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Example 7.85. Let \( G = {\mathrm{{SL}}}_{2}\left( k\right) \), where \( k \) is a field. Let \( {U}_{ + } \) be the strict upper-triangular subgroup, let \( {U}_{ - } \) be the strict lower-triangular subgroup, and let \( T \) be the diagonal subgroup. It is easy to verify the axioms, the only interesting one being (R... | \[ m\left( u\right) \mathrel{\text{:=}} \left( \begin{matrix} 0 & \lambda \\ - {\lambda }^{-1} & 0 \end{matrix}\right) = \left( \begin{matrix} 1 & 0 \\ - {\lambda }^{-1} & 1 \end{matrix}\right) \left( \begin{array}{ll} 1 & \lambda \\ 0 & 1 \end{array}\right) \left( \begin{matrix} 1 & 0 \\ - {\lambda }^{-1} & 1 \end{mat... | Yes |
Lemma 7.90. If \( \alpha \) and \( \beta \) are distinct roots of \( \sum \), then there are elements \( w \in W \) and \( s \in S \) such that \( {w\alpha } = - {\alpha }_{s} \) and \( {w\beta } \in {\Phi }_{ + } \) . | Proof. We claim there are adjacent chambers \( {C}^{\prime },{D}^{\prime } \) with \( {C}^{\prime } \in \beta \smallsetminus \alpha \) and \( {D}^{\prime } \in \alpha \) . Assuming the claim, let \( w \in W \) be the element such that \( w{C}^{\prime } \) is the fundamental chamber 1. [Recall that \( \mathcal{C}\left( ... | Yes |
Lemma 7.91. \( \ker \nu \cap {G}_{s} = T \) for each \( s \in S \) . | Proof. It is immediate from (7.15) that \( \ker \nu = N \cap \mathop{\bigcap }\limits_{{\alpha \in \Phi }}{N}_{G}\left( {U}_{\alpha }\right) \) . Hence\n\n\[ \ker \nu \cap {G}_{s} \leq {N}_{{G}_{s}}\left( {U}_{s}\right) \cap {N}_{{G}_{s}}\left( {U}_{-s}\right) = T \]\n\nby statement (15) in Section 7.8.2. The opposite ... | Yes |
Lemma 7.93. \( N \cap {N}_{G}\left( {U}_{ + }\right) = T \), and hence \( N \cap {B}_{ + } = T \) . | Proof. It suffices to show that\n\n\[ N \cap {N}_{G}\left( {U}_{ + }\right) \leq T = \ker \nu \]\n\nGiven \( n \in N \cap {N}_{G}\left( {U}_{ + }\right) \), set \( w \mathrel{\text{:=}} \nu \left( n\right) \) . For any \( \alpha \in {\Phi }_{ + } \), we then have\n\n\[ {U}_{w\alpha } = n{U}_{\alpha }{n}^{-1} \leq {U}_{... | No |
Corollary 7.94. \( {U}_{-s} \cap {B}_{ + } = \{ 1\} \) for all \( s \in S \) . | Proof. Suppose there exists an element \( v \in {U}_{-s}^{ * } \cap {B}_{ + } \), and consider the element \( m\left( v\right) \in {U}_{s}v{U}_{s} \) that interchanges \( {U}_{\pm s} \) by conjugation; see Section 7.8.2, statement (1). The construction of \( m\left( v\right) \) shows that it is in \( N \) ; hence it is... | Yes |
Lemma 7.98. \( {B}_{ + }s{B}_{ + } = {U}_{s}s{B}_{ + } \) for any \( s \in S \) . | Proof. We have \( {B}_{ + }s{B}_{ + } = {U}_{ + }{Ts}{B}_{ + } = {U}_{ + }{sT}{B}_{ + } = {U}_{ + }s{B}_{ + } \), so we must show that \( {U}_{ + }s{B}_{ + } = {U}_{s}s{B}_{ + } \). In view of the definition of \( {U}_{ + } \), it suffices to show that \( {U}_{\alpha }{U}_{s}s{B}_{ + } \subseteq {U}_{s}s{B}_{ + } \) fo... | Yes |
Corollary 7.99. \( {B}_{ + } \cup {B}_{ + }s{B}_{ + } \) is a subgroup of \( G \) for each \( s \in S \) . | Proof. It suffices to show that \( s{B}_{ + }s{B}_{ + } \subseteq {B}_{ + } \cup {B}_{ + }s{B}_{ + } \) . By the lemma we have\n\n\[ \ns s{B}_{ + }s{B}_{ + } = s{U}_{s}s{B}_{ + } \subseteq {G}_{s}{B}_{ + }, \]\n\nwhere, as usual, \( {G}_{s} \mathrel{\text{:=}} T\left\langle {{U}_{s},{U}_{-s}}\right\rangle \) . We can n... | Yes |
Lemma 7.101. Fix \( w \in W \) .\n\n(1) If \( \alpha ,\beta \in \Phi \left( w\right) \), then \( \left\lbrack {\alpha ,\beta }\right\rbrack \subseteq \Phi \left( w\right) \) . | Proof. (1) This follows easily from the convexity of \( \left\lbrack {\alpha ,\beta }\right\rbrack \) ; see Exercise 7.75. | No |
Lemma 7.102. Fix \( w \in W \) .\n\n(1) Given a reduced decomposition \( w = {s}_{1}\cdots {s}_{n} \), let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be as in part (2) of Lemma 7.101. Then \( {U}_{w} = {U}_{{\alpha }_{1}}\cdots {U}_{{\alpha }_{n}} \) . | Proof. (1) We argue by induction on \( n = l\left( w\right) \) . Assuming, as we may, that \( w \neq \) 1, set \( {w}^{\prime } \mathrel{\text{:=}} {s}_{1}\cdots {s}_{n - 1} \) and \( s \mathrel{\text{:=}} {s}_{n} \) . By Lemma 7.101, \( \Phi \left( w\right) = \Phi \left( {w}^{\prime }\right) \cup \left\{ {\alpha }_{n}... | Yes |
Lemma 7.104. \( {B}_{ + }w{B}_{ + } = {U}_{w}w{B}_{ + } \) for all \( w \in W \) . | Proof. We argue by induction on \( l\left( w\right) \), which may be assumed \( > 0 \) . It suffices to show that \( {B}_{ + }w \subseteq {U}_{w}w{B}_{ + } \) . Writing \( w = s{w}^{\prime } \) with \( l\left( w\right) = l\left( {w}^{\prime }\right) + 1 \), we have\n\n\[ \n{B}_{ + }w = {B}_{ + }s{w}^{\prime } \n\]\n\n\... | Yes |
Lemma 7.105. If \( H \) is a group generated by two nilpotent normal subgroups, then \( H \) is nilpotent. | Proof. Call the two subgroups \( M, N \), so that \( H = {MN} \) . Let \( M \) and \( N \) be nilpotent of class \( c \) and \( d \), respectively. We will show by induction on \( c + d \) that \( H \) is nilpotent of class \( \leq c + d \) . We may assume \( c, d > 0 \) (i.e., \( M \) and \( N \) are both nontrivial).... | Yes |
Proposition 7.106. If \( {U}_{\alpha } \) is nilpotent for all \( \alpha \in \Phi \), then \( {U}_{w} \) is nilpotent for all \( w \in W \) . In particular, \( {U}_{ + } \) is nilpotent. | Proof. The last assertion follows from the first because \( {U}_{ + } = {U}_{{w}_{0}} \) . [Every wall separates 1 from \( {w}_{0} \) .] We prove the first assertion by induction on \( l\left( w\right) \), which may be assumed \( \geq 2 \) . Write \( w = s{w}^{\prime }t \) with \( s, t \in S \) and \( l\left( w\right) ... | Yes |
