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Lemma 5.126. If \( \left( {{\mathcal{C}}^{\prime },{\delta }^{\prime }}\right) \) is another building of type \( \left( {W, S}\right) \), the following are equivalent:\n\n(i) \( {\mathcal{C}}^{\prime } \) and \( {\mathcal{C}}_{ - } \) are isometric.\n\n(ii) There is a type-preserving simplicial isomorphism \( \Delta \l... | Proof. By Remark 5.90, an isometry between \( {\mathcal{C}}^{\prime } \) and \( {\mathcal{C}}_{ - } \) induces a type-preserving simplicial automorphism between \( \Delta \left( {\mathcal{C}}^{\prime }\right) \) and \( \Delta \left( {\mathcal{C}}_{ - }\right) \) . Conversely, a type-preserving automorphism between \( \... | Yes |
The building \( \Delta \left( V\right) \) admits an automorphism that is not type-preserving (and so has \( {\sigma }_{0} \) as its type-change map) if and only if the underlying field \( k \) admits an antiautomorphism. | The answer is classical and well known: \( P\left( V\right) \) admits a correlation if and only if the division rings \( k \) and \( {k}^{\text{op }} \) are isomorphic (or, in other words, if and only if \( k \) admits an antiautomorphism). In view of Corollary 5.127, we can restate this result as follows: The building... | Yes |
Lemma 5.139. Given \( \epsilon \in \{ + , - \}, C \in {\mathcal{C}}_{\epsilon }, D \in {\mathcal{C}}_{-\epsilon } \), and \( s \in S \), let \( w \mathrel{\text{:=}} \) \( {\delta }^{ * }\left( {C, D}\right) \) . Then we have:\n\n(1) \( {\delta }^{ * }\left( {{C}^{\prime }, D}\right) \in \{ w,{sw}\} \) for any \( {C}^{... | Proof. (1) follows from (Tw2) if \( l\left( {sw}\right) < l\left( w\right) \), so assume \( l\left( {sw}\right) > l\left( w\right) \) . Then, by (Tw3), there exists \( {C}^{\prime \prime } \in {\mathcal{C}}_{\epsilon } \) with \( {\delta }_{\epsilon }\left( {{C}^{\prime \prime }, C}\right) = s \) and \( {\delta }^{ * }... | Yes |
Lemma 5.140. Given \( \epsilon \in \{ + , - \}, C, D \in {\mathcal{C}}_{\epsilon } \), and \( E \in {\mathcal{C}}_{-\epsilon } \), let \( w \mathrel{\text{:=}} \) \( {\delta }^{ * }\left( {D, E}\right) \) . (1) If \( \Gamma \) is a gallery of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{n}}\right) \) from \( C \) ... | Proof. (1) follows immediately from Lemma 5.139(1) by an obvious induction on \( n \) . (2) The hypothesis can also be written as \( l\left( w\right) = l\left( {v}^{-1}\right) + l\left( {vw}\right) \), so there is a reduced decomposition \( w = {s}_{1}\cdots {s}_{n} \) such that some initial segment \( {s}_{1}\cdots {s... | Yes |
(1) If \( C, D \in {\mathcal{C}}_{\epsilon } \) and \( {C}^{\prime } \in {\mathcal{C}}_{-\epsilon } \) satisfy \( {\delta }_{\epsilon }\left( {D, C}\right) = {\delta }^{ * }\left( {D,{C}^{\prime }}\right) \), then \( C \) op \( {C}^{\prime } \) . | Proof. (1) If \( w \mathrel{\text{:=}} {\delta }^{ * }\left( {D,{C}^{\prime }}\right) \), then the hypothesis says that \( {\delta }_{\epsilon }\left( {C, D}\right) = {w}^{-1} \) . Now apply Lemma 5.140(2) to get \( {\delta }^{ * }\left( {C,{C}^{\prime }}\right) = 1 \) . | Yes |
Lemma 5.143. For any \( C \in {\mathcal{C}}_{\epsilon } \) and any \( D \in {\mathcal{C}}_{-\epsilon },{\delta }^{ * }\left( {C, D}\right) \) is the unique element of minimal length in the set \( W\left( {C, D}\right) \subseteq W \) defined by\n\n\[ \nW\left( {C, D}\right) \mathrel{\text{:=}} \left\{ {{\delta }_{\epsil... | Proof. Set \( w \mathrel{\text{:=}} {\delta }^{ * }\left( {C, D}\right) \) . Then we have \( w \in W\left( {C, D}\right) \) by Corollary 5.141(1). Now consider any \( {D}^{\prime } \in {\mathcal{C}}_{\epsilon } \), and set \( v \mathrel{\text{:=}} {\delta }_{\epsilon }\left( {{D}^{\prime }, C}\right) \) . By Lemma 5.14... | Yes |
Lemma 5.148. If \( \mathcal{R} \) is a residue of \( {\mathcal{C}}_{\epsilon } \) of type \( J \) and \( \mathcal{S} \) is a residue of \( {\mathcal{C}}_{-\epsilon } \) of type \( K \), then \( {\delta }^{ * }\left( {\mathcal{R},\mathcal{S}}\right) = {W}_{J}{\delta }^{ * }\left( {C, D}\right) {W}_{K} \) for any \( C \i... | Proof. This follows immediately from (Tw3) and Lemma 5.139(1) (and their \ | No |
Lemma 5.149. If \( \mathcal{R} \) is a residue of \( {\mathcal{C}}_{\epsilon } \) of spherical type and \( D \) is a chamber in \( {\mathcal{C}}_{-\epsilon } \), then there is a unique \( {C}_{1} \in \mathcal{R} \) such that \( {\delta }^{ * }\left( {{C}_{1}, D}\right) \) is of maximal length in \( {\delta }^{ * }\left... | Proof. By Lemma 5.148, \( {\delta }^{ * }\left( {\mathcal{R}, D}\right) \) is a coset of the form \( {W}_{J}w \), where \( J \) is the type of \( \mathcal{R} \) . Choose \( {C}_{1} \in \mathcal{R} \) such that \( {\delta }^{ * }\left( {{C}_{1}, D}\right) \) is the element \( {w}_{1}^{ * } \) of maximal length in this c... | Yes |
Proposition 5.152. Let \( \mathcal{R} \) and \( \mathcal{S} \) be opposite residues of spherical type \( J \) in the twin building \( \left( {{\mathcal{C}}_{ + },{\mathcal{C}}_{ - },{\delta }^{ * }}\right) \) . Then the projection maps \( {\operatorname{proj}}_{\mathcal{R}} \) and \( {\operatorname{proj}}_{\mathcal{S}}... | Proof. By assumption, we have \( 1 \in {\delta }^{ * }\left( {\mathcal{R},\mathcal{S}}\right) \) ; hence \( {\delta }^{ * }\left( {\mathcal{R},\mathcal{S}}\right) = {W}_{J} \) by Lemma 5.148. The same lemma also implies that \( {\delta }^{ * }\left( {\mathcal{R}, D}\right) = {\delta }^{ * }\left( {C,\mathcal{S}}\right)... | Yes |
Lemma 5.156. Let \( \left( {{\mathcal{C}}_{ + },{\mathcal{C}}_{ - },{\delta }^{ * }}\right) \) be a thick twin building of type \( \left( {W, S}\right) \) . Then for each \( \epsilon \in \{ + , - \} \) and any two chambers \( C, D \in {\mathcal{C}}_{\epsilon } \), there is a chamber \( E \in {\mathcal{C}}_{-\epsilon } ... | Proof. Choose, by applying Corollary 5.141 for instance, a chamber \( {E}^{\prime } \in {\mathcal{C}}_{-\epsilon } \) that is opposite \( C \) . Set \( w \mathrel{\text{:=}} {\delta }^{ * }\left( {{E}^{\prime }, D}\right) \), and choose a reduced decomposition \( w = {s}_{1}\cdots {s}_{n} \) . By Corollary 5.153 and th... | Yes |
Corollary 5.157. Let \( \left( {{\mathcal{C}}_{ + },{\mathcal{C}}_{ - },{\delta }^{ * }}\right) \) be a thick twin building of type \( \left( {W, S}\right) \) . Then for each \( \epsilon \in \{ + , - \} \) and any spherical subset \( J \subseteq S \), any two residues of \( {\mathcal{C}}_{\epsilon } \) of type \( J \) ... | Proof. Let \( \mathcal{R} \) and \( \mathcal{T} \) be two residues of \( {\mathcal{C}}_{\epsilon } \) of type \( J \) . Choose \( C \in \mathcal{R}, D \in \mathcal{T} \) , and (by Lemma 5.156) \( E \in {\mathcal{C}}_{-\epsilon } \) such that \( E \) op \( C \) and \( E \) op \( D \) . Denote by \( \mathcal{S} \) the \(... | Yes |
Lemma 5.173. Let \( \sum = \left( {{\sum }_{ + },{\sum }_{ - }}\right) \) be a twin apartment and let \( \epsilon = + \) or - .\n\n(1) \( {\mathrm{{op}}}_{\sum } : {\sum }_{\epsilon } \rightarrow {\sum }_{-\epsilon } \) is an isometry. | Proof. (1) Let \( C \) and \( D \) be \( s \) -adjacent chambers of \( {\sum }_{\epsilon } \), and set \( {C}^{\prime } \mathrel{\text{:=}} {\operatorname{op}}_{\sum }\left( C\right) \) . Then \( {\delta }^{ * }\left( {C,{C}^{\prime }}\right) = 1 \) and hence, by Lemma \( {5.139}\left( 1\right) ,{\delta }^{ * }\left( {... | Yes |
