2.1 Root systems and Chevalley groups

Let $\Phi $ be a reduced irreducible crystallographic root system in an n-dimensional real vector space V with inner product $\langle \cdot ,\cdot \rangle $ , with $\alpha _1,\ldots ,\alpha _n$ a choice of simple roots and $\Phi ^+$ the associated positive roots. We will adopt the standard Bourbaki labeling [Reference Bourbaki4] of the simple roots. Let $\alpha ^{\vee }=2\alpha /\langle \alpha ,\alpha \rangle $ . Let $\omega _1,\ldots ,\omega _n$ be the fundamental coweights, defined by $\langle \omega _i,\alpha _j\rangle =\delta _{i,j}$ . Let

$$ \begin{align*}Q=\mathbb{Z}\alpha_1^{\vee}+\cdots+\mathbb{Z}\alpha_n^{\vee}\quad\text{and}\quad P=\mathbb{Z}\omega_1+\cdots+\mathbb{Z}\omega_n \end{align*} $$

be the coroot lattice and coweight lattice, respectively, and note that $Q\subseteq P$ .

Let $\Gamma =\Gamma (\Phi )$ denote the Dynkin diagram of $\Phi $ (with the arrow pointing toward the short root in the case of double and triple bonds). The Coxeter diagram of $\Phi $ is obtained by removing all arrows from $\Gamma $ . The height of a root $\alpha =k_1\alpha _1+\cdots +k_n\alpha _n$ is $\mathrm {ht}(\alpha )=k_1+\cdots +k_n$ . There is a unique root $\varphi \in \Phi $ of maximal height (the highest root of $\Phi $ ). The polar type of $\Phi $ is the subset $\wp \subseteq \{1,2,\ldots ,n\}$ given by

$$ \begin{align*}\wp=\{1\leq i\leq n\mid \langle\alpha_i,\varphi\rangle\neq0\}. \end{align*} $$

See Appendix A for the list of polar types. In particular, note that if $\Phi \neq \mathsf {A}_n$ , then $\wp =\{p\}$ is a singleton set, and in this case, we often refer to the element p as the polar node.

Let $W=\langle s_{\alpha }\mid \alpha \in \Phi \rangle $ be the subgroup of $\mathsf {GL}(V)$ generated by the reflections $s_{\alpha }$ , where

$$ \begin{align*}s_{\alpha}(\lambda)=\lambda-\langle\lambda,\alpha\rangle\alpha^{\vee}\quad\text{for}\ \lambda\in V. \end{align*} $$

Let $S=\{s_1,\ldots ,s_n\}$ , where $s_i=s_{\alpha _i}$ . Then, $(W,S)$ is a spherical Coxeter system. Writing $\ell : W\to \mathbb {Z}_{\geq 0}$ for the usual length function on W, it is a well-known fact that in the simply laced case, $\ell (s_{\alpha })=2\mathrm {ht}(\alpha )-1$ .

Let $w_0$ denote the longest element of $(W,S)$ , and let $\pi _0:\{1,\ldots ,n\}\to \{1,\ldots ,n\}$ be the opposition relation given by $w_0\alpha _i=-\alpha _{\pi _0(i)}$ , for $1\leq i\leq n$ . We typically regard $\pi _0$ as an automorphism of the Dynkin diagram $\Gamma $ , and we say that “opposition is type preserving” if $\pi _0$ is the identity. If $J\subseteq S$ , let $w_J$ be the longest element of the parabolic subgroup $W_J$ generated by J.

The inversion set of $w\in W$ is $ \Phi (w)=\{\alpha \in \Phi ^+\mid w^{-1}\alpha \in -\Phi ^+\}. $ We note that the inversion set of the highest root $\varphi $ is

(2.1) $$ \begin{align} \Phi(s_{\varphi})=\{\alpha\in\Phi^+\mid \langle\alpha,\omega_i\rangle>0\text{ for some}\ i\in\wp\}, \end{align} $$

which follows directly from the equation $s_{\varphi }(\alpha )=\alpha -\langle \alpha ,\varphi ^{\vee }\rangle \varphi $ .

Let $\mathbb {F}$ be a field, and let $G_0=G_0(\Phi ,\mathbb {F})$ be the associated adjoint Chevalley group. Thus, $G_0$ is generated by elements $x_{\alpha }(a)$ with $\alpha \in \Phi $ and $a\in \mathbb {F}$ , and writing (for $\alpha \in \Phi $ and $c\in \mathbb {F}^{\times }$ )

$$ \begin{align*} s_{\alpha}(c)=x_{\alpha}(c)x_{-\alpha}(-c^{-1})x_{\alpha}(c)\quad\text{and}\quad h_{\alpha^{\vee}}(c)=s_{\alpha}(c)s_{\alpha}(1)^{-1}, \end{align*} $$

the following relations hold (for $a,b\in \mathbb {F}$ , $\alpha ,\beta \in \Phi $ with $\beta \neq \pm \alpha $ , and $c,d\in \mathbb {F}^{\times }$ )

$$ \begin{align*} x_{\alpha}(a)x_{\alpha}(b)&=x_{\alpha}(a+b),\\ h_{\alpha^{\vee}}(c)h_{\alpha^{\vee}}(d)&=h_{\alpha^{\vee}}(cd),\\ x_{\alpha}(a)x_{\beta}(b)&=x_{\beta}(b)x_{\alpha}(a)\prod x_{i\alpha+j\beta}(C_{\alpha,\beta}^{i,j}a^ib^j), \end{align*} $$

where the product is taken over $i,j\geq 1$ with $i\alpha +j\beta \in \Phi $ in any fixed order, and the elements $C_{\alpha ,\beta }^{i,j}$ are integers (depending on the order chosen in the product). For example, in the simply laced case, these commutator relations take the form $x_{\alpha }(a)x_{\beta }(b)=x_{\beta }(b)x_{\alpha }(a)$ (if $\alpha +\beta \notin \Phi $ ) or $x_{\alpha }(a)x_{\beta }(b)=x_{\beta }(b)x_{\alpha }(a)x_{\alpha +\beta }(C_{\alpha ,\beta }ab)$ for some integer $C_{\alpha ,\beta }$ (if $\alpha +\beta \in \Phi $ ), and it turns out that in this case $C_{\alpha ,\beta }=\pm 1$ .

The above relations imply the following useful formula (for $\alpha ,\beta \in \Phi $ and $a\in \mathbb {F}$ ):

$$ \begin{align*} s_{\alpha}(1)x_{\beta}(a)s_{\alpha}(1)^{-1}=x_{s_{\alpha}\beta}(\epsilon_{\alpha\beta}a), \end{align*} $$

where $\epsilon _{\alpha \beta }=\pm 1$ depend on some initial choices made in the Lie algebra (see [Reference Carter5, Proposition 4.2.2]). For many calculations, it is sufficient to simply know that $\epsilon _{\alpha \beta }\in \{-1,1\}$ ; however, when more precise knowledge is required, we will adopt the sign conventions from the Groups of Lie Type package in Magma [Reference Bosma, Cannon and Playoust3, Reference Cohen, Murray and Taylor6].

Let $G=G(\Phi ,\mathbb {F})$ be the subgroup of $\mathrm {Aut}(G_0)$ generated by the inner automorphisms $G_0$ and the diagonal automorphisms, as in [Reference Humphreys8, Reference Steinberg22]. Thus, G is generated by $G_0$ and elements $h_{\lambda }(c)$ with $\lambda \in P$ and $c\in \mathbb {F}^{\times }$ , and the following relations hold (for $a\in \mathbb {F}$ , $c,d\in \mathbb {F}^{\times }$ , $\alpha ,\beta \in \Phi $ , and $\lambda ,\mu \in P$ ):

$$ \begin{align*} h_{\lambda}(c)h_{\mu}(d)&=h_{\mu}(c)h_{\lambda}(d)& h_{\lambda}(c)h_{\lambda}(d)&=h_{\lambda}(cd),\\ h_{\lambda}(c)x_{\alpha}(a)h_{\lambda}(c)^{-1}&=x_{\alpha}(ac^{\langle\lambda,\alpha\rangle})&s_{\alpha}(1)h_{\lambda}(d)s_{\alpha}(1)^{-1}&=h_{s_{\alpha}\lambda}(d). \end{align*} $$

For each $\alpha \in \Phi $ , we write $U_{\alpha }=\langle x_{\alpha }(a)\mid a\in \mathbb {F}\rangle $ and $U^+=\langle U_{\alpha }\mid \alpha \in \Phi ^+\rangle $ . Let

$$ \begin{align*} N&=\langle s_{\alpha}(c)\mid \alpha\in\Phi, c\in\mathbb{F}^{\times}\rangle,\\ H&=\langle h_{\lambda}(c)\mid \lambda\in P,c\in\mathbb{F}^{\times}\rangle,\\ B&=\langle U^+,H\rangle=HU^+. \end{align*} $$

The subgroup B is often called the (standard) Borel subgroup. We have $H=B\cap N$ , and $(B,N)$ is a $BN$ -pair in G with Weyl group $N/H\cong W$ , where

$$ \begin{align*}s_{\alpha}(c)H\mapsto s_{\alpha}\quad\text{for all}\ c\in \mathbb{F}^{\times}. \end{align*} $$

We often write $wH$ (or $wB$ ) in place of $nH$ (or $nB$ ) whenever $n\in N$ with $nH\mapsto w$ . In fact, we will frequently write $s_{\alpha }$ in place of $s_{\alpha }(1)$ when there is no risk of confusion; however, note that $s_{\alpha }\in G$ is typically not an involution.

The Bruhat decomposition gives

$$ \begin{align*}G=\bigsqcup_{w\in W}BwB. \end{align*} $$

For subsets $A\subseteq \Phi ^+$ , we write $U_A^+=\langle x_{\alpha }(a)\mid \alpha \in A,\,a\in \mathbb {F}\rangle $ . A subset $A\subseteq \Phi ^+$ is closed if $\alpha ,\beta \in A$ and $\alpha +\beta \in \Phi $ implies that $\alpha +\beta \in A$ . It is a fundamental fact that if $A\subseteq \Phi ^+$ is closed, and if $(\beta _1,\ldots ,\beta _k)$ is a fixed choice of ordering of the elements of A, then each $u\in U_A^+$ has a unique expression as $ u=x_{\beta _1}(a_1)\cdots x_{\beta _k}(a_k) $ for some $a_1,\ldots ,a_k\in \mathbb {F}$ (see [Reference Steinberg23, Lemma 17]). In particular, because the set $A=\Phi (w)$ is closed, the B cosets in $BwB$ are precisely

(2.2) $$ \begin{align} x_{\beta_1}(a_1)\cdots x_{\beta_k}(a_k)wB,\quad\text{with}\ a_1,\ldots,a_k\in\mathbb{F}, \end{align} $$

where $\Phi (w)=\{\beta _1,\ldots ,\beta _k\}$ .