Proposition 7.108. Given \( w \in W \), one obtains a presentation for \( {U}_{w} \) by taking generators and relations for the groups \( {U}_{\alpha } \) with \( \alpha \in \Phi \left( w\right) \) and adding the commutator relations (7.18) for \( \alpha ,\beta \in \Phi \left( w\right) ,\alpha \neq \beta \) . | Sketch of proof. Choose a reduced decomposition \( w = {s}_{1}\cdots {s}_{n} \), let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be as in Lemma 7.101(2), and set \( {U}_{i} \mathrel{\text{:=}} {U}_{{\alpha }_{i}} \) for \( i = 1,\ldots, n \) . In view of Remark 7.103(b), the proof of Proposition 7.106 gives us a semidire... | No |
Theorem 7.115. \( \\left( {G,{B}_{ + }, N, S}\\right) \) is a Tits system with Weyl group | Proof. We have already shown that \( {B}_{ + } \cap N = T \) (Lemma 7.93), and we have exhibited a homomorphism \( \\nu : N \rightarrow W \) with kernel \( T \) (Lemma 7.92), so that \( N/T \\cong W \) . The canonical generating set of \( N/T \) corresponding to \( S \) is \( \\{ \\widetilde{s}T \\mid s \\in S\\} \), w... | Yes |
Corollary 7.117. \( {N}_{G}\left( {U}_{ + }\right) = {B}_{ + } \) and \( {N}_{G}\left( {U}_{ - }\right) = {B}_{ - } \) . | Proof. \( T \) normalizes \( {U}_{ + } \), so \( {B}_{ + } = T{U}_{ + } \trianglelefteq {N}_{G}\left( {U}_{ + }\right) \) . The latter is therefore a standard parabolic subgroup. If it were strictly bigger than \( {B}_{ + } \), it would contain \( \widetilde{s} \) for some \( s \in S \) . But then we would have \( {U}_... | Yes |
Suppose \( \left( {G,{\left( {U}_{\alpha }\right) }_{\alpha \in \Phi }, T}\right) \) is the RGD system associated to a strictly Moufang building \( \Delta \) as in Example 7.83. Then \( \Delta \left( {G,{B}_{ + }}\right) \) is canonically isomorphic to the building \( \Delta \) that we started with. | This follows from Proposition 7.28 and the fact that \( {B}_{ + } \) is the stabilizer of the fundamental chamber \( C \) of \( \Delta \) . We proved this fact in Corollary 7.65(2). [Note that \( {B}_{ + } \), in the context of that corollary, was defined to be the stabilizer of \( C \) ; but the corollary implies that... | Yes |
Corollary 7.120. For each \( w \in W,{U}_{w} \) acts simply transitively on \( {\mathcal{C}}_{w} \mathrel{\text{:=}} \) \( \left\{ {C \in \mathcal{C}\left( \Delta \right) \mid \delta \left( {{C}_{0}, C}\right) = w}\right\} . \) | Note that the stabilizer in \( {U}_{w} \) of \( w{B}_{ + } \) is \( {U}_{w} \cap w{B}_{ + }{w}^{-1} \) . So the triviality of this stabilizer can be rewritten as\n\n\[ \n{U}_{w} \cap w{B}_{ + }{w}^{-1} = \{ 1\}\n\]\n\n(7.19)\n\nfor any \( w \in W \) . Equivalently: | No |
Corollary 7.125. If \( G = \left\langle {{U}_{\alpha } \mid \alpha \in \Phi }\right\rangle \), then\n\n\[ T = \left\langle {m{\left( u\right) }^{-1}m\left( v\right) \mid u, v \in {U}_{s}^{ * }, s \in S}\right\rangle . \]\n\nConsequently, \( N = \left\langle {m\left( u\right) \mid u \in {U}_{s}^{ * }, s \in S}\right\ran... | Proof. Let \( {T}_{1} \mathrel{\text{:=}} \left\langle {m{\left( u\right) }^{-1}m\left( v\right) \mid u, v \in {U}_{s}^{ * }, s \in S}\right\rangle \) . Then, under our assumption that \( G \) is generated by the \( {U}_{\alpha } \), we still have an RGD system if we replace \( T \) by \( {T}_{1} \) . So \( {T}_{1} = T... | Yes |
Proposition 7.127. Let \( {G}_{1} \mathrel{\text{:=}} \left\langle {{U}_{\alpha } \mid \alpha \in \Phi }\right\rangle \) . (1) \( \ker \phi = {C}_{G}\left( {G}_{1}\right) \), where the latter is the centralizer of \( {G}_{1} \) in \( G \) . Moreover, \[ Z\left( G\right) \leq \ker \phi \leq T \] where \( Z\left( G\right... | Proof. (1) We have already observed (Remark 7.124(a)) that \( T \) is the set of \( g \in G \) such that \( g \) fixes \( {\sum }_{0} \) pointwise. So \( \ker \phi \leq T \) . In particular, \( \ker \phi \) normalizes \( {U}_{\alpha } \) for all \( \alpha \in \Phi \) . Since \( \ker \phi \) is normal in \( G \), it fol... | Yes |
Let \( G = {\mathrm{{GL}}}_{n}\left( D\right) \) , where \( n \geq 2 \) and \( D \) is a division ring. Let \( W \) be the symmetric group \( {S}_{n} \) on \( n \) letters with its standard generating set \( S = \left\{ {{s}_{1},\ldots ,{s}_{n - 1}}\right\} \) consisting of adjacent transpositions. Let \( \sum = \sum \... | Let \( T \) be the group \( {T}_{n}\left( D\right) \) of diagonal matrices, i.e.,\n\n\[ T \mathrel{\text{:=}} \left\{ {\operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \mid {\lambda }_{i} \in {D}^{ * }}\right\} . \]\n\nFor \( 1 \leq i, j \leq n \) with \( i \neq j \), let \( {U}_{{\alpha }_{ij}}... | Yes |
Definition 7.137. Let \( V \) be a \( K \) -vector space, possibly infinite-dimensional. Fix \( \epsilon = \pm 1 \) . A function \( B : V \times V \rightarrow K \) is said to be \( \left( {\sigma ,\epsilon }\right) \) -Hermitian if it is linear in the first variable and satisfies\n\n\[ B\left( {y, x}\right) = {\epsilon... | Note that (7.27) implies that \( B \) is \( \sigma \) -linear in the second variable, i.e., it is additive and satisfies \( B\left( {x,{\lambda y}}\right) = {\lambda }^{\sigma }B\left( {x, y}\right) \) for \( \lambda \in K \) and \( x, y \in V \) . | No |
Lemma 8.3. Let \( {\left( {X}_{\alpha }\right) }_{\alpha \in \Phi } \) be a system of pre-root groups in \( {\operatorname{Aut}}_{0}\mathcal{C} \) . Then for each \( \alpha = \left( {{\alpha }_{ + },{\alpha }_{ - }}\right) \) in \( \Phi \) there is an element \( {n}_{\alpha } \in \left\langle {{X}_{\alpha },{X}_{-\alph... | Proof. Let \( C \) and \( D \) be the adjacent chambers in \( {\sum }_{ + } \) with \( C \in {\alpha }_{ + } \) and \( D \in - {\alpha }_{ + } \) . Let \( {C}^{\prime } \mathrel{\text{:=}} {\operatorname{op}}_{\sum }C \) and \( {D}^{\prime } \mathrel{\text{:=}} {\operatorname{op}}_{\sum }D \) . Let \( \mathcal{P} \) (r... | Yes |