Lemma 5.178. Let \( C \) and \( {C}^{\prime } \) be opposite chambers with \( C \in {\mathcal{C}}_{\epsilon } \), and let \( {\sum }_{\epsilon } \) be an apartment of \( {\mathcal{C}}_{\epsilon } \) such that \( {\left( {C}^{\prime }\right) }^{\text{op }} \cap {\sum }_{\epsilon } = \{ C\} \).\n\n(1) \( {\sum }_{\epsilo... | Proof. (1) For each \( w \in W \), let \( {D}_{w} \) be the chamber of \( {\sum }_{\epsilon } \) such that \( {\delta }_{\epsilon }\left( {C,{D}_{w}}\right) = w \) . We have to show that \( {D}_{w} = {C}_{w} \), where \( {C}_{w} \) is defined as in Lemma 5.175. In other words, we have to show that \( {\delta }^{ * }\le... | Yes |
Proposition 5.182. Let \( \sum = \sum \left\{ {C,{C}^{\prime }}\right\} \) be a twin apartment, with \( C \in {\mathcal{C}}_{ + } \) , \( {C}^{\prime } \in {\mathcal{C}}_{ - } \), and \( C \) op \( {C}^{\prime } \) . Let \( \mathcal{R} \) be a spherical residue of \( {\mathcal{C}}_{ + } \) containing \( C \), and let \... | Proof. The first assertion of (1) follows from the fact that \( \mathcal{R} \cap {\sum }_{ + } \) is an apartment of \( \mathcal{R} \) . We then have \( {\delta }^{ * }\left( {{C}^{\prime },{C}_{1}}\right) = {\delta }_{ + }\left( {C,{C}_{1}}\right) = {w}_{0}\left( J\right) \), which is the longest element of \( {W}_{J}... | Yes |
Lemma 5.191. Let \( \sum \) be a twin apartment.\n\n(1) Let \( \alpha \) be a twin root in \( \sum \), described as in (5.18) and (5.19). Let \( \mathcal{P} \) be the panel of \( {\mathcal{C}}_{ + } \) containing \( C \) and \( D \), and let \( {\mathcal{P}}^{\prime } \) be the panel of \( {\mathcal{C}}_{ - } \) contai... | Proof of Lemma 5.191. (1) For any chamber \( X \in {\sum }_{ + } \), Lemma 5.173(4) implies that \( {\delta }^{ * }\left( {{D}^{\prime }, X}\right) = s{\delta }^{ * }\left( {{C}^{\prime }, X}\right) \), where \( s = {\delta }_{ - }\left( {{C}^{\prime },{D}^{\prime }}\right) = {\delta }_{ + }\left( {C, D}\right) \) . He... | Yes |
Proposition 5.193. A pair \( \mathcal{M} = \left( {{\mathcal{M}}_{ + },{\mathcal{M}}_{ - }}\right) \) of nonempty sets in the twin apartment \( \sum \) is convex if and only if it is an intersection of twin roots of \( \sum \) . | Proof. In view of Lemma 5.191(2), it suffices to prove the \ | No |
Corollary 5.194. Let \( \sum \) be a twin apartment.\n\n(1) If \( \alpha \) is a twin root of \( \sum \) described as in (5.18) and (5.19), then \( \alpha \) is the convex hull of the pair \( \left( {C,{D}^{\prime }}\right) \) .\n\n(2) The twin roots of \( \sum \) are precisely the convex hulls of pairs \( \left( {C,{D... | Proof. Both parts of the corollary will follow if we show that for any pair \( \left( {C,{D}^{\prime }}\right) \) as in (2), there is a unique twin root of \( \sum \) containing it. Set \( {C}^{\prime } \mathrel{\text{:=}} \n\n\( {\operatorname{op}}_{\sum }\left( C\right) \) and \( D \mathrel{\text{:=}} {\operatorname{... | Yes |
Lemma 5.198. Let \( \alpha = \left( {{\alpha }_{ + },{\alpha }_{ - }}\right) \) be a twin root, and for \( \epsilon \in \{ + , - \} \), let \( \mathcal{P} \) be a panel in \( {\mathcal{C}}_{\epsilon } \) that contains precisely one chamber \( C \in {\alpha }_{\epsilon } \) . Then there is a bijection \( \mathcal{P} \sm... | Proof. We may assume that \( \epsilon = + \) . Observe first that there is a panel \( {\mathcal{P}}^{\prime } \) of \( {\mathcal{C}}_{ - } \) that is opposite \( \mathcal{P} \) and contains precisely one chamber \( {D}^{\prime } \in {\alpha }_{ - } \) . (Work in a fixed twin apartment, and use the description of twin r... | Yes |
Theorem 5.205. Let \( \left( {\mathcal{C},\delta }\right) \) be a thick spherical building, and let \( C,{C}^{\prime } \) be opposite chambers in \( \mathcal{C} \) . If an automorphism \( \phi \) of \( \mathcal{C} \) fixes \( {E}_{1}\left( C\right) \cup \left\{ {C}^{\prime }\right\} \) pointwise, then \( \phi \) is the... | Proof. Note first that \( \phi \) is actually an isometry, since it fixes every chamber adjacent to \( C \) . (From the simplicial point of view, \( \phi \) is type-preserving.) The proof now consists of two steps. The first step is to remove the apparent asymmetry in the hypothesis.\n\n(a) \( \phi \) fixes every chamb... | Yes |
Corollary 5.207. Let \( \left( {\mathcal{C},\delta }\right) \) be a thick spherical building, and let \( C,{C}^{\prime } \) be opposite chambers in \( \mathcal{C} \) . Then \( \mathcal{C} \) is the convex hull of \( {E}_{1}\left( C\right) \cup \left\{ {C}^{\prime }\right\} \) . | Proof. Let \( \mathcal{D} \) be the convex hull of \( {E}_{1}\left( C\right) \cup \left\{ {C}^{\prime }\right\} \) . \n\n(a) \( \mathcal{D} \) contains \( {E}_{1}\left( {C}^{\prime }\right) \) . \n\nThis follows from step (a) of the proof of Theorem 5.205. [Recall that the nonopposition bijection from \( \mathcal{P} \)... | Yes |
Theorem 5.209. Let \( \left( {\mathcal{C},\delta }\right) \) and \( \left( {{\mathcal{C}}^{\prime },{\delta }^{\prime }}\right) \) be thick, irreducible, spherical buildings of rank at least 3. Let \( \phi : {E}_{2}\left( C\right) \rightarrow {E}_{2}\left( {C}^{\prime }\right) \) be an adjacency-preserving bijection fo... | This is Theorem 4.1.2 of Tits [247]. The original proof was long and technical, but there is a simplified proof (with a slightly different hypothesis on \( \phi \) ) in Weiss [281, Chapter 10], based on ideas of Ronan. This proof makes systematic use of the W-metric point of view, which was not available when Tits wrot... | No |
Theorem 5.210. Let \( \left( {\mathcal{C},\delta }\right) \) and \( \left( {{\mathcal{C}}^{\prime },{\delta }^{\prime }}\right) \) be thick, irreducible, spherical buildings of rank at least 3, and let \( \mathcal{A} \) (resp. \( {\mathcal{A}}^{\prime } \) ) be an apartment of \( \mathcal{C} \) (resp. \( {\mathcal{C}}^... | For the proof, see Tits [247, Proposition 4.16] or Weiss [281, Theorem 10.1]. | No |
Corollary 5.211. Let \( \Delta \) be a thick, irreducible, spherical building of rank at least 3. Let \( \alpha \) be a root of \( \Delta \), and let \( \sum \) and \( {\sum }^{\prime } \) be apartments containing \( \alpha \) . Then there is an automorphism of \( \Delta \) that fixes \( \alpha \) pointwise and maps \(... | Proof. By axiom \( \left( {\mathrm{{B2}}}^{\prime \prime }\right) \) for buildings, there is an isomorphism \( \psi : \sum \rightarrow {\sum }^{\prime } \) that fixes \( \alpha \) pointwise. Now apply Proposition 3.125 to get a chamber \( C \in \alpha \) that is disjoint from \( \partial \alpha \), and let \( \phi : {E... | Yes |
Corollary 5.215. Let \( \mathcal{C} \) be a thick, irreducible,2-spherical twin building of rank at least 3 satisfying (co). Let \( \alpha \) be a twin root of \( \mathcal{C} \), and let \( \sum \) and \( {\sum }^{\prime } \) be twin apartments containing \( \alpha \) . Then there is an automorphism of \( \mathcal{C} \... | Proof. As in the proof of Corollary 5.211, we can find a chamber \( {C}_{ + } \in {\alpha }_{ + } \) that is disjoint from \( \partial {\alpha }_{ + } \), and we then have \( {E}_{2}\left( {C}_{ + }\right) \cap {\sum }_{ + } = {E}_{2}\left( {C}_{ + }\right) \cap {\alpha }_{ + } = \) \( {E}_{2}\left( {C}_{ + }\right) \c... | Yes |