We also note that

(2.3) $$ \begin{align} BwB\cdot BsB&=\begin{cases} BwsB&\text{if}\ \ell(ws)=\ell(w)+1\\ BwB\cup BwsB&\text{if}\ \ell(ws)=\ell(w)-1 \end{cases} \end{align} $$

(see [Reference Carter5, Proposition 8.1.5]).

Let $U^-=\langle U_{\alpha }\mid \alpha \in -\Phi ^+\rangle $ . Throughout this paper, we often need to convert an element in $U^-$ to an expression in $BwB$ for some w. To do so, we make frequent use of the relation (for $a\neq 0$ )

(2.4) $$ \begin{align} x_{-\alpha}(a)=x_{\alpha}(a^{-1})s_{\alpha}(-a^{-1})x_{\alpha}(a^{-1})=x_{\alpha}(a^{-1})s_{\alpha}x_{\alpha}(a)h_{\alpha^{\vee}}(-a) \end{align} $$

(which follows from the definition of $s_{\alpha }(a)$ ). We call this the folding relation, due to connections with path models in algebraic combinatorics (see [Reference Parkinson, Ram and Schwer13]).

We say that $\mathbb {F}$ has “special characteristic” if $\mathrm {char}(\mathbb {F})=2$ for $\Phi =\mathsf {B}_n,\mathsf {C}_n,\mathsf {F}_4$ , or $\mathrm {char}(\mathbb {F})=3$ for $\Phi =\mathsf {G}_2$ . Often these cases behave differently due to additional symmetries being present.

We record some basic lemmas for latter use.

Lemma 2.1 Let $\beta _1,\ldots ,\beta _N\in \Phi ^+$ be mutually perpendicular roots. Then,

$$ \begin{align*} x_{-\beta_1}(a_1)\cdots x_{-\beta_N}(a_N)\in Bs_{\beta_1}\cdots s_{\beta_N}B\quad\text{for all}\ a_1,\ldots,a_N\neq 0. \end{align*} $$

Proof Let $U_k^+=\langle U_{\beta _1},\ldots ,U_{\beta _k}\rangle $ , for $1\leq k\leq N$ . We show, by induction, that

$$ \begin{align*}x_{-\beta_1}(a_1)\cdots x_{-\beta_N}(a_N)\in U_N^+s_{\beta_1}\cdots s_{\beta_N}U_N^+H.\end{align*} $$

The case $N=1$ is the folding relation (2.4). By the induction hypothesis, and the folding relation, for $k>1$ , we have

$$ \begin{align*} x_{\beta_1}(a_1)\cdots x_{\beta_{k}}(a_{k})&=us_{\beta_1}\cdots s_{\beta_{k-1}}u'h\cdot x_{\beta_k}(a_k^{-1})s_{\beta_k}x_{\beta_k}(a_k)h_{\beta_k^{\vee}}(-a_k), \end{align*} $$

for some $u,u'\in U_{k-1}^+$ , $h\in H$ . Because $h x_{\beta _k}(a_k^{-1})s_{\beta _k}x_{\beta _k}(a_k)=x_{\beta _k}(a)s_{\beta _k}x_{\beta _k}(b)h$ for some $a,b\in \mathbb {F}$ and $s_{\beta _1}\cdots s_{\beta _{k-1}}u'x_{\beta _k}(a)=x_{\beta _k}(\pm a)s_{\beta _1}\cdots s_{\beta _{k-1}}u'$ (as $\beta _k$ is orthogonal to $\beta _1,\ldots ,\beta _{k-1}$ ), we have

$$ \begin{align*}x_{\beta_1}(a_1)\cdots x_{\beta_{k}}(a_{k})=ux_{\beta_k}(\pm a)s_{\beta_1}\cdots s_{\beta_{k-1}}u'\cdot s_{\beta_k}x_{\beta_k}(b)hh_{\beta_k^{\vee}}(-a_k).\end{align*} $$

Similarly, $u's_{\beta _k}=s_{\beta _k}u'$ , and hence the result.▪

2.2 Split spherical buildings

We assume that the reader is already familiar with the basic theory of buildings, and our main reference for the general theory is [Reference Abramenko and Brown1]. By a split spherical building, we shall mean a building associated to a Chevalley group via the standard $BN$ -pair construction. It is easiest to define this building as a W-metric space (cf. [Reference Abramenko and Brown1, Chapter 5]), as follows.

Definition 2.4 The split spherical building $\Delta =\Delta _{\Phi }(\mathbb {F})$ associated to $G=G_{\Phi }(\mathbb {F})$ has chamber set $\Delta =G/B$ and Weyl distance function given by

$$ \begin{align*}\delta(gB,hB)=w\quad\text{if and only if}\quad g^{-1}h\in BwB. \end{align*} $$

Chambers $c,d\in \Delta $ are s-adjacent (with $s\in S$ ) if $\delta (c,d)=s$ , and are adjacent if they are s-adjacent for some $s\in S$ .

In particular, if $c=gB$ is a chamber of $\Delta $ , then by (2.2), the set of chambers $d\in \Delta $ with $\delta (c,d)=w$ is precisely

$$ \begin{align*}g\cdot\{x_{\beta_1}(a_1)\cdots x_{\beta_k}(a_k)wB\mid a_1,\ldots,a_k\in\mathbb{F}\}\quad\text{where}\ \Phi(w)=\{\beta_1,\ldots,\beta_k\}. \end{align*} $$

An automorphism of $\Delta $ is an adjacency preserving bijection $\theta :\Delta \to \Delta $ . Each automorphism $\theta $ of $\Delta $ induces an automorphism $\pi _{\theta }$ of the Coxeter diagram by $\delta (c,d)=s$ if and only if $\delta (\theta (c),\theta (d))=\pi _{\theta }(s)$ . We say that $\theta $ is a collineation (or type preserving) if $\pi _{\theta }=\mathrm {id}$ , and a duality if $\pi _{\theta }$ has order $2$ .

By [Reference Tits28, Corollaries 5.9 and 5.10] (and using [Reference Humphreys8, Reference Steinberg22]), every automorphism $\theta $ of $\Delta $ is of the form $\theta =g\circ \pi \circ \sigma $ , where $g\in G$ , $\pi =\pi _{\theta }$ is a Dynkin diagram automorphism, and $\sigma $ is a field automorphism (in the special characteristic case, $\pi $ is a Coxeter diagram automorphism). Note that the “diagonal automorphisms” are already built into G. By the Bruhat decomposition, each element $g\in G$ can be written as $g=uwb$ with $u\in U^+$ , $w\in W$ , and $b\in B$ , and so each automorphism of $\Delta $ can be written as $\theta =uwb\circ \pi \circ \sigma $ . If $\sigma $ is trivial, we say that $\theta $ is linear.

By a root elation, we shall mean an automorphism $\theta $ conjugate to $x_{\alpha }(a)$ for some $\alpha \in \Phi $ and $a\in \mathbb {F}^{\times }$ , and we call $\theta $ a long (respectively short) root elation if $\alpha $ is a long (respectively short) root. By a homology, we shall mean an automorphism $\theta $ conjugate to a nontrivial element $h_{\lambda }(c)$ with $\lambda \in P$ and $c\in \mathbb {F}^{\times }$ .

We often regard $\Delta $ as a simplicial complex in the standard way (cf. [Reference Abramenko and Brown1, Chapter 4]). Let us briefly describe this conversion in a group theoretic way in the split context. For subsets $J\subseteq S$ , let

$$ \begin{align*}P_{J}=\bigcup_{w\in W_{J}}BwB \end{align*} $$

be the standard parabolic subgroup of G of type J. For each nonempty $J\subseteq S$ , the set of “type J-simplices” of the building is the set of cosets $G/P_{S\backslash J}$ , and the simplicial complex structure is given by reverse containment of cosets. For example, in the simplicial complex language, the chamber B “contains” the simplices $P_{S\backslash J}$ for all nonempty $J\subseteq S$ , whereas on the level of cosets, it is, in fact, the parabolic subgroups $P_{S\backslash J}$ that contain the Borel subgroup B.

If $J=\{s\}$ is a singleton, we often write

(2.5) $$ \begin{align} W_s=W_{S\backslash J}\quad\text{and}\quad P_s=W_{S\backslash J} \end{align} $$

for the standard parabolic subgroups of W and G of type $S\backslash J$ . Moreover, if $s=s_i$ , we will often write $W_{s_i}=W_i$ and $P_{s_i}=P_i$ .

Let $\tau (x)\subseteq S$ denote the type of the simplex x of $\Delta $ . Thus, vertices are simplices x with $\tau (x)=\{s\}$ for some $s\in S$ , and chambers are the simplices x with $\tau (x)=S$ . A panel is a codimension $1$ simplex; that is, $\tau (x)=S\backslash \{s\}$ , for some $s\in S$ .

Note that we use the same symbol $\Delta $ for the building regarded as either a set of chambers with a Weyl distance function (as in Definition 2.4) or as a simplicial complex (as outlined above). No confusion will arise from this convention.

2.3 Opposition diagrams and admissible diagrams

Chambers $c,d\in \Delta $ are opposite one another if and only if $\delta (c,d)=w_0$ . If $x,y$ are simplices, with types $J,K$ , respectively, then x and y are opposite one another if and only if $K=\pi _0(J)$ and there exist opposite chambers $c,d$ with $x\subseteq c$ and $y\subseteq d$ . That is, simplices are opposite one another if they have opposite types, and are contained in opposite chambers. In terms of double cosets, chambers $c=gB$ and $d=hB$ are opposite if and only if $g^{-1}h\in Bw_0B$ , and simplices $x=gP_{S\backslash J}$ and $y=hP_{S\backslash K}$ are opposite if and only if $K=\pi _0(J)$ and $g^{-1}h\in P_{S\backslash J}w_0P_{S\backslash K}$ .