Corollary 8.8. \( U \) acts transitively on the set of chambers opposite \( {C}_{ + } \) . | Proof. Given \( D \) op \( {C}_{ + } \), we can find \( u \in U \) with \( {uD} \in {\sum }_{ - } \) . Then \( {uD} \) op \( u{C}_{ + } \) \( = {C}_{ + } \) ; hence \( {uD} = {C}_{ - } \) . | Yes |
Lemma 8.12. There is an order-reversing \( 1 - 1 \) correspondence between convex pairs \( \mathcal{M} \) and convex subsets of \( \Phi \) . It is given by \( \mathcal{M} \mapsto \Psi \left( \mathcal{M}\right) \) | and its inverse is given by \( \Psi \mapsto \mathop{\bigcap }\limits_{{\alpha \in \Psi }}\alpha \) . | Yes |
Given two chambers \( C, D \in {\sum }_{ + } \), set\n\n\[ \Phi \left( {C, D}\right) \mathrel{\text{:=}} \left\{ {\alpha \in \Phi \mid C \in {\alpha }_{ + }, D \notin {\alpha }_{ + }}\right\} .\n\] | Then \( \Phi \left( {C, D}\right) = \Psi \left( \mathcal{M}\right) \), where \( {\mathcal{M}}_{ + } = \{ C\} \) and \( {\mathcal{M}}_{ - } = \left\{ {D}^{\prime }\right\} \), with \( {D}^{\prime } \mathrel{\text{:=}} \) \( {\operatorname{op}}_{\sum }D \) . Hence \( \Phi \left( {C, D}\right) \) is a convex set of roots.... | Yes |
(1) For any twin root \( \\alpha \) and any \( g \\in {\\operatorname{Aut}}_{0}\\mathcal{C} \) ,\n\n\[ g{U}_{\\alpha }{g}^{-1} = {U}_{g\\alpha }.\n\]\n\n(2) Let \( \\alpha \) be a twin root and let \( \\mathcal{P} \) be a boundary panel of \( \\alpha \) . Then the root group \( {U}_{\\alpha } \) acts on the sets \( \\m... | Proof. This is similar to the proof of Lemma 7.25. For (2) one uses Lemma 5.198 instead of Lemma 4.118, and for (3) one needs to recall that the rigidity theorem is valid for twin buildings (see Remark 5.208). | No |
Proposition 8.25. Let \( \mathcal{C} = \left( {{\mathcal{C}}_{ + },{\mathcal{C}}_{ - }}\right) \) be a thick, irreducible,2-spherical twin building of rank at least 3, and let \( \alpha = \left( {{\alpha }_{ + },{\alpha }_{ - }}\right) \) be a twin root of \( \mathcal{C} \) . Assume that \( g \) is an automorphism of \... | Proof. Given an interior panel \( {\mathcal{P}}_{ - } \) of \( \alpha \) contained in \( {\mathcal{C}}_{ - } \), we have to show that \( g \) fixes all chambers in \( {\mathcal{P}}_{ - } \) . In view of Proposition 7.58, it suffices to prove this for \( {\mathcal{P}}_{ - } \) \ | No |
Theorem 8.27. If \( \mathcal{C} \) is a thick, irreducible, 2-spherical twin building of rank at least 3 that satisfies condition (co) of Section 5.11, then \( \mathcal{C} \) is Moufang. | Proof. Given a twin root \( \alpha \) of \( \mathcal{C} \) and two twin apartments \( \sum \) and \( {\sum }^{\prime } \) containing \( \alpha \), we need to find an element \( g \) in the root group \( {U}_{\alpha } \) with \( {g\sum } = {\sum }^{\prime } \) . By Corollary 5.215, there is an automorphism \( g \) of \(... | Yes |
Corollary 8.28. If \( \mathcal{C} \) is as in the theorem, then it is the twin building associated to a twin BN-pair. | Proof. This follows from the theorem together with Proposition 8.19. | No |
Corollary 8.29. If \( \mathcal{C} \) is as in the theorem, then all rank-2 residues of \( \mathcal{C} \) are Moufang. In particular, the Coxeter matrix associated to \( \mathcal{C} \) has all of its entries \( m\left( {s, t}\right) \) in \( \{ 1,2,3,4,6,8\} \) . | Proof. The first assertion follows from the theorem and Proposition 8.21. The second assertion now follows from the result of Tits cited in Remark 7.60. | No |
Proposition 8.39. For all \( \alpha \neq \beta \) in \( \Phi \) such that the pair \( \{ \alpha ,\beta \} \) is prenilpo-tent, | \[ \left\lbrack {{U}_{\alpha },{U}_{\beta }}\right\rbrack \leq {U}_{\left( \alpha ,\beta \right) } \] | No |
Corollary 8.40. Let \( {\alpha }_{1},\ldots ,{\alpha }_{d} \) be the sequence of twin roots associated to a minimal gallery in \( {\sum }_{ + } \) as in Example 8.13, and set \( {U}_{i} \mathrel{\text{:=}} {U}_{{\alpha }_{i}} \) for \( i = 1,\ldots, d \) . If \( d \geq 2 \), then | \[ \left\lbrack {{U}_{1},{U}_{d}}\right\rbrack \leq {U}_{2}\cdots {U}_{d - 1} \] | Yes |
If \( W \) is finite, then \( \{ \alpha ,\beta \} \) is prenilpotent if and only if \( \beta \neq - \alpha \) . | (This follows for instance from Lemma 3.53 and part (3) of Lemma 8.42.) | No |
Suppose \( W \) is a Euclidean reflection group, so that \( \sum \) can be identified with a Euclidean space \( V \) decomposed into simplices by affine hyperplanes. The closed half-spaces determined by the hyperplanes can be identified with the roots. For each root \( \alpha \), let \( {e}_{\alpha } \) be the unit vec... | (1) \( {e}_{\alpha } = - {e}_{\beta } \) and \( \alpha = - \beta \) . Then \( \alpha \cap \beta = \left( {-\alpha }\right) \cap \left( {-\beta }\right) = \partial \alpha \) . Neither intersection contains a chamber, and \( \{ \alpha ,\beta \} \) is not prenilpotent.\n\n(2) \( {e}_{\alpha } = - {e}_{\beta } \) but \( \a... | Yes |
Lemma 8.45. Let \( \{ \alpha ,\beta \} \) be a prenilpotent pair of roots. Then one of the following holds:\n\n(a) \( \{ \alpha ,\beta \} \) is nested, say \( \alpha \subseteq \beta \), in which case\n\n\[ \left\lbrack {\alpha ,\beta }\right\rbrack = \{ \gamma \in \Phi \mid \alpha \subseteq \gamma \subseteq \beta \} .\... | Proof. If each of the four intersections \( \left( {\pm \alpha }\right) \cap \left( {\pm \beta }\right) \) contains a chamber, then we are in case (b). Indeed, the first assertion follows from Lemma 3.164, and the rest is proved exactly as in the proof of Lemma 7.81. [One of course uses the definition of prenilpotence ... | Yes |