Lemma 5.220. Let \( \kappa : {\mathcal{C}}^{\prime } \rightarrow \mathcal{C} \) be a morphism of chamber systems over \( S \) such that for all \( s \in S,\kappa \) maps every \( s \) -panel of \( {\mathcal{C}}^{\prime } \) onto an \( s \) -panel of \( \mathcal{C} \) . Let \( {C}^{\prime } \) be a chamber in \( {\mathc... | Proof. This is immediate from the definitions. | No |
Lemma 5.221. Let \( \kappa : {\mathcal{C}}^{\prime } \rightarrow \mathcal{C} \) be a 1-covering, where \( {\mathcal{C}}^{\prime } \) is a chamber system over \( S \) and \( \mathcal{C} \) is a building of type \( \left( {W, S}\right) \). Assume that any two chambers \( {C}^{\prime },{D}^{\prime } \in {\mathcal{C}}^{\pr... | Proof. (a) \( \kappa \) is surjective.\n\nChoose an arbitrary \( {C}^{\prime } \in {\mathcal{C}}^{\prime } \). Then \( \kappa \) maps \( {R}_{S}\left( {C}^{\prime }\right) \) onto \( {R}_{S}\left( {\kappa \left( {C}^{\prime }\right) }\right) \) by the last assertion of Lemma 5.220. Since \( \mathcal{C} \), being a buil... | Yes |
Proposition 5.223. Let \( \kappa : {\mathcal{C}}^{\prime } \rightarrow \mathcal{C} \) be a 2-covering, where \( {\mathcal{C}}^{\prime } \) is a connected chamber system over \( S \) and \( \mathcal{C} \) is a building of type \( \left( {W, S}\right) \). Then \( {\mathcal{C}}^{\prime } \) is a building of type \( \left(... | Proof. We will verify the assumption of Lemma 5.221. Since \( {\mathcal{C}}^{\prime } \) is connected, it suffices to show that any minimal gallery in \( {\mathcal{C}}^{\prime } \) has reduced type. For this we first observe the following:\n\n(*) Let \( {\Gamma }^{\prime } \) be a gallery of type \( \mathbf{s} \) in \(... | Yes |
Lemma 6.4. Suppose a group \( G \) acts strongly transitively on a set \( \mathcal{A} \) of apartments in a building \( \Delta \) . Then, with the notation above, the following conditions are equivalent:\n\n(i) The subcomplex \( {\Delta }^{\prime } \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{{\sum }^{\prime } \in \m... | Proof. (i) holds if and only if any two chambers of \( {\Delta }^{\prime } \) are contained in an apartment in \( \mathcal{A} \) . We may assume that one of the two chambers is the fundamental chamber \( C \), and the other chamber is then \( {gC} \) for some \( g \in G \) . So a restatement of (i) is that for any \( g... | Yes |
Proposition 6.6. Suppose a group \( G \) acts strongly transitively on a building \( \Delta \) with respect to an apartment system \( \mathcal{A} \). (1) Given two apartments \( \sum ,{\sum }^{\prime } \in \mathcal{A} \) and a type-preserving isomorphism \( \phi : \sum \rightarrow {\sum }^{\prime } \), there is an elem... | Proof. (1) Choose an arbitrary chamber \( C \in \sum \) . By strong transitivity, there is an element \( g \in G \) such that \( {g\sum } = {\sum }^{\prime } \) and \( {gC} = \phi \left( C\right) \) . Then \( g \) agrees with \( \phi \) on \( \sum \) by the standard uniqueness argument. | Yes |
Proposition 6.11. Assume that the action of \( G \) is chamber transitive. Let \( C \) be an arbitrary chamber, and let \( \sum \) an arbitrary apartment (in the complete apartment system) containing \( C \) . Let \( B \) be the stabilizer of \( C \) in \( G \) . Then the action of \( G \) on \( \Delta \) is Weyl trans... | Proof. As we noted above, Weyl transitivity holds if and only if the action of \( B \) is transitive on the \( w \) -sphere centered at \( C \) for each \( w \in W \) . Given \( w \) , there is a unique chamber \( {C}_{w} \in \mathcal{C}\left( \sum \right) \) with \( \delta \left( {C,{C}_{w}}\right) = w \) . (If we ide... | Yes |
Lemma 6.13. Suppose the action of \( G \) on \( \Delta \) is Weyl transitive, and let \( \sum \) be an arbitrary apartment (in the complete system of apartments). Then the set \( {G\sum } \mathrel{\text{:=}} \{ {g\sum } \mid g \in G\} \) is a system of apartments. | Proof. Since \( {G\sum } \) is given to us as a subset of the complete system of apartments, it suffices to show that any two chambers \( C, D \) are contained in some apartment in \( {G\sum } \) . By chamber transitivity we may assume that \( C \in \mathcal{C}\left( \sum \right) \) . Then equation (6.1) gives us an ap... | Yes |
Proposition 6.14. The following conditions are equivalent:\n\n(i) The \( G \) -action on \( \Delta \) is strongly transitive with respect to some apartment system.\n\n(ii) The G-action on \( \Delta \) is Weyl transitive, and there is an apartment \( \sum \) (in the complete system of apartments) such that the stabilize... | Proof. The implication (i) \( \Rightarrow \) (ii) is immediate from the definitions and Corollary 6.12. Conversely, if (ii) holds, then the action is strongly transitive with respect to \( {G\sum } \), which is an apartment system by Lemma 6.13. | Yes |
Proposition 6.15. The following conditions are equivalent for a (type-preserving) action of a group on a spherical building:\n\n(i) The action is strongly transitive.\n\n(ii) The action is Weyl transitive.\n\n(iii) The action is transitive on ordered pairs \( \\left( {C,{C}^{\\prime }}\\right) \) of opposite chambers. | Proof. We already know that (i) \\( \\Rightarrow \\) (ii). To prove (ii) \\( \\Rightarrow \\) (iii), just note that \\( C \\) and \\( {C}^{\\prime } \\) are opposite if and only if \\( \\delta \\left( {C,{C}^{\\prime }}\\right) = {w}_{0} \\), where \\( {w}_{0} \\) is the longest element of \\( W \\) (Section 1.5.2). Fi... | Yes |
Theorem 6.17. Assume that the action of \( G \) on \( \Delta \) is Weyl transitive, and let \( B \) be the stabilizer of a chamber \( C \) . Then there is a bijection \( B \smallsetminus G/B \rightarrow W \) given by \( {BgB} \mapsto \delta \left( {C,{gC}}\right) \) . Hence (6.2) holds, where \( w \mapsto C\left( w\rig... | With the aid of the Bruhat decomposition, we can completely reconstruct the Weyl distance function \( \delta \), and hence the building \( \Delta \), from group-theoretic data. Namely, if we identify \( \mathcal{C} \) with \( G/B \), then \( \delta \) becomes a function\n\n\[ G/B \times G/B \rightarrow W \]\n\nstill de... | No |
Theorem 6.21. Given \( s \in S \) and \( w \in W \), we have\n\n\[ C\left( {sw}\right) \subseteq C\left( s\right) C\left( w\right) \subseteq C\left( {sw}\right) \cup C\left( w\right) . \]\n\n(6.5)\n\nIn particular, \( C\left( s\right) C\left( w\right) \) is either the double coset \( C\left( {sw}\right) \) or else the ... | Proof. Given \( h \in C\left( s\right) \) and \( g \in C\left( w\right) \), we want to know which double coset contains the product \( {hg} \) . In other words, we are given that \( \delta \left( {C,{hC}}\right) = s \) and \( \delta \left( {C,{gC}}\right) = w \), and we want to compute \( \delta \left( {C,{hgC}}\right)... | Yes |
Proposition 6.27. Given \( J \subseteq S \), let \( A \) be the face of \( C \) of cotype \( J \) . Then the stabilizer of \( A \) in \( G \) is\n\n\[ \n{P}_{J} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{w \in {W}_{J}}}C\left( w\right) \n\]\n\n(6.11)\n\nIn particular, the union of double cosets in (6.11) is a subgr... | Proof. Recall that simplices are in 1-1 correspondence with residues (Corollary 4.11). Given \( g \in G \), it follows that \( {gA} = A \) if and only if \( {gC} \) and \( C \) are in the same \( J \) -residue. As we noted in Section 5.3.1 (or Exercise 4.92), this happens if and only if \( \delta \left( {C,{gC}}\right)... | Yes |