Let $\theta $ be an automorphism of $\Delta $ . Recall, from the introduction, that $\operatorname {{\mathrm {Opp}}}(\theta )$ denotes the set of all simplices x such that $x^{\theta }$ is opposite x. The type $\operatorname {{\mathrm {Typ}}}(\theta )$ of $\theta $ is the union of all subsets $J\subseteq S$ such that there exists a type J simplex mapped to an opposite simplex by $\theta $ . The opposition diagram of $\theta $ is the triple $(\Gamma ,\operatorname {{\mathrm {Typ}}}(\theta ),\pi _{\theta })$ .

Less formally, the opposition diagram of $\theta $ is depicted by drawing $\Gamma $ and encircling the nodes of $\operatorname {{\mathrm {Typ}}}(\theta )$ , where we encircle nodes in minimal subsets invariant under $\pi _0\circ \pi _{\theta }$ . We draw the diagram “bent” (in the standard way) if $\pi _0\circ \pi _{\theta }\neq \mathrm {id}$ . For example, consider the diagrams

Diagram $\mathsf {X}$ represents a collineation $\theta $ of an $\mathsf {E}_6$ building with $\operatorname {{\mathrm {Typ}}}(\theta )=\{1,2,6\}$ , and diagram $\mathsf {Y}$ represents a duality of an $\mathsf {E}_6$ building with $\operatorname {{\mathrm {Typ}}}(\theta )=\{1,6\}$ .

An automorphism $\theta $ is domestic if $\operatorname {{\mathrm {Opp}}}(\theta )$ contains no chamber (that is, $\theta $ maps no chamber to an opposite chamber). More generally, if $J\subseteq S$ , then $\theta $ is J-domestic if no type J simplex is mapped onto an opposite simplex by $\theta $ . To avoid trivialities, the definition of J-domesticity is restricted to subsets J with $\pi _0(J)=\pi _{\theta }(J)$ (for if J does not satisfy this, then $\theta $ is J-domestic for trivial reasons).

An automorphism $\theta $ is called capped if the following closure property holds: If there exist type $J_1$ and $J_2$ simplices in $\operatorname {{\mathrm {Opp}}}(\theta )$ , then there exists a type $J_1\cup J_2$ simplex in $\operatorname {{\mathrm {Opp}}}(\theta )$ . Equivalently, $\theta $ is capped if and only if there exists a type $\operatorname {{\mathrm {Typ}}}(\theta )$ simplex in $\operatorname {{\mathrm {Opp}}}(\theta )$ . By [Reference Parkinson and Van Maldeghem15, Theorem 1], every automorphism of a “large” spherical building of rank at least $3$ is capped, where a building is called large if it contains no Fano plane residues (for split buildings, this simply means $|\mathbb {F}|>2$ ).

In [Reference Parkinson and Van Maldeghem15, Reference Parkinson and Van Maldeghem16], we showed that the opposition diagrams of automorphisms of spherical buildings satisfy various restrictive properties, and we used these properties to determine a list of all possible opposition diagrams. We call the diagrams $(\Gamma ,J,\pi )$ in this list admissible Dynkin diagrams (more precisely, in [Reference Parkinson and Van Maldeghem15, Reference Parkinson and Van Maldeghem16], we considered Coxeter diagrams rather than Dynkin diagrams; however, the arguments are nearly identical).

The complete list of admissible Dynkin diagrams of exceptional type is given in Figure 1 (taken from [Reference Parkinson and Van Maldeghem15]). Each admissible diagram $(\Gamma ,J,\pi )$ of exceptional type is denoted by a symbol

$$ \begin{align*}{^t}\mathsf{X}_{n;i}\quad\text{or}\quad {^t}\mathsf{X}_{n;i}^{k}, \end{align*} $$

where:

1. (1) $\mathsf {X}\in \{\mathsf {E},\mathsf {F},\mathsf {G}\}$ is the type of $\Gamma $ , and n is the rank;

2. (2) $t\in \{1,2\}$ is the order of the graph automorphism $\pi _0\circ \pi $ (the “twisting”);

3. (3) i is the number of distinguished orbits contained in J; and

4. (4) k is an additional index occurring only for $\mathsf {F}_4$ and $\mathsf {G}_2$ in the case that a single node is encircled, in which case k is the type of this node.

In the case $t=1$ (that is, when $\pi _0\circ \pi =\mathrm {id}$ ), we usually omit the t from the notation, writing simply ${^1}\mathsf {X}_{n;i}=\mathsf {X}_{n;i}$ . For example, the diagrams $\mathsf {X}$ and $\mathsf {Y}$ given above are $\mathsf {X}={^2}\mathsf {E}_{6;2}$ and $\mathsf {Y}=\mathsf {E}_{6;2}$ . Similar notation is introduced in [Reference Parkinson and Van Maldeghem17] for the classical types (except with a different meaning for the index k).

In special characteristic, one often ignores the arrows on Dynkin diagrams, thus giving the additional admissible (Coxeter) diagrams in Figure 2.

Figure 2:

Additional admissible diagrams for special characteristic.

Summarizing the above discussion, we have the following from [Reference Parkinson and Van Maldeghem15, Reference Parkinson and Van Maldeghem16].

A striking feature of Theorem 2.5 is that there are very few possible opposition diagrams. We note that the analysis in [Reference Parkinson and Van Maldeghem15, Reference Parkinson and Van Maldeghem16] does not prove the converse to Theorem 2.5. That is, a priori, there may be redundancies in the list of admissible diagrams, in the sense that some admissible diagrams may not actually be the opposition diagram of any automorphism of any spherical building. It is a consequence of the work of this paper, combined with [Reference Parkinson and Van Maldeghem17], that no such redundancies exist—more precisely, Corollary 10 holds.

We call an admissible diagram empty if no nodes are encircled, and full if all nodes are encircled. We call an admissible diagram $(\Gamma ,J,\pi )$ type preserving if $\pi =\mathrm {id}$ . By inspection of the list in Figure 1, we note that for each Dynkin diagram $\Gamma $ , the type preserving diagram $(\Gamma ,\wp ,\mathrm {id})$ (with $\wp $ the polar type) is admissible. This diagram is called the polar diagram.

The polar–copolar diagram is the type preserving admissible diagram with $J=\wp \cup \wp ^*$ , where $\wp ^*$ is the polar type of the type $S\backslash \wp $ residue (we call $\wp ^*$ the copolar type; again, by inspection of Figure 1, the triple $(\Gamma ,\wp \cup \wp ^*,\mathrm {id})$ is always an admissible diagram). See Appendix A for the list of polar and copolar types. Alternatively, the polar–copolar diagrams of exceptional type are characterized as the type preserving diagrams in which exactly two orbits of nodes are encircled. Specifically, the polar diagrams of exceptional type are ${^2}\mathsf {E}_{6;1}$ , $\mathsf {E}_{7;1}$ , $\mathsf {E}_{8;1}$ , $\mathsf {F}_{4;1}^1$ , and $\mathsf {G}_{2;1}^2$ , and the polar–copolar diagrams are ${^2}\mathsf {E}_{6;2}$ , $\mathsf {E}_{7;2}$ , $\mathsf {E}_{8;2}$ , $\mathsf {F}_{4;2}$ , and $\mathsf {G}_{2;2}$ .

Every duality of a thick $\mathsf {G}_2$ (respectively $\mathsf {F}_4$ ) building has opposition diagram ${^2}\mathsf {G}_{2;1}$ (respectively ${^2}\mathsf {F}_{4;2}$ ), and no such dualities are domestic (see [Reference Parkinson, Temmermans and Van Maldeghem14, Theorem 2.7] and [Reference Parkinson and Van Maldeghem15, Lemma 4.1]). We note that dualities of split $\mathsf {F}_4$ and $\mathsf {G}_2$ buildings only exist for perfect fields of special characteristic.

2.4 Basic techniques

It is generally rather difficult to prove that a given automorphism is domestic, and more challenging to compute its opposition diagram. In this section, we describe some of the techniques that we will use in this paper.

The displacement of an automorphism $\theta $ is

$$ \begin{align*}\operatorname{{\mathrm{disp}}}(\theta)=\max\{\ell(\delta(c,c^{\theta}))\mid c\in\Delta\}. \end{align*} $$

Thus, $\theta $ is domestic if and only if $\operatorname {{\mathrm {disp}}}(\theta )<\ell (w_0)$ . Moreover, by [Reference Parkinson and Van Maldeghem16, Corollary 2.29], we have, with $J=\operatorname {{\mathrm {Typ}}}(\theta )$ ,

$$ \begin{align*}\operatorname{{\mathrm{disp}}}(\theta)=\begin{cases} \ell(w_{S\backslash J}w_0)&\text{if}\ \theta\ \text{is capped}\\ \ell(w_{S\backslash J}w_0)-1&\text{if}\ \theta\ \text{is uncapped.} \end{cases} \end{align*} $$

Uncapped automorphisms will not play a significant role here, so assume that $\theta $ is capped. By the above comments, and the classification of admissible diagrams, the list of possible displacements of $\theta $ is very restricted. For example, for a capped automorphism of an $\mathsf {E}_7$ building, the displacements for the nonfull nonempty opposition diagrams are:

$$ \begin{align*} 33\text{ for}\ \mathsf{E}_{7;1},\quad 50\text{ for}\ \mathsf{E}_{7;2},\quad 51\text{ for}\ \mathsf{E}_{7;3},\quad 60\text{ for}\ \mathsf{E}_{7;4}. \end{align*} $$

Thus, for example, to show that a capped automorphism $\theta $ of an $\mathsf {E}_7$ building:

1. (1) is not domestic, it is sufficient to show that $\operatorname {{\mathrm {disp}}}(\theta )>60$ ;

2. (2) has opposition diagram $\mathsf {E}_{7;3}$ , it is sufficient to show that $\operatorname {{\mathrm {disp}}}(\theta )<60$ and that there is a type $\{1,6,7\}$ simplex mapped to an opposite (in fact, from the classification of diagrams, it would be sufficient to show that there is a type $7$ vertex mapped to an opposite).

Such arguments will be used on multiple occasions throughout the paper.

The following lemma is useful to compute displacement, and Corollary 2.7 is useful to prove J-domesticity.

