisimple complex algebraic k-group. Let $n=\dim V$, so that $G\cong SO(n,\mathbb {C})$. Its group G(k) of k-points identifies with $SO(V,B)\cong SO(p,q)$ where (p, q) is the signature of (V, B). Let $r\,{:}{=}\,\min (p,q)$ be the Witt index of (V, B). Since $(\Phi ,\Delta )$ is of indefinite type and rank at least three, we have $r\ge 1$ and $p+q\ge 3$. Fix a choice of maximal k-split torus in G and a set $\Pi$ of simple relative roots for the corresponding relative root system for G, which is of type $B_{r}$ if $p\ne q$ and $D_{r}$ (interpreted as $A_{1}\times A_{1}$, $A_{3}$ for $r=2,3$) if $p=q$ (see [7, 23.4]). The standard minimal parabolic k-subgroup $P_{ \Pi }$ identifies (see loc. cit.) with the stabilizer in G of a standard maximal flag, defined over k, of totally isotropic subspaces of $(V_{\mathbb {C}},B_{\mathbb {C}})$. The standard parabolic k-subgroups $P_{\theta }$, where $\theta \subseteq \Pi$, of G are precisely the subgroups of G which contain $P_{ \Pi }$. The standard k-parabolic subgroups all have interpretations similar to that of $P_{ \Pi }$, as stabilizers of standard isotropic flags in $V_{\mathbb {C}}$ defined over k, but there are complications in type D because there are two G-orbits of maximal isotropic $\mathbb {C}$-subspaces of $V_{\mathbb {C}}$.

For purposes here, it suffices to note that there is a standard parabolic k-subgroup of G, which we write as $P_{\theta }$ for some $\theta \subseteq \Pi$, given by the stabilizer of the isotropic line in that standard maximal isotropic flag. (We do not need the explicit description of $\theta$ as a subset of $\Pi$, but it may easily be determined for each type of root system). The corresponding (partial) flag variety $Y_\theta =G(k)/P_\theta (k)$ identifies with the G(k)-orbit in $\mathbb {P}(V)$ of the corresponding (real) line. Now all isotropic lines in (V, B) are in the same $G(k)=SO(V,B)$ orbit, since (by Witt’s theorem) they are in the same orbit for O(V, B), and any one of them is stabilized by the reflection in some non-isotropic vector orthogonal to that line (such a line always exists since $p+q\ge 3$). Hence $Y_{\theta }$ naturally identifies (as homogeneous spaces for G(k)) with the set [Q] of all isotropic lines in $[V]=\mathbb {P}(V)$. The above identification $[Q]=Y_{\theta }$ can be made as analytic manifolds, but it suffices here to make it as topological spaces (which is straightforward since both are compact Hausdorff spaces).

Let $\Gamma \,{:}{=}\,W\cap G(k)=\{w\in W\,|\,l(w) \text { is even}\}$ be the “rotation subgroup” of W, regarded as Zariski dense subgroup of G(k). Denote the projected limit set for $\Gamma$ in $Y_{\theta }$ as $\Lambda _{\theta }$, as in “ A.1”.

Theorem A.3

Assume (V, B) is non-degenerate and $\Delta$ spans V, and make the identification $[Q]=Y_{\theta }$, with the specific $\theta$ defined in the previous paragraphs. Then $\Lambda _{\theta }=[E(\Phi )]$, i.e., the projected limit set $\Lambda _\theta \subseteq Y_{\theta }$ for $\Gamma$ as a Zariski dense subgroup of G(k) identifies with the set of limit roots $E(\Phi )$, as subsets of [Q]. In particular, $\Lambda _{\theta }$ is W-stable.

Proof

Note $\Gamma$ is a normal subgroup of W, which acts on [Q] by restriction of the natural O(V, B)-action given by $g[\mathbb {R}\alpha ]=[\mathbb {R}g\alpha ]$ for any $g\in O(V,B)$ and non-zero $\alpha \in Q$. For any $w\in W$, $w\Lambda _{\theta }$ is a minimal non-empty closed $\Gamma$-invariant subset of [Q] (since $\Gamma =w\Gamma w^{-1}$) and therefore coincides with $\Lambda _{\theta }$ (which is the unique minimal such subset). Hence $\Lambda _{\theta }$ is stable under the W-action on [Q].