Proposition 6.36. Let \( G \) be a group and \( B \) a subgroup. Suppose we are given a group \( W \), a generating set \( S \) consisting of elements of order 2, and a function \( C : W \rightarrow B \smallsetminus G/B \) satisfying (Bru1),(Bru2), and (Bru3). Then the six conditions below are satisfied. In particular,... | Proof. For (1), we must show that \( C\left( v\right) = C\left( w\right) \Rightarrow v = w \) for \( v, w \in W \) . We argue by induction on \( \min \{ l\left( v\right), l\left( w\right) \} \), which we may assume is equal to \( l\left( v\right) \) . The case \( l\left( v\right) = 0 \) is covered by (Bru1), so suppose... | No |
Proposition 6.40. Let \( G \) be a group and \( B \) a subgroup. Suppose we are given a group \( W \), a generating set \( S \) of \( W \), and a function \( C : W \rightarrow B \smallsetminus G/B \) satisfying (Bru1), (Bru2), (Bru3), and (Th). Then (W, S) is a Coxeter system, and hence the given data constitute a thic... | Proof. We show first that elements of \( S \) have order 2. Take \( w = {s}^{-1} \) in (Bru3) to get\n\n\[ B \subseteq C\left( s\right) C\left( {s}^{-1}\right) \subseteq B \cup C\left( {s}^{-1}\right) . \]\n\n(6.17)\n\nThis implies that \( C\left( {s}^{-1}\right) \) meets \( C{\left( s\right) }^{-1} \) and hence is equ... | Yes |
Lemma 6.41. Suppose we are given a thick Bruhat decomposition for \( \left( {G, B}\right) \) . If \( w \in W \) admits a reduced decomposition \( w = {s}_{1}\cdots {s}_{l} \), then the subgroup of \( G \) generated by \( C\left( w\right) \) contains the double cosets \( C\left( {s}_{i}\right) \) for \( i = 1,\ldots, l ... | Proof. The subgroup generated by \( C\left( w\right) = {BgB} \) contains \( g \) and \( B \) . We therefore have\n\n\[ \left\langle {B,{gB}{g}^{-1}}\right\rangle \leq \langle C\left( w\right) \rangle \leq \left\langle {C\left( {s}_{1}\right) ,\ldots, C\left( {s}_{l}\right) }\right\rangle ,\]\n\nwhere the second inclusi... | Yes |
Theorem 6.43. Suppose \( \left( {G, B}\right) \) admits a thick Bruhat decomposition.\n\n(1) The standard parabolic subgroups are precisely the subgroups of \( G \) containing \( B \) .\n\n(2) If \( P \) is a standard parabolic subgroup and \( {gB}{g}^{-1} \leq P \) for some \( g \in G \) , then \( g \in P \) .\n\n(3) ... | Proof. The standard parabolics certainly contain \( B = C\left( 1\right) \) . Conversely, suppose \( P \) is a subgroup containing \( B \) . Then \( P \) is a union of double cosets \( C\left( w\right) \) ; hence it is generated by certain double cosets \( C\left( s\right) \) with \( s \in S \) by Lemma 6.41. But any s... | Yes |
Proposition 6.52. Assume that \( \left( {W, S}\right) \) is a Coxeter system and that\n\n\[ \n{sBw} \subseteq {BswB} \cup {BwB}\n\]\n\n(6.18)\n\nfor all \( s \in S \) and \( w \in W \) . Then the function \( w \mapsto C\left( w\right) \mathrel{\text{:=}} {BwB} \) provides a Bruhat decomposition for \( \left( {G, B}\rig... | Proof. The inclusion (6.18) is a restatement of the second inclusion in (Bru3), since \( C\left( s\right) C\left( w\right) = {BsBwB} \) . The first inclusion in (Bru3) holds trivially in the present setup, as does (Bru1). To get a Bruhat decomposition we must show that the conditions of the proposition imply (Bru2). To... | Yes |
Lemma 6.59. Let \( \left( {B, N}\right) \) be a saturated \( {BN} \) -pair in \( G \), and let \( {N}^{\prime } \) be a subgroup of \( N \) . Then \( \left( {B,{N}^{\prime }}\right) \) is a \( {BN} \) -pair if and only if \( {N}^{\prime }T = N \) or, equivalently, \( {N}^{\prime } \) surjects onto \( W = N/T \) . | Proof. If \( {N}^{\prime } \) surjects onto \( W \), then it is trivial to verify that the BN-pair axioms are still satisfied by \( \left( {B,{N}^{\prime }}\right) \) . [Alternatively, note that \( {N}^{\prime } \) stabilizes the fundamental apartment and is transitive on its chambers, so \( \left( {B,{N}^{\prime }}\ri... | Yes |
Lemma 6.61. If \( H \) is a normal subgroup of \( G \), then either \( H \leq Z \) or else \( {HB} = G \). In other words, either \( H \) acts trivially on \( \Delta \) or else \( H \) acts chamber transitively on \( \Delta \). | Proof. Since \( H \) is normal, \( {HB} \) is a subgroup of \( G \) containing \( B \); hence it is a standard parabolic subgroup (Theorem 6.43). Thus\n\n\[ {HB} = \mathop{\bigcup }\limits_{{w \in {W}_{J}}}C\left( w\right) \]\n\nfor some subset \( J \subseteq S \). We claim that if \( s \in J \) and \( t \in S \smallse... | Yes |
Theorem 6.62. Assume that \( G \) is perfect and \( B \) is solvable. Then every proper normal subgroup of \( G \) is contained in \( Z \) . Hence \( G/Z \) is a simple group. | Proof. Let \( H \) be a proper normal subgroup of \( G \) . By the lemma, either \( H \leq Z \) or \( {HB} = G \) . But the second case is impossible, because it would imply that the (nontrivial) perfect group \( G/H \) is isomorphic to the solvable group \( B/\left( {B \cap H}\right) \) . | Yes |
Lemma 6.63. Let \( U \) be a normal subgroup of \( B \), and let \( {G}_{1} \) be the normal closure of \( U \) in \( G \) . Assume that \( {G}_{1} \) is perfect and that \( U \) is solvable. Then every normal subgroup of \( G \) either is contained in \( Z \) or contains \( {G}_{1} \). | Proof. Let \( H \) be a normal subgroup of \( G \) that is not contained in \( Z \) . Then \( {HB} = G \) by Lemma 6.61. Since \( U \) is normal in \( B,{HU} \) is normal in \( {HB} = G \) , so \( {HU} \geq {G}_{1} \) . We now have\n\n\[ \nU/\left( {U \cap H}\right) \cong {HU}/H = H{G}_{1}/H \cong {G}_{1}/\left( {{G}_{... | Yes |
Theorem 6.64. Let \( U \) be a normal subgroup of \( B \) such that \( B = {UT} \), and let \( {G}_{1} \) be the normal closure of \( U \) in \( G \) . Assume that \( {G}_{1} \) is perfect and that \( U \) is solvable. Then every subgroup of \( G \) that is normalized by \( {G}_{1} \) either is contained in \( Z \) or ... | Proof. Recall that by Weyl transitivity, \( \Delta = \mathop{\bigcup }\limits_{{b \in B}}{b\sum } \) (Proposition 6.11). Since \( B = {UT} \), this can be rewritten as\n\n\[ \Delta = \mathop{\bigcup }\limits_{{u \in U}}{u\sum } \]\n\n(6.20)\n\nIn particular, \( U \) does not act trivially on \( \Delta \), since that wo... | Yes |
Lemma 6.69. With the notation above, we have:\n\n(1) \( N \) is the stabilizer of \( {\sum }_{ + } \) as well as the stabilizer of \( {\sum }_{ - } \) .\n\n(2) \( T \) is the pointwise fixer of each \( {\sum }_{\epsilon } \), i.e.,\n\n\[ T = \left\{ {g \in G \mid {gC} = C\text{ for all }C \in {\sum }_{ + }}\right\} ,\]... | Proof. (1) follows immediately from the fact that \( {\sum }_{\epsilon } \), for \( \epsilon = + \) or \( - \), has only one twin partner with which it can form a twin apartment, namely \( {\sum }_{-\epsilon } \) (see Proposition \( {5.179}\left( 2\right) \) ). And if an element of \( N \) fixes \( {\sum }_{\epsilon } ... | Yes |