Lemma 2.6 Let $\theta $ be an automorphism of a thick spherical building $\Delta $ , and let $N=\operatorname {{\mathrm {disp}}}(\theta )$ . Let c be any chamber. Suppose that either:

1. (1) each panel of $\Delta $ has at least four chambers, or

2. (2) $\theta $ is an involution, or

3. (3) $\theta $ induces opposition and $N=\ell (w_0)$ .

Then, $\theta $ is necessarily capped, and there exists a chamber d with $\delta (c,d)=w_0$ and $\ell (\delta (d,d^{\theta }))=N$ . In particular,

$$ \begin{align*}\operatorname{{\mathrm{disp}}}(\theta)=\max\{\ell(\delta(d,d^{\theta}))\mid d\in \Delta\text{ with}\ \delta(c,d)=w_0\}. \end{align*} $$

The following proposition gives a useful technique for proving domesticity. We refer to this technique as the “standard technique.”

Proposition 2.8 Let $\Delta =\Delta _{\Phi }(\mathbb {F})$ be split, and let $\theta \in G=G_{\Phi }(\mathbb {F})$ . Suppose that the hypothesis of Lemma 2.6 is satisfied for the automorphism $\theta $ . If there is $w_1\in W$ with

$$ \begin{align*}w_1^{-1}w_0^{-1}u^{-1}\theta uw_0w_1\in B\quad\text{for all}\ u\in U^+, \end{align*} $$

then $ \operatorname {{\mathrm {disp}}}(\theta )\leq 2\ell (w_1)-1$ . Thus, in particular, if $\ell (w_1)\leq \ell (w_0)/2$ , then $\theta $ is domestic.

Proof Each chamber $gB$ of the building $G/B$ can be written uniquely as $uwB$ for some $w\in W$ and some $u\in \langle U_{\alpha }\mid \alpha \in \Phi (w)\rangle $ . Then, $\delta (gB,\theta gB)$ is the unique element $v\in W$ such that $w^{-1}u^{-1}\theta uw\in BvB$ . If the hypothesis of Lemma 2.6 is satisfied, then the displacement of $\theta $ is achieved for some chamber $gB$ opposite the base chamber B. These chambers are of the form $uw_0B$ with $u\in U^+$ . By the hypothesis, we have

$$ \begin{align*}w_0^{-1}u^{-1}\theta uw_0\in w_1Bw_1^{-1}\subseteq Bw_1B\cdot Bw_1^{-1}B. \end{align*} $$

In particular, if $w=\delta (uw_0B,\theta uw_0B)$ , then $BwB\subseteq Bw_1B\cdot Bw_1^{-1}B$ . Thus,

(2.6) $$ \begin{align} \operatorname{{\mathrm{disp}}}(\theta)\leq\max\{\ell(w)\mid BwB\subseteq Bw_1B\cdot Bw_1^{-1}B\}. \end{align} $$

Writing $w_1=w_2s$ with $\ell (w_2s)=\ell (w_2)+1$ , by (2.3), we have

$$ \begin{align*}Bw_1B\cdot Bw_1^{-1}B=(Bw_2sB\cdot Bw_2^{-1}B)\cup (Bw_2B\cdot Bw_2^{-1}B),\end{align*} $$

and it follows from (2.3) that $ \operatorname {{\mathrm {disp}}}(\theta )\leq 2\ell (w_1)-1$ .▪

In the case that $\theta \in U^+$ , the following lemma is helpful in finding an element $w_1\in W$ as in Proposition 2.8. If $A\subseteq \Phi $ , let $ A_{\geq }=\{\beta \in \Phi \mid \beta \geq \alpha \text { for some}\ \alpha \in A\}, $ where $\alpha \leq \beta $ if and only if $\beta -\alpha $ is a nonnegative linear combination of positive roots. Let $\pi _0$ be the automorphism of $\Phi $ given by $\pi _0(\alpha )=-w_0\alpha $ (thus, $\pi _0$ is the automorphism induced by the opposition diagram automorphism).

Proof Let $F=\langle U_{\alpha }\mid \alpha \in A_{\geq }\rangle $ . If $\beta \in \Phi ^+$ and $\alpha \in A_{\geq }$ , then by commutator relations, we have, for all $a\in \mathbb {F}$ , $ x_{\beta }(a)^{-1}U_{\alpha }x_{\beta }(a)\subseteq U_{\alpha }U_{\alpha +\beta }U_{2\alpha +\beta }\cdots \subseteq F$ (with the convention that $U_{\gamma }=\{1\}$ if $\gamma \notin \Phi $ ). It follows that $x_{\beta }(a)^{-1}Fx_{\beta }(a)\subseteq F$ . Thus, by induction, $u^{-1}Fu\subseteq F$ for all $u\in U$ , and because $\theta \in F$ , we have $u^{-1}\theta u\in F$ as required. Thus, if $w_1\in W$ is such that $\pi _0(A_{\geq })\subseteq \Phi (w_1)$ , then

$$ \begin{align*}w_0^{-1}u^{-1}\theta u w_0\in \langle U_{-\alpha}\mid \alpha\in \pi_0(A_{\geq})\rangle =w_1\langle U_{-w_1^{-1}\alpha}\mid \alpha\in \pi_0(A_{\geq})\rangle w_1^{-1}, \end{align*} $$

which is contained in $w_1Bw_1^{-1}$ , and hence the result.▪

2.5 Parapolar spaces and Lie incidence geometries

At certain points of this paper (in particular, in Section 6), we will work with Lie incidence geometries $\mathsf {X}_{n,J}(\mathbb {F})$ , for example, the long root geometries $\mathsf {E}_{6,2}(\mathbb {F})$ , $\mathsf {E}_{7,1}(\mathbb {F})$ , $\mathsf {E}_{8,8}(\mathbb {F})$ , and $\mathsf {F}_{4,1}(\mathbb {F})$ , and the geometries $\mathsf {E}_{6,1}(\mathbb {F})$ , $\mathsf {E}_{7,7}(\mathbb {F})$ , and $\mathsf {F}_{4,4}(\mathbb {F})$ . We note the similarity of notation with that used for opposition diagrams; however, no confusion should arise.

In general, the Lie incidence geometry $\mathsf {X}_{n,J}(\mathbb {F})$ , with $J\subseteq S$ , is defined from the building $\Delta =\Delta _\Phi (\mathbb {F})$ , with $\Phi $ the root system of type $\mathsf {X}_n$ , as the point-line geometry with point set, the set of simplices (or flags) of type J of $\Delta $ , and a typical line is the set of flags of type J incident with a flag of type $S\backslash \{j\}$ for some $j\in J$ . If $J=\wp $ , then we call the geometry $\mathsf {X}_{n,J}(\mathbb {F})$ a long root geometry.

The geometries listed in the first paragraph of this subsection are all examples of parapolar spaces (see [Reference Shult21, Chapter 13] for the basic terminology and definition). In particular, these point-line geometries $\mathscr {G}=(\mathcal {P},\mathcal {L})$ contain symplecta (or symps for short), being convex subsets that are nondegenerate polar spaces of rank at least $2$ . If all symplecta have the same rank $r\geq 2$ , then $\mathscr {G}$ is said to have symplectic rank r. Recall that in any incidence geometry, $x^{\perp }$ denotes the set of all points collinear to the point x. In addition, each symp $\xi $ is, by convexity, determined by any pair $\{x,y\}$ of noncollinear points, and we denote $\xi =\xi (x,y)$ . If $\mathsf {X}_n=\mathsf {E}_6,\mathsf {E}_7,\mathsf {E}_8,\mathsf {F}_4$ , then $\mathsf {X}_{n,\wp }(\mathbb {F})$ has symplectic rank $4,5,7,3$ , respectively. Moreover, the symps precisely correspond to the residues of vertices of type $\wp ^*$ .

In the parapolar spaces $\mathsf {E}_{6,2}(\mathbb {F})$ , $\mathsf {E}_{7,1}(\mathbb {F})$ , $\mathsf {E}_{8,8}(\mathbb {F})$ , $\mathsf {F}_{4,1}(\mathbb {F})$ , and $\mathsf {F}_{4,4}(\mathbb {F})$ , there are precisely five possible “distances” between two points $x,y$ . Either (1) $x=y$ , (2) x and y are collinear, (3) $\{x,y\}$ lies in a symplecton, (4) $|x^{\perp }\cap y^{\perp }|=1$ , or (5) $|x^{\perp }\cap y^{\perp }|=0$ . In case (3), we say that $x,y$ are symplectic (or at symplectic distance), and we denote by $x^{\perp \!\!\!\perp }$ the set of points at symplectic distance from x. In case (4), we say that $x,y$ are special (or a special pair, or at special distance), and we denote by $x^{\Join }$ the set of points at special distance from x. Finally, in case (5), the points $x,y$ are opposite each other (in the building theoretic sense).

The parapolar spaces $\mathsf {F}_{4,1}(\mathbb {F})$ and $\mathsf {F}_{4,4}(\mathbb {F})$ are also called metasymplectic spaces. In the parapolar spaces $\mathsf {E}_{6,1}(\mathbb {F})$ and $\mathsf {E}_{7,7}(\mathbb {F})$ , there are no special pairs, and so in these spaces, there are only four possible distances between two points (these spaces are called strong parapolar spaces).

An imaginary line of a long root geometry $\mathscr {G}$ is a set of the form $x^T\cup \{y\}$ , where $x,y$ are opposite points, and T is the full collineation group of central elations with center y. Note that if $\mathscr {G}$ is a polar space of rank at least $3$ , or if $\mathscr {G}$ is a Moufang quadrangle, then $x^T\cup \{y\}=(\{x,y\}^{\perp })^{\perp }$ for opposite points $x,y$ .

Let us briefly describe the long root geometry parapolar spaces in more algebraic terms. Let $\mathsf {X}_n=\mathsf {E}_n$ ( $n=6,7,8$ ) or $\mathsf {F}_4$ , and let $p\in S$ be the polar node. Recall the notation (2.5). Let $M_p$ denote the set of minimal length coset representatives of cosets in $W/W_p$ , and let $R_p$ denote the set of minimal length representatives for the double cosets in $W_p\backslash W/W_p$ . The points of the long root geometry $\mathsf {X}_{n,p}(\mathbb {F})$ are the cosets in $G/P_p$ . We have $ G=\bigsqcup _{w\in R_p}P_pwP_p, $ and the Weyl-distance $\delta (g_1P_p,g_2P_p)$ between points $g_1P_p$ and $g_2P_p$ is defined to be the unique element $\delta (g_1P_p,g_2P_p)=w\in R_p$ with $g_1^{-1}g_2\in P_pwP_p$ .