Lemma 8.50. \( {B}_{ + }s{B}_{ + } = {U}_{s}s{B}_{ + } \) for all \( s \in S \) . | Proof. As before, it suffices to show that \( {U}_{\alpha }{U}_{s}s{B}_{ + } \subseteq {U}_{s}s{B}_{ + } \) for all \( \alpha \neq {\alpha }_{s} \) in \( {\Phi }_{ + } \) . If the pair \( \left\{ {\alpha ,{\alpha }_{s}}\right\} \) is prenilpotent, then the proof of Lemma 7.98 remains valid. So suppose \( \left\{ {\alph... | Yes |
Proposition 8.56. The system of groups \( {\left( {U}_{\alpha }\right) }_{\alpha \in \Phi } \) acting on \( {\mathcal{C}}_{ + } \) satisfies the following properties:\n\n(1) For each \( \alpha \in \Phi ,{U}_{\alpha } \) fixes the root \( \alpha \) of \( {\sum }_{ + } \) pointwise.\n\n(2) For each \( \alpha \in \Phi \) ... | Proof. Parts (1) and (2) are proved as in the proof of Theorem 7.116, and (3) is just one of our axioms. | No |
Lemma 8.58. If a building \( \mathcal{C} \) admits a system \( {\left( {U}_{\alpha }\right) }_{\alpha \in \Phi } \) of subgroups of \( {\operatorname{Aut}}_{0}\mathcal{C} \) satisfying properties (1)-(3) of Proposition 8.56, then for every \( \alpha \in \Phi \) and every interior panel \( \mathcal{P} \) of \( \alpha \)... | Proof. The proof of Proposition 7.79 goes through with one minor change: In that proof we had a pair of opposite roots determined by an interior panel of \( \alpha \), and we arbitrarily called one of them \( \beta \) and applied the commutator relations. In the present setting we instead choose \( \beta \) so that the... | No |
Proposition 8.59. Fix \( w \in W \) .\n\n(1) Given a reduced decomposition \( w = {s}_{1}\cdots {s}_{n} \), let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be as in Lemma 7.101 (2). Then every \( u \in {U}_{w} \) admits a unique decomposition \( u = {u}_{1}\cdots {u}_{n} \) with \( {u}_{i} \in {U}_{{\alpha }_{i}} \) for ... | Proof. We have \( {U}_{w} = {U}_{{\alpha }_{1}}\cdots {U}_{{\alpha }_{n}} \) by the analogue of Lemma 7.102(1). So (1) and the first assertion of (2) will follow if we can prove the following claim: Given any chamber \( C \in {\mathcal{C}}_{w} \), there are unique elements \( {u}_{i} \in {U}_{{\alpha }_{i}}\left( {1 \l... | Yes |
Lemma 8.65. For all \( w,{w}_{1},{w}_{2} \in W \), we have:\n\n(1) \( {U}_{w}^{\prime } = w{U}_{{w}^{-1}}{w}^{-1} \) .\n\n(2) If \( l\left( {{w}_{1}{w}_{2}}\right) = l\left( {w}_{1}\right) + l\left( {w}_{2}\right) \), then \( {U}_{{w}_{1}{w}_{2}}^{\prime } = {U}_{{w}_{1}}^{\prime }{w}_{1}{U}_{{w}_{2}}^{\prime }{w}_{1}^... | Proof. (1) follows immediately from the definition of \( {U}_{w}^{\prime } \) and the easily checked equation \( - \Phi \left( w\right) = {w\Phi }\left( {w}^{-1}\right) \) .\n\n(2) is already known for the groups \( {U}_{w} \) (see Lemma 7.102 and the remarks in Section 8.6.3). The proof does not require (RGD3), so it ... | Yes |
Lemma 8.67. The groups \( {U}_{w}^{\prime }\left( {w \in W}\right) \) generate \( {U}_{ - } \) . | Proof. Given \( \alpha \in {\Phi }_{ + } \), choose any \( w \in - \alpha \) . Then we have \( \alpha \in \Phi \left( w\right) \) (see Definition 7.100); hence \( {U}_{\alpha }^{\prime } \leq {U}_{w}^{\prime } \) . Since \( {U}_{ - } = \left\langle {{U}_{\alpha }^{\prime } \mid \alpha \in {\Phi }_{ + }}\right\rangle \)... | No |
Lemma 8.68. The chamber system \( {\mathcal{C}}^{\prime } \) is connected. | Proof. In the following, we will use the notation \( C - D \) as a schematic representation of a gallery between two chambers \( C, D \in {\mathcal{C}}^{\prime } \). We will also identify \( {U}_{ - } \) with \( {U}_{ - }/{U}_{w}^{\prime } \) when \( w = 1 \), i.e., we will write \( x \) instead of \( x\{ 1\} \) for \(... | Yes |
Lemma 8.69. \( \kappa \) is a \( {U}_{ - } \) -equivariant morphism of chamber systems over \( S \) . | Proof. \( {U}_{ - } \) -equivariance is obvious. To prove that \( \kappa \) preserves \( s \) -equivalence for \( s \in S \), suppose \( x{U}_{v}^{\prime }{ \sim }_{s}y{U}_{w}^{\prime } \) for some \( x, y \in {U}_{ - } \) and \( v, w \in W \) . In other words,\n\n\[ v \in \{ w,{ws}\} \;\text{ and }\;{y}^{-1}x \in {U}_... | Yes |
Lemma 8.70. If \( w,{w}_{1},{w}_{2} \in W \) are elements such that \( {U}_{w}^{\prime }{w}_{1}{B}_{ + } \cap {w}_{2}{B}_{ + } \neq \varnothing \) , then \( {w}_{1} = {w}_{2} \) . | Proof. By Lemma \( {8.65}\left( 1\right) \), we can rewrite the assumption as\n\n\[ w{U}_{{w}^{-1}}{w}^{-1}{w}_{1}{B}_{ + } \cap {w}_{2}{B}_{ + } \neq \varnothing . \]\n\nThis implies that \( {B}_{ + }{w}^{-1}{w}_{1}{B}_{ + } = {B}_{ + }{w}^{-1}{w}_{2}{B}_{ + } \) and hence, by the Bruhat decomposition, \( {w}^{-1}{w}_... | Yes |
Lemma 8.71. For all \( w,{w}_{1},{w}_{2} \in W \), we have:\n\n(1) \( {U}_{w}^{\prime } \cap {B}_{ + } = \{ 1\} \) .\n\n(2) If \( l\left( {{w}_{1}{w}_{2}}\right) = l\left( {w}_{1}\right) + l\left( {w}_{2}\right) \), then \( {U}_{{w}_{1}{w}_{2}}^{\prime } \cap {w}_{1}{B}_{ + }{w}_{1}^{-1} = {U}_{{w}_{1}}^{\prime } \) .\... | Proof. (1) follows from Lemma 8.65(1) and equation (8.12) (with \( w \) replaced by \( \left. {w}^{-1}\right) \) .\n\n(2) Lemma 8.65 yields \( {w}_{1}^{-1}{U}_{{w}_{1}{w}_{2}}^{\prime }{w}_{1} = {U}_{{w}_{1}^{-1}}{U}_{{w}_{2}}^{\prime } \) . Since \( {U}_{{w}_{1}^{-1}} \leq {U}_{ + } \), we obtain\n\n\[ \n{w}_{1}^{-1}{... | Yes |