Lemma 6.70. The following conditions are equivalent:\n\n(i) For any \( w \in W, G \) acts transitively on\n\n\[ \left\{ {\left( {C, D}\right) \in {\mathcal{C}}_{ + } \times {\mathcal{C}}_{ - } \mid {\delta }^{ * }\left( {C, D}\right) = w}\right\} .\n\]\n\n(ii) For \( \epsilon = + \) or \( -, G \) acts transitively on \... | Proof. The equivalence of (i) and (ii) is routine, and trivially (i) \( \Rightarrow \) (iii). Suppose now that (iii) is satisfied. Then (by Proposition 5.179) \( G \) acts transitively on \( \mathcal{A} \). And if \( g \in G \) satisfies \( g\left( {{C}_{ + },{C}_{ - }}\right) = \left( {C,{C}^{\prime }}\right) \) for s... | Yes |
Corollary 6.72. Suppose that \( G \) acts strongly transitively on \( \mathcal{C} \) . (1) For \( \epsilon \in \{ + , - \} \) , \( G \) acts strongly transitively on the building \( {\mathcal{C}}_{\epsilon } \) with respect to the apartment system \( {\mathcal{A}}_{\epsilon } \) . In particular, \( G \) acts Weyl trans... | Proof. The first assertion of (1) is immediate from the definition of strong transitivity via condition (v) of Lemma 6.70, and the second assertion follows from the first by Corollary 6.12. We already remarked after Lemma 6.69 that \( W \) can be identified with \( N/T \) if \( N \) acts transitively on \( {\sum }_{\ep... | Yes |
Lemma 6.73. Suppose a group \( G \) acts strongly transitively on a twin building. Given two twin apartments \( \sum ,{\sum }^{\prime } \), there is an element \( g \in G \) such that \( {g\sum } = \) \( {\sum }^{\prime } \) and \( g \) fixes every chamber in \( \sum \cap {\sum }^{\prime } \) . | Proof. This is immediate if \( \sum \cap {\sum }^{\prime } = \varnothing \), so we may assume without loss of generality that \( {\sum }_{ + } \cap {\sum }_{ + }^{\prime } \) contains a chamber \( C \) . By strong transitivity, there exists \( g \in G \) such that \( g{\sum }_{ + } = {\sum }_{ + }^{\prime } \) and \( {... | Yes |
Corollary 6.76. If \( G \) acts strongly transitively on a twin building, then\n\n\[ \n{B}_{ + }s \cap {B}_{ - } = \varnothing \n\]\n\nfor all \( s \in S \) . | Proof. This just says that the double cosets \( {B}_{ + }s{B}_{ - } \) and \( {B}_{ + }{B}_{ - } \) are different. | No |
Proposition 6.77. Assume that \( G \) acts strongly transitively on \( \mathcal{C} \) . For \( \epsilon = \pm \) , \( w \in W \), and \( s \in S \), we have:\n\n(1) \( {B}_{\epsilon }s{B}_{\epsilon }w{B}_{-\epsilon } \subseteq {B}_{\epsilon }\{ {sw}, w\} {B}_{-\epsilon } \) .\n\n(2) If \( l\left( {sw}\right) < l\left( ... | Proof. We may assume \( \epsilon = + \) . Given \( h \in {B}_{ + }s{B}_{ + } \) and \( g \in {B}_{ + }w{B}_{ - } \), we have \( {\delta }_{ + }\left( {{C}_{ + }, h{C}_{ + }}\right) = s \) and \( {\delta }^{ * }\left( {h{C}_{ + },{hg}{C}_{ - }}\right) = w \) . Hence \( {\delta }^{ * }\left( {{C}_{ + },{hg}{C}_{ - }}\rig... | Yes |
Lemma 6.80. Let \( \left( {G,{B}_{ + },{B}_{ - }, N, S}\right) \) be a twin Tits system. For each \( \epsilon = \pm \) and all \( s \in S \) and \( w \in W \) , \[ {B}_{\epsilon }s{B}_{\epsilon }w{B}_{-\epsilon } \subseteq {B}_{\epsilon }\{ s, w\} {B}_{-\epsilon }. \] Equality holds if \( l\left( {sw}\right) > l\left( ... | Proof. We may assume that \( \epsilon = + \) and that \( l\left( {sw}\right) > l\left( w\right) \) . Since \( \left( {{B}_{ + }, N}\right) \) is a BN-pair, we have \( {B}_{ + }s{B}_{ + }s{B}_{ + } = {B}_{ + } \cup {B}_{ + }s{B}_{ + } \) . Combining this with the assumption \( l\left( {sw}\right) > l\left( w\right) \) a... | Yes |
Proposition 6.81. Let \( \left( {G,{B}_{ + },{B}_{ - }, N, S}\right) \) be a twin Tits system with Weyl group \( W \) . Then\n\n\[ G = \mathop{\coprod }\limits_{{w \in W}}{B}_{\epsilon }w{B}_{-\epsilon } \]\n\nfor each \( \epsilon \in \{ + , - \} \) . | Proof. We may assume that \( \epsilon = + \) . The proof is similar to the proof of the Bruhat decomposition in Section 6.2.2, except that we do not have an assumption analogous to (Bru1). So we begin by proving that \( {B}_{ + }w{B}_{ - } \neq \) \( {B}_{ + }{B}_{ - } \) for \( w \neq 1 \) in \( W \) . Choose \( s \in... | Yes |
(1) Let \( \left( {{B}_{ + },{B}_{ - }, N}\right) \) be a saturated twin \( {BN} \) -pair in a group \( G \) . If \( {N}^{\prime } \) is an arbitrary subgroup of \( G \), then \( \left( {{B}_{ + },{B}_{ - },{N}^{\prime }}\right) \) is a twin \( {BN} \) -pair if and only if \( {N}^{\prime }T = N \) . | Proof. (1) If \( {N}^{\prime }T = N \), it is trivial that \( \left( {{B}_{ + },{B}_{ - },{N}^{\prime }}\right) \) satisfies the TBN axioms. Conversely, suppose that \( \left( {{B}_{ + },{B}_{ - },{N}^{\prime }}\right) \) is a second twin BN-pair with the same groups \( {B}_{ \pm } \) . By Remark 6.83, our two twin BN-... | Yes |
Theorem 6.94. If \( \Delta \) is a finite thick building, then every connected component of its Coxeter diagram is of type \( {A}_{n},{C}_{n},{D}_{n},{E}_{n},{F}_{4},{G}_{2} \), or \( {I}_{2}\left( 8\right) \) . | Here are a few words about the proof. First, it suffices to consider the case that \( \Delta \) is irreducible, by which we mean that its Coxeter diagram is connected. For in the general case, \( \Delta \) can be decomposed as a join of irreducible buildings, one for each component of the diagram. Next, it suffices to ... | No |
Proposition 6.109. A discrete valuation ring \( A \) is a principal ideal domain, and every nonzero ideal is generated by \( {\pi }^{n} \) for some \( n \geq 0 \) . In particular, \( {\pi A} \) is the unique nonzero prime ideal of \( A \) . | Proof. Let \( I \) be a nonzero ideal and let \( n \mathrel{\text{:=}} \min \{ v\left( a\right) \mid a \in I\} \) . Then \( I \) contains \( {\pi }^{n} \), and every element of \( I \) is divisible by \( {\pi }^{n} \) ; hence \( I = {\pi }^{n}A \) . | Yes |
Lemma 6.118. Let \( G \) be a topological group and \( {G}^{\prime } \) a dense subgroup.\n\n(1) If \( U \) is an open subgroup of \( G \), then \( {G}^{\prime } \) maps onto \( G/U \) under the quotient map \( G \rightarrow G/U \) . | Proof. For (1), observe that every coset \( {gU} \) is a nonempty open set, so it meets \( {G}^{\prime } \) . | No |
Proposition 6.119. Suppose a group \( G \) acts Weyl transitively on a building \( \Delta \) . Assume that \( G \) is a topological group and that the stabilizer \( B \) of some chamber is an open subgroup. If \( {G}^{\prime } \) is a dense subgroup of \( G \), then the action of \( {G}^{\prime } \) on \( \Delta \) is ... | Proof. Consider the diagonal action of \( G \) on \( \mathcal{C} \times \mathcal{C} \), where \( \mathcal{C} = \mathcal{C}\left( \Delta \right) \) . Note that every stabilizer is an open subgroup of \( G \), being an intersection of two conjugates of \( B \) . To show that the action of \( {G}^{\prime } \) is Weyl tran... | Yes |
Proposition 6.120. Let \( D \) be a noncommutative 4-dimensional division algebra over a field \( F \) of characteristic \( \neq 2 \) . Then \( D \) is isomorphic to a quaternion algebra. | This is proved in Bourbaki [43, Section 11.2, Proposition 1] and Lam [150, Theorem III.5.1]. Since we will need to refer to the proof in Exercise 6.121 below, we sketch it here. We follow Bourbaki's proof, which uses standard facts from Wedderburn theory. Lam's proof is slightly more elementary.\n\nSketch of proof. The... | Yes |