In each case, $R_p$ contains precisely five elements $e,s_p,w_1,w_2,s_{\varphi }$ , arranged in increasing length, corresponding to the five possible distances between points. Explicitly, the points $x=g_1P_p$ and $y=g_2P_p$ are collinear (respectively symplectic, at special distance, opposite) if $\delta (x,y)=s_p$ (respectively $w_1$ , $w_2$ , $s_{\varphi }$ ). Similar remarks hold for the metasymplectic space $\mathsf {F}_{4,4}(\mathbb {F})$ , and also the strong parapolar spaces $\mathsf {E}_{6,1}(\mathbb {F})$ and $\mathsf {E}_{7,7}(\mathbb {F})$ (where only four distances are possible, and hence $w_2$ is omitted).

3.1 Long root elations

To prove Theorem 1, we must first show that long root elations have polar opposition diagram, and conversely that every automorphism with polar opposition diagram is necessarily a long root elation. We prove the first statement in Theorem 3.1 in a more general context of Moufang spherical buildings, and the second statement is proved in Theorem 3.4.

Let $\Delta $ be an irreducible Moufang spherical building. Recall (for example, from [Reference Timmesfeld27]) that if $\Delta $ is not a generalized octagon, then one can associate a crystallographic (not necessarily reduced) root system $\Phi $ to $\Delta $ in such a way that the root subgroup $U_{\varphi }$ , with $\varphi $ the highest root of $\Phi $ , is contained in the center of the positive root subgroup $U^+=\langle U_{\alpha }\mid \alpha \in \Phi ^+\rangle $ . In the case that $\Phi $ has only one root length, we call all roots long. Let $\wp \subseteq S$ denote the polar type of the Dynkin diagram of $\Phi $ . If $\Delta $ is a generalized octagon, then by [Reference Tits29], one may associate a noncrystallographic (and nonreduced) root system $\Phi $ , and again, there is a “highest root” $\varphi $ such that $U_{\varphi }$ is contained in the center of $U^+$ (the root $\varphi $ is $\alpha _4^{\prime }$ in [Reference Tits29], and the polar type corresponds to “points” of the octagon). In all cases, let W be the Weyl group of $\Phi $ (so W is a dihedral group of order $16$ in the octagon case).

Theorem 3.1 Let $\Delta $ be an irreducible Moufang spherical building with associated root system $\Phi $ and Weyl group W as above. Let $\alpha \in \Phi $ be a long root, and let $\theta \in U_{\alpha }\backslash \{1\}$ . Then, the collineation $\theta $ of $\Delta =G/B$ has polar opposition diagram. Moreover,

(3.1) $$ \begin{align} \{\delta(c,c^{\theta})\mid c\in \Delta\}=\{1\}\cup\{s_{\alpha}\mid \alpha\in W\varphi\}. \end{align} $$

Proof Because $\alpha $ is a long root, it is in the W-orbit of the highest root, and it follows from standard root group datum (RGD) properties (see [Reference Abramenko and Brown1, Section 7.8]) that $\theta $ is conjugate to an element of $U_{\varphi }\backslash \{1\}$ . Thus, after conjugation, we may assume that $\theta $ is central in $U^+$ . A chamber $gB$ is opposite its image $\theta gB$ if and only if $\delta (gB,\theta gB)=w_0$ , if and only if $g^{-1}\theta g\in Bw_0B$ . By the Bruhat decomposition, each chamber $gB$ can be written as $ gB=uwB$ for some $u\in U^+$ and $w\in W$ . Because $\theta $ is central in $U^+$ , we have $ w^{-1}u^{-1}\theta uw=w^{-1}\theta w\in U_{w^{-1}\varphi } $ . Thus, if $w^{-1}\varphi \in \Phi ^+$ , we have $\delta (gB,\theta gB)=1$ , and if $w^{-1}\varphi \in -\Phi ^+$ , then $w^{-1}\theta w\in U_{w^{-1}\varphi }\subseteq Bs_{w^{-1}\varphi }B$ , and so $\delta (gB,\theta gB)=s_{w^{-1}\varphi }$ . Equation (3.1) follows.

Because $\theta $ is a nontrivial collineation, there is some simplex mapped onto an opposite simplex. Let $J\subseteq S$ be the type of such a simplex x, and write $J^{\prime }=S\backslash J$ . Thus, for each chamber $c\in \Delta $ containing x, we have $\delta (c,c^{\theta })\in W_{J^{\prime }}w_0W_{J^{\prime }}$ . Because J, and hence also $J^{\prime }$ , are stable under opposition, we have $W_{J^{\prime }}w_0W_{J^{\prime }}=w_0W_{J^{\prime }}$ . It follows from (3.1) that there is a root $\alpha \in W\varphi $ with $s_{\alpha }=w_0w$ for some $w\in W_{J^{\prime }}$ . Because $W_{J^{\prime }}$ is a proper parabolic subgroup of W, each $w\in W_{J^{\prime }}$ maps the highest root $\varphi $ to a positive root, and because $w_0$ maps all positive roots to negative roots, we have $s_{\alpha }\varphi \in -\Phi ^+$ . Because $\varphi $ is the highest root of $\Phi $ , this forces $\alpha =\pm \varphi $ , and so $w_0^{-1}s_{\varphi }\in W_{J^{\prime }}$ . For all $t\in S$ , we have

$$ \begin{align*}w_0^{-1}s_{\varphi}(\alpha_t)=-\alpha_{\pi_0(t)}+\langle\alpha_t,\varphi^{\vee}\rangle\varphi \end{align*} $$

(where $\pi _0(t)=w_0tw_0^{-1}$ ), and because $w(\Phi _{J^{\prime }})\subseteq \Phi _{J^{\prime }}$ for all $w\in W_{J^{\prime }}$ , we deduce that $\langle \alpha _t,\varphi ^{\vee }\rangle =0$ for all $t\in {J^{\prime }}=S\backslash J$ . It follows that $J=\wp $ .▪

We now prove Corollary 2, which is restated below for convenience.

We now prove the converse to Theorem 3.1 for split buildings of types $\mathsf {E}_n$ and $\mathsf {F}_4$ . Let $\Phi $ be a root system of type $\mathsf {E}_n$ for $n=6,7,8$ , or of type $\mathsf {F}_4$ . Let $\varphi $ be the highest root. Let $i_0$ be the polar node. Let $\Phi _1$ be the polar subsystem, generated by the simple roots $\{\alpha _i\mid i\neq i_0\}$ , and let $W_1$ be the parabolic subgroup generated by $\{s_i\mid i\neq i_0\}$ . Let $w_0$ be the longest element, and let $w_1$ be the longest element of $W_1$ . Let $j_0$ be the unique node joined to the polar node in the Dynkin diagram. Write $\pi =\alpha _{i_0}$ and $\pi ^{\prime }=\alpha _{j_0}$ . Explicitly, $(\pi ,\pi ^{\prime })=(\alpha _2,\alpha _4)$ , $(\alpha _1,\alpha _3)$ , $(\alpha _8,\alpha _7)$ , and $(\alpha _1,\alpha _2)$ for types $\mathsf {E}_6$ , $\mathsf {E}_7$ , $\mathsf {E}_8$ , and $\mathsf {F}_4$ , respectively. Let $\omega =\omega _{i_0}$ be the fundamental coweight corresponding to the polar node (thus $\langle \omega ,\alpha _{i}\rangle = \delta _{i,i_0}$ ). We note the following facts:

1. (1) $\omega =\varphi ^{\vee }$ (because $\langle \varphi ^{\vee },\alpha _{i}\rangle =\delta _{i,i_0}$ ).

2. (2) $\varphi $ is the unique root whose coefficient of $\pi $ is $2$ (because $2=\langle \varphi ,\varphi ^{\vee }\rangle =\langle \varphi ,\omega \rangle $ ).

3. (3) $\varphi -\alpha _i\in \Phi $ if and only if $i=i_0$ (by (2)).

4. (4) The elements $\varphi -\pi $ and $\varphi -\pi -\pi ^{\prime }$ are roots, but $\varphi -\pi ^{\prime }$ is not a root (by (2)).

5. (5) $s_{\varphi }=w_1w_0$ (as both have inversion set $\Phi ^+\backslash \Phi _1=\{\alpha \in \Phi ^+\mid \langle \alpha ,\omega \rangle \in \{1,2\}\}$ ).

Proof By the classification of admissible diagrams, if $\operatorname {{\mathrm {Typ}}}(\theta )=\wp $ , then $\theta $ is necessarily type preserving, and capped. Thus, $\operatorname {{\mathrm {disp}}}(\theta )=\ell (s_{\varphi })$ , and because $\theta $ is capped, $\ell (\delta (gB,\theta gB))=\ell (s_{\varphi })$ if and only if $\delta (gB,\theta gB)=s_{\varphi }$ . Moreover, after replacing $\theta $ by a conjugate, we may assume that the base chamber B is mapped to Weyl distance $s_{\varphi }=w_1w_0$ . Because the stabilizer of B is transitive on each w-sphere centered at B, we may assume that B is mapped to the chamber $x_{\varphi }(1)s_{\varphi }B$ . By the folding relation, we have $x_{\varphi }(1)s_{\varphi }B=x_{-\varphi }(1)B$ . The condition $\theta (B)=x_{-\varphi }(1)B$ gives

$$ \begin{align*}\theta=x_{-\varphi}(1)uh\sigma\quad\text{for some}\ u\in U^+, h\in H,\ \text{and}\ \sigma\in\mathrm{Aut}(\mathbb{F}). \end{align*} $$

We will now determine $u,h$ , and $\sigma $ . The primary strategy is to show that if these elements do not take certain particular forms, then one can find elements $g\in G$ such that $g^{-1}\theta g\in BwB\sigma $ with $\ell (w)>\ell (w_1w_0)$ , which contradicts the fact that $\operatorname {{\mathrm {Typ}}}(\theta )=\wp $ . A useful observation is that if $w=s_{\varphi }v$ with $v\in W_1$ , then $\ell (w)=\ell (s_{\varphi })+\ell (v)$ (because $s_{\varphi }=w_1w_0=w_0w_1$ ). We now proceed with the analysis.