Since $[E({\Phi })]$ is a non-empty, closed, $\Gamma$-invariant subset of [Q], one has $\Lambda _{\theta }\subseteq [E(\Phi )]$, by minimality of $\Lambda _{\theta }$ amongst sets with those properties. But then $\Lambda _{\theta }$ is a non-empty, closed W-invariant subset of $[E(\Phi )]$, and the minimality of the W-action (Theorem 3.1) on $[E(\Phi )]$ forces equality in the inclusion.

Corollary A.4

If (V, B) is degenerate and $\Delta$ spans V, any non-empty, closed W-invariant subset of [Q] contains $[E(\Phi )]$.

Proof

This follows since the previous theorem implies it holds with $[E(\Phi )]$ replaced by $\Lambda _{\theta }$ and W by its subgroup $\Gamma$.

Remark A.5

• (1) Although [5] and (our extension of) [4] easily imply as above the existence of a unique non-empty closed W-invariant subset of [Q] on which the W-action is minimal, we do not know how to prove that set identifies with $[E(\Phi )]$ except as above, i.e., by use of our Theorem 3.1. In particular, we do not have a way to relate the two notions of limit sets ($\Lambda _\theta$ and $E(\Phi )$) directly from their definitions, without using their characterizations via minimality.

• (2) We do not know how to prove Corollary A.4 without use of [5]. The related result we have (Theorem 4.10) assumes that the root system is weakly hyperbolic and states only that any non-empty, closed W-invariant subset of [Q], that is also contained in $[{{\mathrm{\mathrm{conv}}}}(\Delta )]$, is equal to $[E(\Phi )]$.

• (3) For $\Phi$ non-degenerate, Theorem A.3 provides an interpretation of $[E(\Phi )]$ as a projected limit set, and [5] then yields many additional facts about $E(\Phi )$ which seem likely to have significant applications (see for example Remark A.6(3)).

• (4) It is an interesting question whether other projected limit sets $\Lambda _{\theta '}$ for $\Gamma$, and especially the limit set $\Lambda$ itself, can be given interpretations similar to those in the theorem in terms of the root system. The corresponding flag varieties $Y_{\theta '}$ involve flags containing higher dimensional totally isotropic spaces, and such isotropic subspaces already appear naturally in the study of limit roots (see for instance Proposition 6.8).

1.6 A.6

We now consider the situation for possibly degenerate root bases. Let us explain a classical way (after Krammer) to obtain from a degenerate root system a non-degenerate one, with the same Coxeter group. Let $\pi :V\rightarrow V/V^\perp$ be the natural map. The restriction of $\pi$ to U identifies U isomorphically with $V/ V^\perp$ as real vector space, and we further identify $(U ,B_{U} )$ with the quotient of (V , B) by its radical $V^\perp$. Since $(\Phi ,\Delta )$ is of indefinite type, one has $\mathrm{cone}(\Delta )\cap V^\perp =\{0\}$, and there is a (non-degenerate, spanning) based root system $(\Psi , \Sigma )$ for $(U ,B_{U} )$ where $\Psi =\pi (\Phi )$ and $\Sigma =\pi (\Delta )$ (see [31, 6.1]). The Coxeter system attached to $(\Psi , \Sigma )$ identifies canonically with W, with its natural action on the quotient space $U =V/V^\perp$.