Lemma 6.125. \( G \mathrel{\text{:=}} {\mathrm{{SL}}}_{1}\left( D\right) \) is dense in \( {G}_{p} \mathrel{\text{:=}} {\mathrm{{SL}}}_{1}\left( {D}_{p}\right) \) . | Proof. This is a special case of the weak approximation theorem [188, Chapter 7], but we will give a direct proof. The main point is to construct many elements of \( G \), which we do by the following \ | No |
Lemma 6.126. \( T \) is dense in \( {T}_{p} \) . | Proof. We use the same normalization trick as in the proof of Lemma 6.125. Namely, we construct elements of \( T \) by starting with an arbitrary \( x = \) \( {x}_{1} + {x}_{2}{e}_{2} \in {D}^{ * } \) and forming \( {x}^{\prime } \mathrel{\text{:=}} {x}^{2}/N\left( x\right) \) . Computing the images of such elements \(... | Yes |
Proposition 6.127. The action of \( G \) on \( {\Delta }_{p} \) is Weyl transitive. | Proof. Since \( G \) is dense in \( {G}_{p} \) by Lemma 6.125, and since the \( B \) of the BN-pair in \( {G}_{p} \) is an open subgroup, this follows from Proposition 6.119. | Yes |
Proposition 6.129. The following conditions are equivalent:\n\n(i) -1 has a square root in \( D \) .\n\n(ii) \( G \) contains an element that stabilizes the fundamental apartment \( \sum \) and acts as a reflection on it.\n\n(iii) The action of \( G \) on \( {\Delta }_{p} \) is strongly transitive with respect to some ... | Proof. If (i) holds, then \( \beta = - 1 \) by our choices at the beginning of this subsection. The quaternion \( {e}_{3} \) is therefore in \( G = {\mathrm{{SL}}}_{1}\left( D\right) \) and maps to \( \left( \begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}\right) \in {\mathrm{{SL}}}_{2}\left( {\mathbb{Q}}_{p}\right) \) . T... | Yes |
Lemma 7.4. Assume that \( \Delta \) is spherical and that \( {\left( {X}_{\alpha }\right) }_{\alpha \in \Phi } \) is a system of subgroups of \( {\operatorname{Aut}}_{0}\Delta \) satisfying condition (1) of Definition 7.2. Then the following conditions are equivalent:\n\n(i) The system \( {\left( {X}_{\alpha }\right) }... | Proof. Let \( \alpha \) be a root and let \( P \) be a panel in \( \partial \alpha \) . Then Lemma 4.118 implies that the action of \( {X}_{\alpha } \) is transitive on \( \mathcal{C}\left( {P,\alpha }\right) \) if and only if it is transitive on \( \mathcal{A}\left( \alpha \right) \) . Thus (ii) \( \Rightarrow \) (iii... | Yes |
Lemma 7.5. If \( \Delta \) is spherical and \( {\left( {X}_{\alpha }\right) }_{\alpha \in \Phi } \) is a system of subgroups of \( {\operatorname{Aut}}_{0}\Delta \) satisfying (1) and (2), then it also satisfies (3). | Proof. Choose a panel \( P \in \partial \alpha \), and let \( C \) and \( D \) be the chambers of \( \sum \) having \( P \) as a face, with \( C \in \alpha \) and \( D \in - \alpha \) . By thickness and condition (2), we can find an element \( x \in {X}_{\alpha } \) such that \( {xD} \neq D \) ; see Figure 7.1.\n\n![85... | Yes |
Proposition 7.8. If \( \Delta \) is spherical and \( G \) is a strongly transitive group of type-preserving automorphisms of \( \Delta \), then \( G \) contains a system of pre-root groups. | Proof. For each \( \alpha \in \Phi \), let \( {X}_{\alpha } \) be the pointwise fixer of \( \alpha \) in \( G \), i.e.,\n\n\[ \n{X}_{\alpha } = {\operatorname{Fix}}_{G}\left( \alpha \right) \mathrel{\text{:=}} \{ g \in G \mid {gA} = A\text{ for all }A \in \alpha \} .\n\]\n\nThen \( {X}_{\alpha } \) acts transitively on... | Yes |
Lemma 7.9. \( \Delta = \mathop{\bigcup }\limits_{{u \in U}}{u\sum } \) . | Proof. It suffices to show that the union on the right contains every chamber \( C \in \mathcal{C}\left( \Delta \right) \) . We argue by induction on \( d\left( {C,\mathcal{C}\left( \sum \right) }\right) \), which may be assumed \( > 0 \) . Choose a gallery \( {D}_{0},\ldots ,{D}_{l} = C \) of minimal length \( l \math... | Yes |
Proposition 7.11. If \( \Delta \) is a thick, irreducible, spherical building of rank at least 3, then \( \Delta \) is pre-Moufang. In particular, \( \Delta \cong \Delta \left( {G, B}\right) \) for some group \( G \) with a \( {BN} \) -pair. | Proof. Let \( G = {\operatorname{Aut}}_{0}\Delta \) . As in the proof of Proposition 7.8, it suffices to show, for each root \( \alpha \) of \( \sum \), that \( {X}_{\alpha } \mathrel{\text{:=}} {\operatorname{Fix}}_{G}\left( \alpha \right) \) acts transitively on the set of apartments containing \( \alpha \) . This is... | Yes |
Lemma 7.14. There is an order-reversing \( 1 - 1 \) correspondence between convex chamber subcomplexes of \( \sum \) and convex sets of roots. It associates to a convex chamber subcomplex \( K \subseteq \sum \) the set of roots \( \Psi \left( K\right) \) . Its inverse is given by \( \Psi \mapsto \mathop{\bigcap }\limit... | Proof. For any subset \( K \subseteq \sum \), the set \( \Psi \left( K\right) \) does not change if we replace \( K \) by its convex hull, which is a convex chamber subcomplex of \( \sum \) (see Section 3.6.6). So the function \( K \mapsto \Psi \left( K\right) \) is a surjection from the set of convex chamber subcomple... | Yes |
Given two chambers \( C, D \in \sum \), set\n\n\[ \Phi \left( {C, D}\right) \mathrel{\text{:=}} \{ \alpha \in \Phi \mid C \in \alpha, D \notin \alpha \} . \]\n\nThen \( \Phi \left( {C, D}\right) = \Psi \left( \left\{ {C,{D}^{\prime }}\right\} \right) \), where \( {D}^{\prime } \mathrel{\text{:=}} {\operatorname{op}}_{\... | Hence \( \Phi \left( {C, D}\right) \) is a convex set of roots. It contains precisely \( d = d\left( {C, D}\right) \) roots, one for each wall of \( \sum \) that separates \( C \) from \( D \) . We can enumerate these roots by choosing a minimal gallery \( C = {C}_{0},\ldots ,{C}_{d} = D \) from \( C \) to \( D \) . If... | No |
Lemma 7.17. Every convex set of roots admits an admissible ordering. | Proof. Let \( \Psi \) be convex, and write \( \Psi = \Psi \left( K\right) \) with \( K \) a convex chamber subcomplex. We may assume that \( \Psi \neq \varnothing \), so that \( K \neq \sum \) . Choose a pair of adjacent chambers \( D,{D}^{\prime } \) with \( D \in K \) and \( {D}^{\prime } \notin K \), and let \( {\al... | Yes |
Proposition 7.20. Let \( \Psi \) be a convex set of roots in the fundamental apartment \( \sum \), let \( {\alpha }_{1},\ldots ,{\alpha }_{m} \) be an admissible ordering of \( \Psi \), and set \( {X}_{i} \mathrel{\text{:=}} {X}_{{\alpha }_{i}} \) for \( i = 1,\ldots, m \) .\n\n(1) \( {X}_{\Psi }T \) is a subgroup of \... | Proof of Proposition 7.20. We prove (1) and (3) simultaneously, by induction on \( m = \left| \Psi \right| \) . We may assume that the subset \( K \) in (3) is a convex chamber subcomplex. To see this, let \( L \) be the convex hull of \( K \) in \( \Delta \) (Definition 4.116). Then \( L \) is a convex chamber subcomp... | Yes |
Lemma 7.22. Assume that the \( {X}_{\alpha } \) satisfy (1) and (2). Let \( \Gamma : {C}_{0},\ldots ,{C}_{l} \) be a minimal gallery in \( \sum \) . For \( 1 \leq i \leq l \) let \( {\alpha }_{i} \) be the root of \( \sum \) such that \( {C}_{i - 1} \in {\alpha }_{i} \) and \( {C}_{i} \notin {\alpha }_{i} \), and set \... | Proof. Since \( \Gamma \) is minimal, it crosses each wall separating \( {C}_{0} \) from \( {C}_{l} \) exactly once. Hence \( {C}_{0},\ldots ,{C}_{i - 1} \in {\alpha }_{i} \) and \( {C}_{i},\ldots ,{C}_{l} \notin {\alpha }_{i} \) for each \( 1 \leq i \leq l \) . In particular, \( {C}_{0} \in {\alpha }_{i} \) for all \(... | Yes |