Claim 1: We have $u\in U_{\Phi \backslash \Phi _1}^+$ . Write $u=u_1u_2$ with $u_1\in U_{\Phi _1}^+$ and $u_2\in U_{\Phi \backslash \Phi _1}^+$ . Then,

$$ \begin{align*}w_{1}^{-1}\theta w_{1}= x_{-\varphi}(\pm 1)u_1^-u_2^{\prime}h^{\prime}\sigma\quad\text{with}\ u_1^-=w_{1}^{-1}u_1w_{1}\in U_{\Phi_1}^-, u_2^{\prime}\in U_{\Phi\backslash\Phi_1}^+, h^{\prime}\in H. \end{align*} $$

Because $u_1^-\in BW_{1}B$ , we have $u_1^-\in BvB$ for some $v\in W_{1}$ . But because $\ell (s_{\varphi }v)=\ell (s_{\varphi })+\ell (v)$ , we have

$$ \begin{align*}w_{1}^{-1}\theta w_{1}\in Bs_{\varphi}B\cdot BvB\sigma=Bs_{\varphi}vB\sigma, \end{align*} $$

and so $v=1$ (as $\operatorname {{\mathrm {disp}}}(\theta )=\ell (s_{\varphi })$ ). So $u_1^-\in B\cap U_{\Phi _1}^-$ , and so $u_1^-=1$ ; hence, $u_1=1$ .

Claim 2: We have $h=h_{\omega }(c)$ for some $c\in \mathbb {F}^{\times }$ . Write $h=h_{\omega _1}(c_1)\cdots h_{\omega _n}(c_n)$ . Let $i\neq i_0$ . Then,

$$ \begin{align*}x_{-\alpha_i}(-1)\theta x_{-\alpha_i}(1)=x_{-\varphi}(1)x_{-\alpha_i}(-1)ux_{-\alpha_i}(c_i^{-1})h\sigma=x_{-\varphi}(1)x_{-\alpha_i}(c_i^{-1}-1)u^{\prime}h\sigma, \end{align*} $$

with $u^{\prime }\in U^+$ (here, we have used the fact, from Claim 1, that $x_{\alpha _i}(a)$ does not appear as a factor in u). Thus, if $c_i\neq 1$ , the folding relation gives

$$ \begin{align*}x_{-\alpha_i}(-1)\theta x_{-\alpha_i}(1)\in Bs_{\varphi}B\cdot Bs_iB\sigma=Bs_{\varphi}s_iB\sigma, \end{align*} $$

a contradiction as before. Hence, $c_i=1$ for all $i\neq i_0$ , and hence the claim.

Claim 3: We have $\sigma =\mathrm {id}$ . Suppose not. Let $i\neq i_0$ , and let $a\in \mathbb {F}$ with $a^{\sigma }\neq a$ . Then,

$$ \begin{align*}h_{\alpha_i^{\vee}}(a)^{-1}\theta h_{\alpha_i^{\vee}}(a)=x_{-\varphi}(1)uh_{\omega_i}(c)h_{\alpha_i^{\vee}}(a^{\sigma}a^{-1})\sigma. \end{align*} $$

Claim 2 now gives a contradiction.

Claim 4: We have $u\in U_{\varphi }$ . Suppose not, and write $\theta =x_{-\varphi }(1)uh_{\omega }(c)$ with $ u=x_{\beta _k}(a_k)\cdots x_{\beta _1}(a_1) $ in decreasing root height. By assumption, $\beta _1\neq \varphi $ . If $\beta _1\neq \pi $ (the polar simple root), then there exists $\alpha \in \Phi _1^+$ with $\beta _1-\alpha \in \Phi ^+$ . Then,

$$ \begin{align*}x_{-\alpha}(-b)ux_{-\alpha}(b)=u^{\prime}x_{\beta_1-\alpha}(\pm a_1b), \end{align*} $$

with $u^{\prime }$ a product of roots in $\Phi ^+\backslash \Phi _1$ of height at least $\mathrm {ht}(\beta _1-\alpha )$ . Continuing in this way, there exists an element $g\in U_{\Phi _1}^-$ with

$$ \begin{align*}g^{-1}ug=u^{\prime}x_{\pi+\pi^{\prime}}(b)x_{\pi}(a)\quad\text{with}\ a\neq 0\ \text{and}\ b\in \mathbb{F}, \end{align*} $$

where $u^{\prime }$ is a product of elements $x_{\beta }(\cdot )$ with $\beta \in \Phi ^+\backslash (\Phi _1\cup \{\pi ,\pi +\pi ^{\prime }\})$ . We have $g^{-1}x_{-\varphi }(1)g=x_{-\varphi }(1)$ (as $-\varphi +\alpha \notin \Phi $ for all $\alpha \in \Phi _1$ ), and $h_{\omega }(c)gh_{\omega }(c)^{-1}=g$ (as $\langle \omega ,\alpha \rangle =0$ for all $\alpha \in \Phi _1$ ), and hence

$$ \begin{align*} g^{-1}\theta g&=g^{-1}x_{-\varphi}(1)uh_{\omega}(c)g=x_{-\varphi}(1)g^{-1}ugh_{\omega}(c)=x_{-\varphi}(1)u^{\prime}x_{\pi+\pi^{\prime}}(b)x_{\pi}(a)h_{\omega}(c). \end{align*} $$

Let $d\in \mathbb {F}$ with $d\neq 0$ , and write $g_1=x_{-\pi -\pi ^{\prime }}(d)$ . Then,

$$ \begin{align*} Bg_1^{-1}g^{-1}\theta gg_1B&=Bx_{-\pi-\pi^{\prime}}(-d)x_{-\varphi}(1)u^{\prime}x_{\pi+\pi^{\prime}}(b)x_{\pi}(a)h_{\omega}(c)x_{-\pi-\pi^{\prime}}(d)B\\ &= Bx_{-\varphi}(1)x_{-\pi-\pi^{\prime}}(-d)u^{\prime}x_{\pi+\pi^{\prime}}(b)x_{\pi}(a)x_{-\pi-\pi^{\prime}}(dc^{-1})B\\ &= Bx_{-\varphi}(1)x_{-\pi-\pi^{\prime}}(-d)u^{\prime}x_{\pi+\pi^{\prime}}(b)x_{-\pi-\pi^{\prime}}(dc^{-1})x_{-\pi^{\prime}}(\pm adc^{-1})B, \end{align*} $$

where we have used the commutator relation

$$ \begin{align*}x_{\pi}(a)x_{-\pi-\pi^{\prime}}(dc^{-1})=x_{-\pi-\pi^{\prime}}(dc^{-1})x_{-\pi^{\prime}}(\pm adc^{-1})x_{\pi}(a).\end{align*} $$

Note that

$$ \begin{align*} u^{\prime}x_{\pi+\pi^{\prime}}(b)x_{-\pi-\pi^{\prime}}(dc^{-1})=x_{\pi+\pi^{\prime}}(b)x_{-\pi-\pi^{\prime}}(dc^{-1})u''\quad\text{where}\ u''\in U^+ \end{align*} $$

(this follows from the fact that $u'$ is a product of elements $x_{\beta }(\cdot )$ with $\beta \in \Phi ^+\backslash (\Phi _1\cup \{\pi ,\pi +\pi '\})$ , and for such $\beta $ , if $\beta -\pi -\pi '\in \Phi $ , then $\beta -\pi -\pi '\in \Phi ^+$ ). Therefore,

$$ \begin{align*} Bg_1^{-1}g^{-1}\theta gg_1B&=Bx_{-\varphi}(1)x_{-\pi-\pi'}(-d)x_{\pi+\pi'}(b)x_{-\pi-\pi'}(dc^{-1})u''x_{-\pi'}(\pm adc^{-1})B\\ &=Bs_{\varphi}x_{\varphi}(1)x_{-\pi-\pi'}(-d)x_{\pi+\pi'}(b)x_{-\pi-\pi'}(dc^{-1})u''x_{-\pi'}(\pm adc^{-1})B. \end{align*} $$

From the commutator relations, we have

$$ \begin{align*} &x_{\varphi}(1)x_{-\pi-\pi'}(-d)x_{\pi+\pi'}(b)x_{-\pi-\pi'}(dc^{-1})\\ &\quad x_{-\pi-\pi'}(-d)x_{\pi+\pi'}(b)x_{-\pi-\pi'}(dc^{-1})x_{\varphi-\pi-\pi'}(a')x_{\varphi}(b') \end{align*} $$

for some $a',b'\in \mathbb {F}$ , and hence

$$ \begin{align*} Bg_1^{-1}g^{-1}\theta gg_1B&= Bs_{\varphi}x_{-\pi-\pi'}(-d)x_{\pi+\pi'}(b)x_{-\pi-\pi'}(dc^{-1})u'''x_{-\pi'}(\pm adc^{-1})B \end{align*} $$

for some $u'''\in U^+$ .