Lemma 8.72. Let \( J \) be a spherical subset of \( S \) . Then we have:\n\n(1) The \( {U}_{ - } \) -orbit of any \( J \) -residue in \( {\mathcal{C}}^{\prime } \) contains a \( J \) -residue of the form \( {R}_{J}\left( {U}_{w}^{\prime }\right) \), where \( w \) is a \( J \) -reduced element of \( W \) .\n\n(2) If \( ... | Proof. (1) By the definition of \( {\mathcal{C}}^{\prime } \) it is clear that any \( J \) -residue is \( {U}_{ - } \) -equivalent to one of the form \( {R}_{J}\left( {U}_{{w}^{\prime }}^{\prime }\right) \) for some \( {w}^{\prime } \in W \) . Denote by \( w \) the \( J \) -reduced element of \( {w}^{\prime }{W}_{J} \)... | Yes |
Corollary 8.74. The map \( \kappa \) is an isomorphism of chamber systems over \( S \) . | We cooked up \( {\mathcal{C}}^{\prime } \) so that the chamber \( {U}_{w}^{\prime } \), which maps to \( w{B}_{ + } \), would have stabilizer \( {U}_{w}^{\prime } \) . Since \( \kappa \) is \( {U}_{ - } \) -equivariant and the stabilizer of \( w{B}_{ + } \) in \( {U}_{ - } \) is \( {U}_{ - } \cap w{B}_{ + }{w}^{-1} \),... | No |
Corollary 8.77. For all \( s \in S,{U}_{s} \nleqslant {U}_{ - } \) . | So the stronger axiom \( \left( {\mathrm{{RGD3}}}^{\prime \prime }\right) \) mentioned in Remark 8.48(a) is a consequence of the RGD axioms we started with. Replacing (RGD3) by (RGD3 \( {}^{\prime \prime } \) ), we thus get an equivalent set of axioms. Since there is now symmetry with respect to + and -, the system \( ... | No |
Theorem 8.80. \( \\left( {G,{B}_{ + },{B}_{ - }, N, S}\\right) \) is a saturated twin Tits system with Weyl group \( N/T \\cong W \) . | Proof. We have to check the following (see Definitions 6.78 and 6.84):\n\n(a) \( N \\cap {B}_{ + } = N \\cap {B}_{ - } = {B}_{ + } \\cap {B}_{ - } \\trianglelefteq N \) .\n\nWe have shown that all intersections are equal to \( T \), and we already know that \( T \\trianglelefteq N \) .\n\n(b) \( \\left( {G,{B}_{ + }, N... | Yes |
Theorem 8.84. One obtains a presentation of \( {U}_{ + } \) by taking generators and relations for each \( {U}_{\alpha }\left( {\alpha \in {\Phi }_{ + }}\right) \) and adding the commutator relations (8.22). | We will deduce this from Theorem 8.85 below, which exhibits \( {U}_{ + } \) as a direct limit of the subgroups \( {U}_{w}\left( {w \in W}\right) \) defined in Section 8.6.3. We assume that the reader is familiar with the notion of direct limit, as defined for instance in [217, Section I.1.1]. Consider the system consis... | No |
Theorem 8.85. The canonical map \( \mathop{\lim }\limits_{\overrightarrow{w}}{U}_{w} \rightarrow {U}_{ + } \) is an isomorphism. | Proof of Theorem 8.84. Let \( V \) be the group defined by the presentation in the statement of the theorem. It comes equipped with maps \( {i}_{\alpha } : {U}_{\alpha } \rightarrow V \) \( \left( {\alpha \in {\Phi }_{ + }}\right) \), and there is a surjection \( p : V \rightarrow {U}_{ + } \) such that the composite \... | Yes |
Lemma 8.88. The chamber system \( \widetilde{\mathcal{C}} \) is connected. | Now define \( \widetilde{\kappa } : \widetilde{\mathcal{C}} \rightarrow {\mathcal{C}}_{ + } \) by \( \widetilde{\kappa }\left( {x{\widetilde{U}}_{w}}\right) \mathrel{\text{:=}} p\left( x\right) w{B}_{ + } \) for \( x \in \widetilde{U} \) and \( w \in W \) . This is well defined because \( p\left( {\widetilde{U}}_{w}\ri... | No |
Lemma 8.89. \( \widetilde{\kappa } \) is a \( \widetilde{U} \) -equivariant morphism of chamber systems over \( S \) . | Proof. Note that \( {y}^{-1}x \in {\widetilde{U}}_{w} \cup {\widetilde{U}}_{ws} \Rightarrow p{\left( y\right) }^{-1}p\left( x\right) \in {U}_{w}^{\prime } \cup {U}_{ws}^{\prime } \) . The proof of Lemma 8.69 now goes through with no change. | No |
Corollary 8.90. Given \( w,{w}_{1},{w}_{2} \in W \) satisfying\n\n\[ l\left( {w{w}_{1}{w}_{2}}\right) = l\left( w\right) + l\left( {w}_{1}\right) + l\left( {w}_{2}\right) ,\]\n\nwe have\n\n\[ {\widetilde{U}}_{w{w}_{1}} \cap w{\widetilde{U}}_{{w}_{1}{w}_{2}}{w}^{-1} = w{\widetilde{U}}_{{w}_{1}}{w}^{-1}. \] | Proof. Under the hypothesis of the corollary, all the groups occurring in (8.18) are subgroups of \( {U}_{w{w}_{1}{w}_{2}}^{\prime } \) . Apply \( {i}_{w{w}_{1}{w}_{2}} \) to both sides to obtain (8.23). | No |
Lemma 8.91. Let \( J \) be a spherical subset of \( S \) . Then we have:\n\n(1) The \( \widetilde{U} \) -orbit of any \( J \) -residue in \( \widetilde{\mathcal{C}} \) contains a \( J \) -residue of the form \( {R}_{J}\left( {\widetilde{U}}_{w}\right) \), where \( w \) is a \( J \) -reduced element of \( W \) .\n\n(2) ... | Proof. The proof of Lemma 8.72 goes through with the following minor changes. First, the definition of \( {f}_{n} \) in (3) uses the remarks we made about conjugation above. It is well defined and surjective as in the proof of Lemma 8.72. The proof of injectivity is also the same, except that we apply Corollary 8.90 in... | Yes |
Proposition 8.92. \( \widetilde{\kappa } : \widetilde{\mathcal{C}} \rightarrow {\mathcal{C}}_{ + } \) is a 2-covering. | Proof. Step (a) is the same as in the proof of Proposition 8.73 (since \( p\left( {\widetilde{U}}_{s}\right) = \) \( \left. {U}_{s}^{\prime }\right) \) . Step (b) is also basically the same: We just start with \( x, y \in {\widetilde{U}}_{J} \) but then derive the same conclusions from \( p\left( x\right) v{B}_{ + } = ... | Yes |
Corollary 8.93. \( \widetilde{\kappa } \) is an isomorphism of chamber systems over \( S \) . | This quickly yields what we have been aiming for:\n\nProof of Theorem 8.85. It suffices to show that \( p : \widetilde{U} \rightarrow {U}_{ - } \) is injective. Given \( x \in \ker p \), we have \( \widetilde{\kappa }\left( x\right) = p\left( x\right) {B}_{ + } = {B}_{ + } = \widetilde{\kappa }\left( 1\right) \) . So t... | No |