(1) For any root \( \alpha \) and any \( g \in \operatorname{Aut}\Delta \) ,\n\n\[ g{U}_{\alpha }{g}^{-1} = {U}_{g\alpha } \] | Proof. Assertion (1) is straightforward and is left to the reader, | No |
Proposition 7.32. Let \( A \) be a simplex of \( \Delta \) and let \( {\Delta }^{\prime } \mathrel{\text{:=}} {\operatorname{lk}}_{\Delta }A \) . (1) If \( \Delta \) is Moufang, then so is \( {\Delta }^{\prime } \) . | Proof. (1) Consider a root \( {\alpha }^{\prime } \) of \( {\Delta }^{\prime } \) and a panel \( {P}^{\prime } \in \partial {\alpha }^{\prime } \), and set \( \mathcal{D} \mathrel{\text{:=}} \) \( \mathcal{C}\left( {{P}^{\prime },{\alpha }^{\prime }}\right) \) . With \( \alpha \) as above, we have a transitive action o... | No |
Lemma 7.33. Let \( \Delta \) be an arbitrary thick building, let \( \alpha \) be a root of \( \Delta \), let \( C \) be a chamber of \( \alpha \), and let \( A \) be a face of \( C \) such that \( A \notin \partial \alpha \) . Assume that \( {\Delta }^{\prime } \mathrel{\text{:=}} {\operatorname{lk}}_{\Delta }A \) is s... | Proof. Note that \( \alpha \cap {\Delta }^{\prime } \) is an apartment of \( {\Delta }^{\prime } \), since \( \alpha \cap {\Delta }^{\prime } = \sum \cap {\Delta }^{\prime } \) for any apartment \( \sum \) of \( \Delta \) containing \( \alpha \) . So \( g \) fixes an apartment of \( {\Delta }^{\prime } \) pointwise as ... | Yes |
Corollary 7.34. Suppose the Coxeter diagram of \( \Delta \) has no isolated nodes. Then for every root \( \alpha \) of \( \Delta \) and every vertex \( v \in \alpha \smallsetminus \partial \alpha \), the root group \( {U}_{\alpha } \) fixes \( {\operatorname{lk}}_{\Delta }v \) pointwise. In other words, \( {U}_{\alpha ... | Proof. The hypothesis on the Coxeter diagram implies that there is a chamber \( C \in \alpha \) such that \( v \) is a vertex of \( C \) and \( C \) has no panel in \( \partial \alpha \) (Example 3.128). Now apply the lemma, with \( A \) being the vertex \( v \) . | No |
Proposition 7.37. Let \( \Delta \) be a Moufang building whose Coxeter diagram has no isolated nodes. If \( {\Delta }^{\prime } \) is a thick subbuilding of \( \Delta \), then \( {\Delta }^{\prime } \) is also Moufang. If, moreover, \( \sum \) is an apartment of \( {\Delta }^{\prime },\alpha \) is a root of \( \sum \),... | Our proof is inspired by the proof of [264, Lemma 5.2.2], which treats the rank-2 case. We will use the following lemma: | No |
Lemma 7.38. Let \( \Delta \) be a thick spherical building, and let \( {\Delta }^{\prime } \) be a thick sub-building. If \( C \) and \( {C}^{\prime } \) are opposite chambers of \( {\Delta }^{\prime } \), then \( {\Delta }^{\prime } \) is the convex hull in \( \Delta \) of \( {E}_{1}\left( C\right) \cap {\Delta }^{\pr... | Proof. This follows from Corollary 5.207 applied to \( {\Delta }^{\prime } \), together with the fact that \( {\Delta }^{\prime } \) is convex in \( \Delta \) (see Theorem 4.66). | Yes |
Corollary 7.39. Suppose a Moufang building \( {\Delta }^{\prime } \) can be embedded as a sub-complex of a Moufang building \( \Delta \) of the same type. If the Coxeter diagram has no isolated nodes, then the root groups of \( {\Delta }^{\prime } \) are isomorphic to subgroups of the root groups of \( \Delta \) . | Proof. \( {\Delta }^{\prime } \) is a subbuilding of \( \Delta \) by Proposition 4.63, so we may apply Proposition 7.37. | No |
(1) \( {U}_{ij} \) fixes every vertex of \( \Delta \) that is joinable to an interior vertex of \( {\alpha }_{ij} \) . In particular, \( {U}_{ij} \) fixes \( {\alpha }_{ij} \) pointwise.\n\n(2) Let \( P \) be a panel in \( \partial {\alpha }_{ij} \) . Then \( {U}_{ij} \) acts simply transitively on \( \mathcal{C}\left(... | Proof. (1) Let \( Y \) be a vertex of \( \Delta \) that is joinable to an interior vertex \( {Y}^{\prime } \) of \( {\alpha }_{ij} \), and consider a nontrivial element \( \tau \in {U}_{ij} \) . Thus \( \tau = {\tau }_{e, f} \) with \( {eD} = {L}_{i} \) and \( \ker f = {\bigoplus }_{k \neq j}{L}_{k} \) . According to o... | Yes |
Lemma 7.47. If \( H \) is a wall of \( \sum \) and \( A \) is a simplex disjoint from \( H \), then \( {L}_{A} \cap H \) is a convex subcomplex of \( \sum \) . | Proof. Consider the star of \( A \), as defined in Example 3.133(b). According to that example, st \( A \) is a convex subcomplex of \( \sum \), hence so is st \( A \cap H \) . Now note that st \( A \cap H = {L}_{A} \cap H \), since \( A \) is disjoint from \( H \) . | Yes |
Lemma 7.48. Let \( A \) be a simplex in \( \sum \) of codimension 2. Suppose \( K \) is a \( 0 \) -dimensional subcomplex of \( {L}_{A} \) such that \( K \) is convex as a subcomplex of \( \sum \) . If \( K \) has more than one vertex, then we have:\n\n(1) \( {L}_{A} \) is finite, and \( K \) contains exactly two verti... | Proof. (1) Let \( x \) be a vertex of \( K \), and consider the wall \( H \) of \( \sum \) containing the panel \( A \cup \{ x\} \) . Then \( H \supseteq \operatorname{supp}x \supseteq K \), so \( H \cap {L}_{A} \) is a wall of \( {L}_{A} \) containing \( K \) ; see Figure 7.6. Since \( K \) has more than one vertex, t... | Yes |
Lemma 7.49. Let \( {\sum }^{\prime } \) be a Coxeter complex, and let \( u, v \) be vertices of \( {\sum }^{\prime } \) such that \( \operatorname{supp}u = \operatorname{supp}v \) and \( d\left( {u, v}\right) = 1 \), where \( d\left( {-, - }\right) \) denotes gallery distance in \( {\sum }^{\prime } \). Then \( {\sum }... | Proof. To deduce the second statement from the first, one need only check that in a join \( {\sum }_{1} * {\sum }_{2} \) of Coxeter complexes, the walls are the subcomplexes of the form \( {H}_{1} * {\sum }_{2} \) and \( {\sum }_{1} * {H}_{2} \), where \( {H}_{i} \) is a wall of \( {\sum }_{i} \) for \( i = 1,2 \). We ... | No |
Lemma 7.56. Suppose the Coxeter diagram of \( \Delta \) has no isolated nodes. If \( g \) fixes \( {E}_{1}\left( C\right) \) pointwise for every \( C \in {\alpha }^{\left( 1\right) } \), then \( g \) fixes \( {\operatorname{st}}_{\Delta }P \) pointwise for every panel \( P \in \alpha \smallsetminus \partial \alpha \) . | Proof. Let \( P \) be a panel in \( \alpha \smallsetminus \partial \alpha \), and choose a chamber \( C \geq P \) in \( \alpha \) . If \( C \) is in \( {\alpha }^{\left( 1\right) } \), then \( g \) fixes \( {\operatorname{st}}_{\Delta }P \) pointwise by hypothesis. Otherwise, Corollary 7.52 implies that there is a cham... | Yes |
Proposition 7.57. Suppose \( \Delta \) is irreducible and of rank at least 3. If \( g \) fixes \( {E}_{1}\left( C\right) \) pointwise for some chamber \( C \in {\alpha }^{\left( 1\right) } \), then \( g \) fixes \( {\operatorname{st}}_{\Delta }P \) pointwise for any panel \( P \in \alpha \smallsetminus \partial \alpha ... | Proof. In view of Lemma 7.56, it suffices to show that \( g \) fixes \( {E}_{1}\left( {C}^{\prime }\right) \) pointwise for every chamber \( {C}^{\prime } \in {\alpha }^{\left( 1\right) } \) . Recall now that \( {\alpha }^{\left( 2\right) } \neq \varnothing \) by Proposition 3.125.\n\nSo two applications of Theorem 7.5... | Yes |