Let $V_{1}$ be the fixed affine subspace of V transverse to $\Phi$ and $V_{0}$ be its translate through 0, as in 2.1. The map $v \mapsto [\{v\}]$ for $v\in V_{1}$ identifies $V_{1}$ homeomorphically with $[V_{1}]=\mathbb {P}(V){\setminus } \mathbb {P}(V_{0})$, an open subset of $\mathbb {P}(V)$. The rule $[\{v\}]\mapsto [\{\pi (v)\}]$ for v in $\mathrm{cone}(\Delta )\cap V_{1}$ defines a continuous surjective map $[\mathrm{cone}(\Delta )]\rightarrow [\mathrm{cone}( \Sigma )]$ of compact Hausdorff spaces (hence it is a closed, proper quotient map). Since v is isotropic if and only if $\pi (v)$ is isotropic, it easily follows that this map restricts to a surjective continuous (closed, proper, quotient) map $[E(\Phi )]\rightarrow [E(\Psi )]$ (where $E(\Phi )$ and $E(\Psi )$ are the sets of limit roots for $\Phi$ and $\Psi$ respectively) and that this latter map is in addition W-equivariant.

By Theorem 3.1(b), the W-actions on $[E(\Phi )]$ and $[E(\Psi )]$ are minimal. One easily sees that minimality on $[E(\Phi )]$ directly implies that on $[E(\Psi )]$, but we do not know any direct argument for the converse implication. Therefore, results on the limit sets for non-degenerate root systems do not easily extend to degenerate ones.

1.7 A.7

We give a simple example to show that the above map $[E(\Phi )]\rightarrow [E(\Psi )]$ is not bijective in general. Let $(\Phi ,\Delta )$ be the standard based root system attached to the following Coxeter graph, in which vertices are labeled by the corresponding simple roots:

Denote the associated quadratic space as (V, B). One easily checks that $\alpha +\beta -\delta -\epsilon$ is in $V^{\perp }$. One has $\widehat{\alpha +\beta }=\lim _{n\rightarrow \infty }\widehat{(s_{\alpha }s_{\beta })^{n}\alpha }\in E(\Phi )$ and similarly $\widehat{\delta +\epsilon }\in E(\Phi )$. Hence the above map $[E(\Phi )]\rightarrow [E(\Psi )]$ sends the distinct elements $[\{\alpha +\beta \}]$ and $[\{\delta +\epsilon \}]$ of $[E(\Phi )]$ to the same element of $[E(\Psi )]$.

Note also that although $(\Phi ,\Delta )$ is a standard based root system, we are not able to deduce the minimality of the W-action on $E(\Phi )$ from Benoist’s results.

Remark A.6

• (1) We do not know if, for degenerate, spanning $(\Phi ,\Delta )$, with $V^\perp$ defined as the radical of (V, B), any closed non-empty W-invariant subset of $[Q]{\setminus } [V^\perp ]$ contains $[E(\Phi )]$ (though the corresponding statement with $[Q]{\setminus } [ V^\perp ]$ replaced by [Q] obviously fails in general).

• (2) In the case $\Phi$ is degenerate, “ A.6” gives a W-equivariant surjection from $E(\Phi )$ to some $E(\Psi )$ with $\Psi$ non-degenerate. This construction may allow one to transfer some of the properties known in the non-degenerate case to the degenerate case, but not all. Remark (3) below illustrates both this point and Remark A.5(3).

• (3) We sketch another proof of faithfulness of the W-action on $E(\Phi )$ in the setting of Theorem 6.12 ($\Phi$ indefinite of rank at least 3, and irreducible) as follows. From “ A.6” one always has a surjective W-equivariant map from $E(\Phi )$ to some $E(\Psi )$ where $\Psi$ is non-degenerate. So it is sufficient to prove the faithfulness property in the non-degenerate case.

Thus, assume now that $\Phi$ is non-degenerate. Using the notations and result of Theorem A.3, [Q] identifies to $Y_\theta$ and $E(\Phi )$ to $\Lambda _\theta$. Using the projection $Y \rightarrow Y_{\theta }$, the Zariski density of $\Lambda$ in Y (see [5, Lemma3.6]) implies that of $\Lambda _{\theta }$ in $Y_{\theta }$. Therefore, if $w\in W$ acts as the identity on $[E(\Phi )]$, it acts as the identity on [Q]. This implies w fixes each isotropic line in V. Since $\Phi$ is irreducible and non-degenerate, this readily implies that w acts as the identity on V and hence $w=1$ by faithfulness of the W-action on $\Phi$.