Corollary 10.11. With \( W \) as in case (2) of the theorem, the stabilizers of the points of \( \bar{C} \) are precisely the proper standard subgroups of \( W \), and all of these are finite. | Proof. Given \( x \in \bar{C} \), let \( {S}_{x} \) be as in statement (f) in Section 10.1.3. If \( A \) is the face of \( C \) containing \( x \), then \( {S}_{x} \) can also be described as the set of \( s \in S \) such that the \( {H}_{s} \) -component of the sign vector of \( A \) is 0 . Since \( C \) is a simplex,... | Yes |
Proposition 10.17. There exist points \( x \in V \) such that the stabilizer \( {W}_{x} \) maps isomorphically onto \( \bar{W} \), so that \( W = T \rtimes {W}_{x} \) . | Proof. Let \( \overline{\mathcal{H}} \) be the set of linear hyperplanes \( H \) such that \( H \) is parallel to some element of \( \mathcal{H} \) . Then \( \bar{W} \) is generated by \( \left\{ {{s}_{H} \mid H \in \overline{\mathcal{H}}}\right\} \) . Since \( \bar{W} \) is essential, it follows from Chapter 1 that \(... | Yes |
Proposition 10.19. A point \( x \in V \) is special if and only if every hyperplane in \( \mathcal{H} \) is parallel to a hyperplane in \( \mathcal{H} \) passing through \( x \) . | Proof. If the condition on hyperplanes is satisfied, then the map \( {W}_{x} \rightarrow \bar{W} \) is surjective and hence an isomorphism, as in the proof of Proposition 10.17. So \( x \) is special. For the converse, it is convenient to assume (without loss of generality) that \( x = 0 \) . Suppose, then, that 0 is a... | Yes |
Lemma 10.21. \( L \) is a discrete subgroup of the additive group of \( V \), i.e., there is a neighborhood \( U \) of 0 in \( V \) such that \( U \cap L = \{ 0\} \) . The quotient group \( V/L \) , with the quotient topology, is compact. | Proof. Pick a chamber \( C \) and a point \( y \in C \) . Since the transforms \( {wC} \) for \( w \in W \) are all disjoint, the same is true of the translates \( C + l \) for \( l \in L \) . So if we set \( U \mathrel{\text{:=}} \{ v \in V \mid y + v \in C\} \), then \( U \) is a neighborhood of 0 in \( V \) such tha... | Yes |
Lemma 10.22. If \( L \) is a discrete subgroup of the additive group of a finite-dimensional vector space \( V \), then \( L = \mathbb{Z}{e}_{1} \oplus \cdots \oplus \mathbb{Z}{e}_{r} \) for some linearly independent vectors \( {e}_{1},\ldots ,{e}_{r} \). If, in addition, \( V/L \) is compact, then \( L \) is a lattice... | Proof. The second assertion follows immediately from the first. The following proof of the first assertion is taken from Pontryagin [189, Chapter 3, Section 19, Example 33], where a more general result is proved. We argue by induction on \( \dim V \). If \( L = 0 \) there is nothing to prove, so assume \( L \neq 0 \). ... | Yes |
Theorem 10.30. Every Euclidean reflection group with the origin as a special point is the affine Weyl group of an irreducible root system. | For more information about root systems and affine Weyl groups, see Bourbaki [44] or Humphreys [133]. We close our discussion by returning to the description of a typical \( W \) -chamber \( C \) (Section 10.1.5), and restating it in the language of root systems.\n\nWe may assume that the origin is one of the vertices ... | No |
Let \( \Phi \) be the root system of type \( {\mathrm{A}}_{n - 1} \), with fundamental \( \bar{W} \) -chamber | The simple roots are \( {\alpha }_{i} \mathrel{\text{:=}} {e}_{i} - {e}_{i + 1}\left( {i = 1,\ldots, n - 1}\right) \), and the highest root is\n\n\[ \widetilde{\alpha } = {\alpha }_{1} + \cdots + {\alpha }_{n - 1} = {e}_{1} - {e}_{n}. \]\n\nThe resulting \( W \) -chamber as in Proposition 10.31 is gotten by adjoining t... | Yes |
Proposition 10.40. The bilinear form \( B \) is positive semidefinite with a 1-dimensional radical, spanned by a vector \( v = \mathop{\sum }\limits_{{s \in S}}{\lambda }_{s}{e}_{s} \) such that \( {\lambda }_{s} > 0 \) for all \( s \) . | Proof. We may assume that \( W \) is given as a Euclidean reflection group acting on a Euclidean vector space \( {V}^{\prime } \), and that \( S \) is the set of reflections with respect to the walls of a fundamental chamber \( {C}^{\prime } \) . Let \( {\left( {e}_{s}^{\prime }\right) }_{s \in S} \) be the canonical u... | Yes |
Lemma 10.41. The generator \( s \) acts on \( {E}_{0} \) as the orthogonal reflection with respect to \( {\eta }_{s}{}^{ \bot } \), where\n\n\[ \n{\eta }_{s}{}^{ \bot } = \left\{ {\xi \in {E}_{0} \mid {B}^{\prime \prime }\left( {\xi ,{\eta }_{s}}\right) = 0}\right\} = \left\{ {\xi \in {E}_{0} \mid \left\langle {\xi ,{e... | We claim that any \( s \in S \) acts on \( E \) as the reflection with respect to the hyperplane \( {H}_{s} \mathrel{\text{:=}} \left\{ {\xi \in E \mid \left\langle {\xi ,{e}_{s}}\right\rangle = 0}\right\} \) . Note first that \( {H}_{s} \) is indeed an affine hyperplane in \( E \) . To see this, we must check that the... | Yes |
Any Euclidean Coxeter system \( \left( {W, S}\right) \) has a canonical realization as a Euclidean reflection group, acting on an affine hyperplane \( E \) in \( {V}^{ * } \), where \( V = {\mathbb{R}}^{S} \). The cells into which \( E \) is decomposed by the reflecting hyperplanes are precisely the intersections with ... | The only thing that we have not yet proved is the description of the cells in terms of those of the Tits cone. Recall that the fundamental chamber \( C \) in the Tits cone is a simplicial cone with walls given by \( \left\langle {-,{e}_{s}}\right\rangle = 0\left( {s \in S}\right) \). Thus our hyperplanes \( {H}_{s} \) ... | Yes |