Proposition 7.58. Suppose \( \Delta \) is irreducible and of rank at least 3. If \( g \) fixes \( {\operatorname{st}}_{\Delta }P \) pointwise for every panel \( P \in \alpha \) such that \( {\operatorname{codim}}_{\Delta }\left( {P \cap \partial \alpha }\right) = 2 \), then \( g \) fixes \( {\operatorname{st}}_{\Delta ... | Proof. Let \( {C}_{0} \) be any chamber of \( \alpha \) with a panel in \( \partial \alpha \), and choose (by Corollary 7.52) a chamber \( {C}_{1} \in {\alpha }^{\left( 1\right) } \) adjacent to \( {C}_{0} \) . We will show that \( g \) fixes \( {E}_{1}\left( C\right) \) pointwise, so that an application of Proposition... | Yes |
Theorem 7.59. If \( \Delta \) is a thick, irreducible, spherical building of rank at least 3, then \( \Delta \) is Moufang. | Proof. Let \( \alpha \) be a root of \( \Delta \), and let \( \sum \) and \( {\sum }^{\prime } \) be apartments containing \( \alpha \) . By Corollary 5.211, there is an automorphism \( g \) of \( \Delta \) such that \( {g\sum } = {\sum }^{\prime }, g \) fixes \( \alpha \) pointwise, and \( g \) fixes \( {E}_{2}\left( ... | Yes |
Lemma 7.62. \( {U}_{ + } \cap T = \{ 1\} \) . | Proof. Let \( P \) be a panel of the fundamental chamber \( C \), let \( \alpha \) be the positive root with \( P \in \partial \alpha \), and consider the action of \( {U}_{ + } \) on the set of chambers \( \mathcal{C}\left( {P,\alpha }\right) = {\mathcal{C}}_{P} \smallsetminus \{ C\} \) . We claim that if an element \... | Yes |
Corollary 7.63. \( {B}_{ + } \cap {U}_{ - } = \{ 1\} \) . | Proof. By (7.8) we have \( {B}_{ + } \cap {U}_{ - } \leq T \cap {U}_{ - } \) . Now note that \( T \cap {U}_{ - } = \{ 1\} \) by Lemma 7.62, applied with the roles of \( C \) and \( {C}^{0} \) reversed. | Yes |
Proposition 7.64. Let \( \Psi \) be a convex set of roots in the fundamental apartment \( \sum \), let \( {\alpha }_{1},\ldots ,{\alpha }_{m} \) be an admissible ordering of \( \Psi \), and set \( {U}_{i} \mathrel{\text{:=}} {U}_{{\alpha }_{i}} \) for \( i = 1,\ldots, m \) .\n\n(1) \( {U}_{\Psi } = {U}_{1}\cdots {U}_{m... | Proof. Since \( T \) acts trivially on \( \sum \), it follows from Lemma \( {7.25}\left( 1\right) \) that \( T \) normalizes \( {U}_{\alpha } \) for each \( \alpha \in \Phi \) . In particular, \( T \) normalizes \( {U}_{\Psi } \) . Note next that we can take the fundamental chamber \( C \) to be in \( K \), so that \( ... | Yes |
Corollary 7.66. Given \( w \in W \), choose a reduced decomposition \( w = {s}_{1}\cdots {s}_{l} \) , let \( \Gamma \) be the corresponding gallery from \( C \) to \( {wC} \), and let \( {\alpha }_{1},\ldots ,{\alpha }_{l} \) be the sequence of roots associated to \( \Gamma \) as in Example 7.15. Set \( {U}_{i} \mathre... | Proof. (1) is just Proposition \( {7.64}\left( 1\right) \), specialized to \( \Psi = \Phi \left( w\right) \) . To prove (2), note first that the stabilizer of \( w{C}^{0} \) in \( {B}_{ + } \) is the fixer of the set \( K = \left\{ {C, w{C}^{0}}\right\} \) , so it is \( {U}_{w}T \) by Proposition 7.64(2). The stabilize... | Yes |
Corollary 7.67. \( {U}_{w} \) acts simply transitively on the \( w \) -sphere\n\n\[ \n{\mathcal{C}}_{w} \mathrel{\text{:=}} \{ D \in \mathcal{C}\left( \Delta \right) \mid \delta \left( {C, D}\right) = w\} .\n\]\n\nIn particular, \( {U}_{ + } \) acts simply transitively on\n\n\[ \n{C}^{\text{op }} \mathrel{\text{:=}} \{... | Proof. For any \( D \in {\mathcal{C}}_{w} \), there is a gallery from \( C \) to \( D \) having type \( \left( {{s}_{1},\ldots ,{s}_{l}}\right) \) , so transitivity follows immediately from Lemma 7.22. Simple transitivity follows from the uniqueness assertion of that lemma combined with Corollary \( {7.66}\left( 1\righ... | Yes |
Corollary 7.68. For any \( w \in W,{B}_{ + }w{B}_{ + } = {U}_{w}w{B}_{ + } \) . | Proof. For any \( b \in {B}_{ + } \), the chamber \( {bwC} \) is in \( {\mathcal{C}}_{w} \), so there is an element \( u \in {U}_{w} \) such that \( {bwC} = {uwC} \), i.e., \( {bw}{B}_{ + } = {uw}{B}_{ + } \) . Thus \( {B}_{ + }{wB} \subseteq {U}_{w}w{B}_{ + } \) . | Yes |
Proposition 7.72. Given \( \\alpha ,\\beta \\in \\Phi \) with \( \\alpha \\neq \\pm \\beta \), we have\n\n\[ \n\\left\\lbrack {{U}_{\\alpha },{U}_{\\beta }}\\right\\rbrack \\leq {U}_{\\left( \\alpha ,\\beta \\right) } \\mathrel{\\text{:=}} \\left\\langle {{U}_{\\gamma } \\mid \\gamma \\in \\left( {\\alpha ,\\beta }\\ri... | Proof. We first show that \( \\left\\lbrack {\\alpha ,\\beta }\\right\\rbrack \) and \( \\left( {\\alpha ,\\beta }\\right) \) are convex sets of roots, so that the results of the previous subsection about the groups \( {U}_{\\Psi } \) are applicable. The convexity of \( \\left\\lbrack {\\alpha ,\\beta }\\right\\rbrack ... | Yes |
Corollary 7.73. Let \( {\alpha }_{1},\ldots ,{\alpha }_{d} \) be the roots associated to a minimal gallery in \( \sum \) as in Example 7.15, and set \( {U}_{i} \mathrel{\text{:=}} {U}_{{\alpha }_{i}} \) for \( i = 1,\ldots, d \) . If \( d \geq 2 \), then\n\n\[\n\left\lbrack {{U}_{1},{U}_{d}}\right\rbrack \leq {U}_{2}\c... | Proof. It is easy to check that \( \left\lbrack {{\alpha }_{1},{\alpha }_{d}}\right\rbrack \subseteq \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{d}}\right\} \) (see Exercise 7.75); hence \( \left( {{\alpha }_{1},{\alpha }_{d}}\right) \subseteq \left\{ {{\alpha }_{2},\ldots ,{\alpha }_{d - 1}}\right\} \) . The corollary n... | No |
Proposition 7.79. Let \( \Delta \) be a spherical building with a family of subgroups \( {\left( {X}_{\alpha }\right) }_{\alpha \in \Phi } \) of Aut \( \Delta \), where \( \Phi \) is the set of roots of an apartment \( \sum \) . If the \( {X}_{\alpha } \) satisfy \( \left( 1\right) ,\left( 2\right) \), and \( \left( {3... | Proof. Let \( \alpha \) and \( P \) be as in (3), and consider any chamber \( C \in {\mathcal{C}}_{P} \) . Let \( D \) and \( E \) be the chambers of \( \sum \) in \( {\mathcal{C}}_{P} \) . They are necessarily both in \( \alpha \) . We have to show that \( {X}_{\alpha } \) fixes \( C \) . In view of (1), we may assume... | Yes |
Lemma 7.81. Let \( \alpha \) and \( \beta \) be roots of \( \sum \) such that \( \alpha \neq \pm \beta \), let \( A \) be a maximal simplex of \( \partial \alpha \cap \partial \beta \), and let \( {\alpha }^{\prime } \mathrel{\text{:=}} \alpha \cap {L}_{A} \) and \( {\beta }^{\prime } \mathrel{\text{:=}} \beta \cap {L}... | Proof. Given \( \gamma \in \left\lbrack {\alpha ,\beta }\right\rbrack \), we have \( \gamma \supseteq \alpha \cap \beta \) . Applying the opposition involution, we see that also \( - \gamma \supseteq \left( {-\alpha }\right) \cap \left( {-\beta }\right) \) ; hence \( \partial \gamma \supseteq \partial \alpha \cap \part... | Yes |
Example 7.83. Let \( \Delta \) be a strictly Moufang building with fundamental apartment \( {\sum }_{0} \) and fundamental chamber \( {C}_{0} \in {\sum }_{0} \). We identify \( {\sum }_{0} \) with \( \sum \left( {W, S}\right) \), where \( W \) is the group of type-preserving automorphisms of \( {\sum }_{0} \) and \( S ... | \[ T \mathrel{\text{:=}} {\operatorname{Fix}}_{G}\left( \sum \right) \] Then \( \left( {G,{\left( {U}_{\alpha }\right) }_{\alpha \in \Phi }, T}\right) \) is an RGD system of type \( \left( {W, S}\right) \). Axiom (RGD0) follows from the fact that \( \Delta \) is thick, and (RGD1) is Proposition 7.72. The first part of ... | Yes |
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