There are now two cases to consider. If $b=0$ , then, because $s_{\varphi }(-\pi -\pi ')\in \Phi ^+$ , we have

$$ \begin{align*} Bg_1^{-1}g^{-1}\theta gg_1B&= Bs_{\varphi}x_{-\pi-\pi'}(-d+dc^{-1})u'''x_{-\pi'}(\pm adc^{-1})B\\ &=Bs_{\varphi}u'''x_{-\pi'}(\pm adc^{-1})B\\ &\subseteq Bs_{\varphi}B\cdot Bs_{\pi'}B\\ &=Bs_{\varphi}s_{\pi'}B, \end{align*} $$

a contradiction. If $b\neq 0$ , then, again using $s_{\varphi }(-\pi -\pi ')\in \Phi ^+$ , we have $Bs_{\varphi }x_{-\pi -\pi '}(-d)=Bs_{\varphi }x_{-\pi -\pi '}(dc^{-1})$ , and so

$$ \begin{align*} Bg_1^{-1}g^{-1}\theta gg_1B&= Bs_{\varphi}x_{-\pi-\pi'}(dc^{-1})x_{\pi+\pi'}(b)x_{-\pi-\pi'}(dc^{-1})u'''x_{-\pi'}(\pm adc^{-1})B. \end{align*} $$

Choosing $d=-b^{-1}c$ gives $Bs_{\varphi }x_{-\pi -\pi '}(dc^{-1})x_{\pi +\pi '}(b)x_{-\pi -\pi '}(dc^{-1})=Bs_{\varphi }s_{\pi +\pi '}$ , and hence

$$ \begin{align*} Bg_1^{-1}g^{-1}\theta gg_1B&= Bs_{\varphi}s_{\pi+\pi'}u'''x_{-\pi'}(\pm adc^{-1})B\subseteq Bs_{\varphi}s_{\pi+\pi'}B\cdot Bs_{\pi'}B. \end{align*} $$

We have $\Phi (s_{\varphi }s_{\pi +\pi '})=(\Phi (s_{\varphi })\backslash \{\varphi ,\varphi -\pi \})\cup \{\pi '\}$ , and it follows that $\ell (s_{\varphi }s_{\pi +\pi '})=\ell (s_{\varphi })-1$ and $\ell (s_{\varphi }s_{\pi +\pi '}s_{\pi '})=\ell (s_{\varphi })$ . Thus,

$$ \begin{align*} Bg_1^{-1}g^{-1}\theta gg_1B&=Bs_{\varphi}s_{\pi+\pi'}s_{\pi'}B. \end{align*} $$

Thus, the chamber $gg_1B$ is mapped to distance $\ell (s_{\varphi })$ by $\theta $ , contradicting the second sentence of the proof (as $s_{\varphi }s_{\pi +\pi '}s_{\pi '}\neq s_{\varphi }$ ). This completes the proof of the claim.

Claim 5: We have $\theta =x_{-\varphi }(1)x_{\varphi }(-(c-1)^2)h_{\omega }(c)$ for some $c\in \mathbb {F}^{\times }$ . From Claims 1–4, we have

$$ \begin{align*}\theta=x_{-\varphi}(1)x_{\varphi}(a)h_{\omega}(c)\quad\text{for some}\ a\in \mathbb{F}\ \text{and}\ c\in\mathbb{F}^{\times}. \end{align*} $$

We will now make a careful commutator relation calculation, using the consistent sign conventions:

$$ \begin{align*} x_{\varphi}(a)x_{-\varphi+\pi}(b)&=x_{-\varphi+\pi}(b)x_{\varphi}(a)x_{\pi}(ab),\\ x_{\pi}(a)x_{-\pi-\pi'}(b)&=x_{-\pi-\pi'}(b)x_{\pi}(a)x_{-\pi'}(-ab),\\ x_{\varphi}(a)x_{-\pi-\pi'}(b)&=x_{-\pi-\pi'}(b)x_{\varphi}(a)x_{\varphi-\pi-\pi'}(-ab),\\ x_{\varphi-\pi-\pi'}(a)x_{-\varphi+\pi}(b)&=x_{-\varphi+\pi}(b)x_{\varphi-\pi-\pi'}(a)x_{-\pi'}(ab). \end{align*} $$

Let $g=x_{-\varphi +\pi }(1)x_{-\pi -\pi '}(1)$ . Then,

$$ \begin{align*} Bg^{-1}\theta gB&=Bx_{-\pi-\pi'}(-1)x_{-\varphi+\pi}(-1)x_{-\varphi}(1)x_{\varphi}(a)h_{\omega}(c)x_{-\varphi+\pi}(1)x_{-\pi-\pi'}(1)B\\ &=Bx_{-\varphi}(1)x_{-\pi-\pi'}(-1)x_{-\varphi+\pi}(-1)x_{\varphi}(a)x_{-\varphi+\pi}(c^{-1})x_{-\pi-\pi'}(c^{-1})B\\ &= Bs_{\varphi}x_{\varphi}(1)x_{-\pi-\pi'}(-1)x_{-\varphi+\pi}(c^{-1}-1)x_{\varphi}(a)x_{\pi}(ac^{-1})x_{-\pi-\pi'}(c^{-1})B. \end{align*} $$

We have

$$ \begin{align*} x_{\varphi}(a)x_{\pi}(ac^{-1})x_{-\pi-\pi'}(c^{-1})B&=x_{\varphi}(a)x_{-\pi-\pi'}(c^{-1})x_{-\pi'}(-ac^{-2})B\\ &=x_{-\pi-\pi'}(c^{-1})x_{\varphi}(a)x_{\varphi-\pi-\pi'}(-ac^{-1})x_{-\pi'}(-ac^{-2})B\\ &=x_{-\pi-\pi'}(c^{-1})x_{\varphi}(a)x_{-\pi'}(-ac^{-2})B\\ &=x_{-\pi-\pi'}(c^{-1})x_{-\pi'}(-ac^{-2})B, \end{align*} $$

and hence

$$ \begin{align*} &Bg^{-1}\theta gB\\ &= Bs_{\varphi}x_{\varphi}(1)x_{-\pi-\pi'}(-1)x_{-\varphi+\pi}(c^{-1}-1)x_{-\pi-\pi'}(c^{-1})x_{-\pi'}(-ac^{-2})B\\ &= Bs_{\varphi}x_{\varphi}(1)x_{-\pi-\pi'}(c^{-1}-1)x_{-\varphi+\pi}(c^{-1}-1)x_{-\pi'}(-ac^{-2})B\\ &= Bs_{\varphi}x_{-\pi-\pi'}(c^{-1}-1)x_{\varphi}(1)x_{\varphi-\pi-\pi'}(1-c^{-1})x_{-\varphi+\pi}(c^{-1}-1)x_{-\pi'}(-ac^{-2})B\\ &= Bs_{\varphi}x_{\varphi}(1)x_{-\varphi+\pi}(c^{-1}-1)x_{\varphi-\pi-\pi'}(1-c^{-1})x_{-\pi'}(-(c^{-1}-1)^2)x_{-\pi'}(-ac^{-2})B\\ &= Bs_{\varphi}x_{\varphi}(1)x_{-\varphi+\pi}(c^{-1}-1)x_{-\pi'}(-ac^{-2}-(c^{-1}-1)^2)B\\ &= Bs_{\varphi}x_{-\varphi+\pi}(c^{-1}-1)x_{\varphi}(1)x_{\pi}(c^{-1}-1)x_{-\pi'}(-ac^{-2}-(c^{-1}-1)^2)B\\ &= Bs_{\varphi}x_{\varphi}(1)x_{\pi}(c^{-1}-1)x_{-\pi'}(-ac^{-2}-(c^{-1}-1)^2)B. \end{align*} $$

Thus, if $ac^{-2}+(c^{-1}-1)^2\neq 0$ , we have

$$ \begin{align*}Bg^{-1}\theta gB= Bs_{\varphi}B\cdot Bs_{\pi'}B=Bs_{\varphi}s_{\pi'}B, \end{align*} $$

a contradiction. Thus, $a=-(c-1)^2$ , completing the proof of the claim.

Claim 6: $\theta $ is a long root elation. Let $g=x_{\varphi }(-c)x_{-\varphi }(1)s_{\varphi }$ . Then, from Claim 5, we have

$$ \begin{align*} g\theta g^{-1}&=x_{\varphi}(-c)x_{-\varphi}(1)s_{\varphi}x_{-\varphi}(1)x_{\varphi}(-(c-1)^2)h_{\omega}(c)s_{\varphi}^{-1}x_{-\varphi}(-1)x_{\varphi}(c). \end{align*} $$

Using the relations $s_{\varphi }x_{\pm \varphi }(a)s_{\varphi }^{-1}=x_{\mp \varphi }(-a)$ and $s_{\varphi }h_{\omega }(c)s_{\varphi }^{-1}=h_{\omega }(c^{-1})$ , we have

$$ \begin{align*} g\theta g^{-1}&=x_{\varphi}(-c)x_{-\varphi}(1)x_{\varphi}(-1)x_{-\varphi}((c-1)^2)h_{\omega}(c^{-1})x_{-\varphi}(-1)x_{\varphi}(c)\\ &=x_{\varphi}(-c)x_{-\varphi}(1)x_{\varphi}(-1)x_{-\varphi}((c-1)^2)x_{-\varphi}(-c^2)x_{\varphi}(c^{-1})h_{\omega}(c^{-1})\\ &=x_{\varphi}(-c)[x_{-\varphi}(1)x_{\varphi}(-1)x_{-\varphi}(1)]x_{-\varphi}(-2c)x_{\varphi}(c^{-1})h_{\omega}(c^{-1}). \end{align*} $$

Now, $x_{-\varphi }(1)x_{\varphi }(-1)x_{-\varphi }(1)=s_{\varphi }(-1)=h_{\omega }(-1)s_{\varphi }$ , and so

$$ \begin{align*} g\theta g^{-1}&=x_{\varphi}(-c)h_{\omega}(-1)s_{\varphi}x_{-\varphi}(-2c)x_{\varphi}(c^{-1})h_{\omega}(c^{-1})\\ &=x_{\varphi}(-c)h_{\omega}(-1)x_{\varphi}(2c)x_{-\varphi}(-c^{-1})s_{\varphi}h_{\omega}(c^{-1})\\ &=x_{\varphi}(c)x_{-\varphi}(-c^{-1})s_{\varphi}h_{\omega}(-c^{-1}). \end{align*} $$

We have $x_{\varphi }(c)x_{-\varphi }(-c^{-1})=s_{\varphi }(c)x_{\varphi }(-c)=h_{\omega }(c)s_{\varphi }x_{\varphi }(-c)=x_{-\varphi }(c^{-1})h_{\omega }(c)s_{\varphi }$ , and because $s_{\varphi }^2=h_{\omega }(-1)$ , we have

$$ \begin{align*} g\theta g^{-1}&=x_{-\varphi}(c^{-1})h_{\omega}(c)h_{\omega}(-1)h_{\omega}(-c^{-1})=x_{-\varphi}(c^{-1}). \end{align*} $$

Thus, $\theta $ is conjugate to the long root elation $x_{-\varphi }(c^{-1})$ , completing the proof.▪

3.2 A geometric characterization of root elations for $\mathsf {E}_6$ and $\mathsf {E}_7$ buildings

Theorems 3.1 and 3.4 imply an interesting geometric characterization of root elations for types $\mathsf {E}_6$ and $\mathsf {E}_7$ (see Theorem 3.6 below). In the following lemma, and again in the following subsection, we make use of [Reference Humphreys9, Section 10.3, Lemma B], which says that if $\lambda \in P$ is dominant (that is, $\lambda \in \mathbb {Z}_{\geq 0}\omega _1+\cdots +\mathbb {Z}_{\geq 0}\omega _n$ ), and if $w\in W$ with $w\lambda =\lambda $ , then $w\in W_J$ , where $J=\{s\in S\mid s\lambda =\lambda \}$ .