Theorem 10.55. An irreducible Coxeter system \( \left( {W, S}\right) \) arises from a hyperbolic reflection group with simplicial chambers if and only if it has the following properties:\n\n(1) The canonical bilinear form \( B \) on \( {\mathbb{R}}^{S} \) is nondegenerate but not positive definite.\n\n(2) Each proper s... | Note that it suffices to consider \( J \) of the form \( S \smallsetminus \{ s\} \) in (2), in which case \( {W}_{J} \) is a vertex stabilizer. Thus the discussion in the previous subsection explains geometrically why it is of positive type. For a proof of the theorem, see Humphreys [133, Section 6.8] or Bourbaki [44, ... | No |
Theorem 10.56. Let \( D \) be a connected modified diagram with vertex set \( S \) , and let \( A \) be the corresponding symmetric matrix. Then \( D \) arises from a hyperbolic reflection group acting on \( {\mathbb{H}}^{n} \) if and only if the following conditions hold:\n\n(1) Let \( B \) be the bilinear form on \( ... | This criterion, which is due to Vinberg, follows from [274, Theorem 2.1 and Proposition 4.2]. The geometric meaning of (2) is that the fundamental chamber \( P \) must have at least one vertex, possibly at infinity, and the geometric meaning of (3) is that any edge emanating from a vertex must lead to another vertex. N... | Yes |
Proposition 11.5. For any two points \( x, y \) in a \( \operatorname{CAT}\left( 0\right) \) space \( X \), there is a unique geodesic \( \left\lbrack {x, y}\right\rbrack \) joining them. It is characterized by \[ \left\lbrack {x, y}\right\rbrack = \{ z \in X \mid d\left( {x, y}\right) = d\left( {x, z}\right) + d\left(... | Proof. Any geodesic joining \( x \) and \( y \) is contained in the set on the right side of (11.4). It therefore suffices to show that the right side is contained in the left, with \( \left\lbrack {x, y}\right\rbrack \) as in the definition of \ | No |
Proposition 11.7. Let \( X \) be a \( \operatorname{CAT}\left( 0\right) \) space. Then the map \( \left( {x, y, t}\right) \mapsto \) \( {p}_{t}\left( {x, y}\right) \) is continuous as a function of \( x, y, t \) . In particular, \( X \) is contractible. | Proof. Fix \( x, y, t \) and apply the inequality (11.3) to \( z = {p}_{{t}^{\prime }}\left( {{x}^{\prime },{y}^{\prime }}\right) \) for \( \left( {{x}^{\prime },{y}^{\prime },{t}^{\prime }}\right) \) close to \( \left( {x, y, t}\right) \) . Since \( d\left( {z, x}\right) \) is close to \( d\left( {z,{x}^{\prime }}\rig... | Yes |
Proposition 11.8. Consider two triangles in the Euclidean plane, with vertices \( x, y, z \) and \( x, y,{z}^{\prime } \) . Let \( p \) be an arbitrary point on the common side \( \left\lbrack {x, y}\right\rbrack \) . If \( d\left( {{z}^{\prime }, x}\right) \leq d\left( {z, x}\right) \) and \( d\left( {{z}^{\prime }, y... | See Figure 11.2 for an illustration, and see Exercise 11.11 for an alternative proof that is slightly longer but provides better intuition as to why the result is true. | No |
Lemma 11.17. Given \( y \in \bar{C} \) there is a \( \delta > 0 \) with the following property: For any \( x \in X \) of type \( y,\operatorname{st}x \) contains the closed ball of radius \( \delta \) centered at \( x \) . | Proof of the lemma. Choose an apartment \( E \) containing \( C \), and let \( \mathcal{H} \) be the locally finite collection of walls that defines the simplicial decomposition of \( E \) . Let \( \delta \) be the minimum distance from \( y \) to a wall \( H \in \mathcal{H} \) not containing \( y \) . Then for any \( ... | Yes |
Theorem 11.23. Let \( G \) be a group of isometries of a complete \( \operatorname{CAT}\left( 0\right) \) space \( X \) . If \( G \) stabilizes a nonempty bounded subset of \( X \), then \( G \) fixes a point of \( X \) . | The Bruhat-Tits proof of Theorem 11.23 consists in associating to every nonempty bounded subset \( A \subseteq X \) a point \( c = c\left( A\right) \in X \) which, intuitively, is some sort of \ | No |
Theorem 11.26. If \( X \) is a complete \( \operatorname{CAT}\left( 0\right) \) space, then every nonempty bounded subset \( A \) admits one and only one circumcenter. | Proof of Theorem 11.26. For any two points \( x, y \in X \), we can apply the inequality (NC) with \( z \in A \) to get\n\n\[ \n{r}^{2}\left( {m, A}\right) \leq \frac{1}{2}\left( {{r}^{2}\left( {x, A}\right) + {r}^{2}\left( {y, A}\right) }\right) - \frac{1}{4}{d}^{2}\left( {x, y}\right) ,\n\]\n\nwhere \( m \) is the mi... | Yes |
Lemma 11.28. Let \( X \) be a metric space with property (NC), and let \( Y \) be a midpoint-convex subset. Then \( {r}_{Y}\left( A\right) = {r}_{X}\left( A\right) \) for any nonempty bounded subset \( A \subseteq Y \) . | Proof. Given any \( x \in X \) and any \( r > r\left( {x, A}\right) \), we must find \( y \in Y \) such that \( r\left( {y, A}\right) \leq r \) . This is easy if there is a \( y \in Y \) with \( d\left( {x, y}\right) = d\left( {x, Y}\right) \) ; in this case we have \( r\left( {y, A}\right) \leq r\left( {x, A}\right) <... | No |
Lemma 11.32. Let \( G \) be a group with a Euclidean BN-pair. The following conditions on a subset \( F \subseteq G \) are equivalent:\n\n(i) \( F \) is contained in a finite union of double cosets \( {BwB} \).\n\n(ii) For some \( x \in X \), the set \( {Fx} \mathrel{\text{:=}} \{ {gx} \mid g \in F\} \) is a bounded su... | Proof. (i) \( \Rightarrow \) (ii): It suffices to consider the case that \( F \) is a double coset \( {BwB} \) . Let \( C \) be the fundamental chamber of \( X \) ; it is fixed pointwise by \( B \) . Let \( \widetilde{w} \) be a representative of \( w \) in \( N \) . Then for any \( g = b\widetilde{w}{b}^{\prime } \in ... | No |