3.3 Distances attained by long root elations

Here, we prove Corollary 3. Let $\Delta $ be an irreducible Moufang spherical building other than a generalized octagon, and recall (as in Section 3.1) that one may associate a crystallographic root system $\Phi $ to $\Delta $ . Let $\Phi _r$ denote the associated reduced root system (thus, $\Phi _r=\Phi $ if $\Phi $ is reduced, and $\Phi _r$ is the $\mathsf {C}_n$ subsystem consisting of the middle and long length roots in the nonreduced $\mathsf {BC}_n$ case). Consider the long root geometry $\mathscr {G}$ . Let $P=P_{S\backslash \wp }$ be the standard parabolic subgroup of G of type $S\backslash \wp $ , and let $W'=W_{S\backslash \wp }$ . The points of $\mathscr {G}$ are the cosets in $G/P$ , and we have $ G=\bigsqcup _{w\in R(\wp )}PwP, $ where $R(\wp )$ is the set of minimal length double coset representatives for $W'\backslash W/W'$ . The Weyl distance $\delta (g_1P,g_2P)$ between points $g_1P$ and $g_2P$ is defined to be the unique element $\delta (g_1P,g_2P)=w\in R(\wp )$ with $g_1^{-1}g_2\in PwP$ . Points $g_1P$ and $g_2P$ are (i) collinear if $\delta (g_1P,g_2P)=s$ for some $s\in \wp $ (thus, in type $\mathsf {A}$ , there are two “flavors” of collinearity), and (ii) opposite if $\delta (g_1P,g_2P)=w_{S\backslash \wp }w_0$ . Note that $w_{S\backslash \wp }w_0$ is the minimal length representative of $W'w_0W'$ , and that $w_{S\backslash \wp }w_0=s_{\varphi }$ (by comparing inversion sets).

Proof Let $W'=W_{S\backslash \wp }$ , and let $D(\theta )=\{\delta (gP,\theta gP) \mid gP\in G/P\}\subseteq R(\wp )$ be the set of distances realized by $\theta $ . From Theorem 3.1, we see that $D(\theta )$ consists precisely of the identity, along with the minimal length representatives of the double cosets $W's_{\alpha }W'$ with $\alpha $ a long root.

Consider the $\mathsf {A}_n$ case. We claim that

(3.2) $$ \begin{align} \Phi^+=\Phi_{S\backslash\wp}^+\sqcup(W'\cdot\alpha_1)\sqcup (W'\cdot\alpha_n)\sqcup\{\varphi\}. \end{align} $$

The result follows from this claim, because if $\alpha \in \Phi _{S\backslash \wp }^+$ , then $s_{\alpha }\in W'$ , if $\alpha \in W'\cdot \alpha _1$ , then $s_{\alpha }=ws_1w^{-1}$ for some $w\in W'$ , and so $W's_{\alpha }W'=W's_1W'$ , if $\alpha \in W'\cdot \alpha _n$ , then $W's_{\alpha }W'=W's_nW'$ , and if $\alpha =\varphi $ , then $W's_{\alpha }W'=W'w_{S\backslash \wp }w_0W'$ .

To prove (3.2), we first claim that

$$ \begin{align*} W'\cdot\alpha_1=\{\alpha\in\Phi^+\mid \langle\alpha,\omega_1\rangle=1\text{ and }\langle\alpha,\omega_n\rangle=0\}. \end{align*} $$

Denote the right-hand side by X. If $\alpha =w\alpha _1$ with $w\in W'$ , then $\langle \alpha ,\omega _1\rangle =\langle \alpha _1,w^{-1}\omega _1\rangle =\langle \alpha _1,\omega _1\rangle =1$ , because $w^{-1}\omega _1=\omega _1$ for all $w\in W'$ . It follows that $\alpha \in \Phi ^+$ , and similarly, we have $\langle \alpha ,\omega _n\rangle =0$ , and so $W'\cdot \alpha _1\subseteq X$ . By the orbit-stabilizer theorem, we have

$$ \begin{align*}|W'\cdot\alpha_1|=|W'|/|\mathrm{stab}_{W'}(\alpha_1)|=|W_{\mathsf{A}_{n-2}}|/|W_{\mathsf{A}_{n-3}}|=n-1, \end{align*} $$

where the stabilizer computation follows from the fact that if $w\in W'$ , then $w\alpha _1=\alpha _1$ if and only if $w\omega _2=\omega _2$ (as $\alpha _1=2\omega _1-\omega _2$ and $w\omega _1=\omega _1$ ), if and only if $w\in W_{S\backslash \{1,2,n\}}$ (by [Reference Humphreys9, Section 10.3, Lemma B]). Because $|X|=n-1$ , we have $W'\cdot \alpha _1=X$ , and hence the claim.

Dually, we have $W'\cdot \alpha _n=\{\alpha \in \Phi ^+\mid \langle \alpha ,\omega _1\rangle =0\text { and }\langle \alpha ,\omega _n\rangle =1\}$ . Because every positive root $\alpha $ either has $\langle \alpha ,\omega _1\rangle =\langle \alpha ,\omega _n\rangle =0$ (in which case $\alpha \in \Phi _{S\backslash \wp }^+$ ), or $\langle \alpha ,\omega _1\rangle =1$ and $\langle \alpha ,\omega _n\rangle =0$ (in which case $\alpha \in W'\cdot \alpha _1$ ), or the dual situation (with $\alpha \in W'\cdot \alpha _n$ ), or $\langle \alpha ,\omega _1\rangle =\langle \alpha ,\omega _n\rangle =1$ (in which case $\alpha =\varphi $ ), the claim (3.2) follows.

Consider the $\mathsf {B}_n$ case with $n\geq 3$ , and let $\Phi _L$ be the set of long roots. Let $Y=\{\alpha \in \Phi _L^+\mid \langle \alpha ,\omega _2\rangle =0\}$ and $X=\Phi _L^+\backslash (Y\cup \{\varphi \})$ . Thus, $\Phi _L^+=X\sqcup Y\sqcup \{\varphi \}$ . If $\alpha \in Y$ , then $W's_{\alpha }W'=W'$ . By inspection of the root system, we have

$$ \begin{align*}X=\{\alpha\in\Phi_L^+\mid \langle\alpha,\omega_2\rangle=1\text{ and }\langle\alpha,\omega_n\rangle\in\{0,2\}\}. \end{align*} $$

From this description, it is clear that $W'\cdot \alpha _2\subseteq X$ , and a similar orbit-stabilizer calculation as in the $\mathsf {A}_n$ case gives $X=W'\cdot \alpha _2$ . Hence the result in this case.

The $\mathsf {B}_2$ and $\mathsf {C}_n$ cases are immediate, as the polar node corresponds to a short root in these cases, and the $\mathsf {D}_n$ case is very similar to the $\mathsf {B}_n$ case.

Consider the cases $\mathsf {E}_n$ ( $n=6,7,8$ ) and $\mathsf {F}_4$ . Let $\wp =\{p\}$ , $Y=\{\alpha \in \Phi _L^+\mid \langle \alpha ,\omega _p\rangle =0\}$ , and $X=\Phi _L^+\backslash (Y\cup \{\varphi \})$ . Then, $s_{\alpha }\in W'$ for all $\alpha \in Y$ , and by inspection of the root systems, we have $ X=\{\alpha \in \Phi _L^+\mid \langle \alpha ,\omega _p\rangle =1\}, $ from which it follows that $W'\cdot \alpha _p\subseteq X$ . Then, $ |W'\cdot \alpha _p|=|W'|/|\mathrm {stab}_{W'}(\alpha _p)|, $ and we compute $|\mathrm {stab}_{W'}(\alpha _p)|=|W_{\mathsf {A}_2\times \mathsf {A}_2}|$ , $|W_{\mathsf {A}_5}|$ , $|W_{\mathsf {E}_6}|$ , $|W_{\mathsf {A}_2}|$ in the cases $\mathsf {E}_6$ , $\mathsf {E}_7$ , $\mathsf {E}_8$ , $\mathsf {F}_4$ (respectively). For example, in the $\mathsf {E}_6$ case, if $w\in W'$ , then one has $w\alpha _2=\alpha _2$ if and only if $w\omega _4=\omega _4$ (as $\alpha _2=2\omega _2-\omega _4$ and $w\omega _2=\omega _2$ for all $w\in W'$ ), if and only if $w\in W_{S\backslash \{2,4\}}$ . Thus, $|W'\cdot \alpha _p|=20,32,56,8$ in the cases $\mathsf {E}_6$ , $\mathsf {E}_7$ , $\mathsf {E}_8$ , $\mathsf {F}_4$ , and thus $W'\cdot \alpha _p=X$ in all cases.

The cases $\mathsf {G}_2$ and Ree–Tits octagons are elementary from the geometry of these generalized polygons, because the fixed elements of $\theta $ form a ball centered at a point with radius $3$ (for $\mathsf {G}_2$ ) or $4$ (for octagons) in the incidence graph.

Finally, we note that in all cases, $|D(\theta )|<|R(\wp )|$ , and so there exists at least one $w\in R(\wp )$ such that no point is mapped onto distance w by $\theta $ . For example, in type $\mathsf {A}_n$ , we have $|R(\wp )|=7$ and $|D(\theta )|=4$ , and in the cases $\mathsf {E}_n$ and $\mathsf {F}_4$ , we have $|R(\wp )|=5$ and $|D(\theta )|=3$ .▪

The following corollary stems from a question asked to us by Barbara Baumeister.

3.4 Short root elations

We now record the situation for short root elations of split buildings. In this case, there is some dependence on the characteristic of the underlying field. The proof for the $\mathsf {F}_4$ case is postponed to Section 7. For $i\leq n$ , let $\mathsf {B}_{n;i}^1$ (respectively $\mathsf {C}_{n;i}^1$ ) denote the admissible $\mathsf {B}_n$ (respectively $\mathsf {C}_n$ ) diagram $(\Gamma ,\{1,\ldots ,i\},\mathrm {id})$ .