Theorem 11.34. Let \( G \) be a group with a Euclidean BN-pair. The following conditions on a subgroup \( H \leq G \) are equivalent:\n\n(i) \( H \) is bounded.\n\n(ii) \( H \) fixes a point of \( X \) .\n\n(iii) \( H \) fixes a vertex of \( X \) .\n\n(iv) \( H \) is contained in a maximal parabolic subgroup. | Proof. It is immediate that (iv) \( \Leftrightarrow \) (iii) \( \Leftrightarrow \) (ii) \( \Rightarrow \) (i). The content of the theorem, then, is that (i) \( \Rightarrow \) (ii), and this follows from the fixed-point theorem. | No |
Lemma 11.37. If \( P \) and \( Q \) are distinct maximal parabolics, then \( P \cap Q \) is parabolic if and only if \( P \cap Q \) is a maximal (proper) subgroup of \( P \) . | Proof. Let \( x \) (resp. \( y \) ) be the vertex fixed by \( P \) (resp. \( Q \) ). If \( P \cap Q \) is parabolic, then \( x \) and \( y \) are joinable and \( P \cap Q \) is the stabilizer of the edge \( A \) that they determine. Any subgroup \( {P}^{\prime } \) with \( P > {P}^{\prime } > P \cap Q \) would be parab... | Yes |
Theorem 11.43. Let \( \mathcal{B} \) be the collection of bounded subsets \( Y \subseteq X \) such that \( Y \) is contained in an apartment. Then \( \mathcal{B} \) is independent of the system of apartments \( \mathcal{A} \) . In fact, \( \mathcal{B} \) consists of all subsets \( Y \subseteq X \) such that \( Y \) is ... | The proof of the theorem requires a result about Euclidean Coxeter complexes analogous to Lemma 4.69. We need some terminology before we can state it. | No |
Lemma 11.46. With the notation above, suppose \( {C}_{1} \) and \( {C}_{2} \) are chambers in \( E \) such that \( {\bar{C}}_{1} \) meets \( {\mathfrak{C}}_{1}^{\prime } \) and \( {\bar{C}}_{2} \) meets \( {\mathfrak{C}}_{2}^{\prime } \) . If \( C \) is any chamber of \( E \) such that \( \bar{C} \) meets \( {\overline... | Proof. We must show that no wall (i.e., element of \( \mathcal{H} \) ) separates \( C \) from both \( {C}_{1} \) and \( {C}_{2} \) . Let \( H \) be a wall, defined by a linear equation \( f = c \) . We may choose \( f \) such that \( f > 0 \) on \( \mathfrak{D} \), in which case we will say that the closed half-space \... | Yes |
Corollary 11.47. Let \( {\mathfrak{C}}_{1} \) and \( {\mathfrak{C}}_{2} \) be arbitrary sectors in \( E \) with opposite directions. Given any bounded subset \( Y \) of \( E \), there are subsectors \( {\mathfrak{C}}_{1}^{\prime } \subseteq {\mathfrak{C}}_{1} \) and \( {\mathfrak{C}}_{2}^{\prime } \subseteq {\mathfrak{... | Proof. Let \( \mathfrak{D} \) be the direction of \( {\mathfrak{C}}_{1} \), so that \( - \mathfrak{D} \) is the direction of \( {\mathfrak{C}}_{2} \) . Observe first that we can find sectors \( x + \mathfrak{D} \) and \( y - \mathfrak{D} \) as in Lemma 11.46, with \( Y \subseteq {\mathfrak{C}}_{1} \cap {\mathfrak{C}}_{... | Yes |
Lemma 11.48. Let \( E \) be the geometric realization of a Euclidean Coxeter complex, and let \( C \) and \( D \) be chambers of \( E \) . Then there are sectors \( {\mathfrak{C}}_{1},{\mathfrak{C}}_{2} \) in \( E \) with the following property: For any subsectors \( {\mathfrak{C}}_{1}^{\prime } \subseteq {\mathfrak{C}... | Proof. Choose a point \( x \in D \) and a direction \( \mathfrak{D} \) such that \( x + d \) belongs to \( C \) for some \( d \in \mathfrak{D} \) . Let \( {\mathfrak{C}}_{1} \) be a sector with direction \( \mathfrak{D} \), and let \( {\mathfrak{C}}_{2} \) be a sector with direction \( - \mathfrak{D} \) . Then any subs... | Yes |
Theorem 11.53. Let \( Y \) be a subset of \( X \). Assume either that \( Y \) is convex or that \( Y \) has nonempty interior. If \( Y \) is isometric to a subset of \( {\mathbb{R}}^{n} \), then \( Y \) is contained in an apartment. | To deduce Theorem 11.52 from Theorem 11.53, suppose \( E \) is isometric to \( {\mathbb{R}}^{n} \). Then \( E \) is easily seen to be convex in \( X \); this follows from the characterization of geodesics \( \left\lbrack {x, y}\right\rbrack \) given in Theorem 11.16. So Theorem 11.53 implies that \( E \) is contained i... | No |
Lemma 11.54. Let \( Y \) be a subset of \( X \) that contains a nonempty open subset \( U \) of a chamber \( D \) . If \( Y \) is isometric to a subset of \( {\mathbb{R}}^{n} \), then there is a unique isometry \( \alpha \) from \( Y \) into \( {\left. E\text{such that}\alpha \right| }_{U} = {\left. {\tau }_{D}\right| ... | Proof. Suppose first that there exists an isometry \( \alpha \) from \( Y \) into \( E \) such that \( {\left. \alpha \right| }_{U} = {\left. \tau \right| }_{U} \), where \( \tau \mathrel{\text{:=}} {\tau }_{D} \) . Then \( \rho {\alpha }^{-1} : \alpha \left( Y\right) \rightarrow E \) fixes the open set \( \tau \left( ... | Yes |
Lemma 11.56. If \( Y \) is a subset of \( X \) that contains a closed chamber and is isometric to a subset of \( {\mathbb{R}}^{n} \), then \( Y \) is contained in an apartment. | Proof. Let \( \bar{D} \) be a closed chamber contained in \( Y \) . By Lemma 11.54 we can find an isometry \( \alpha \) from \( Y \) into \( E \) that maps \( \bar{D} \) onto the closed fundamental chamber of \( E \) by the map \( {\tau }_{D} \) . By repeated applications of Lemma 11.55 we can successively adjoin close... | Yes |
Proposition 11.62. If \( \mathfrak{A} \) is a conical cell in some apartment \( E \), then \( \mathfrak{A} \) is a conical cell in every apartment \( {E}^{\prime } \) that contains it. | Proof. This follows easily from the fact that there is an isomorphism \( E \rightarrow {E}^{\prime } \) fixing \( E \cap {E}^{\prime } \) pointwise (Proposition 4.101). In the present Euclidean context, we can give an alternative proof of that proposition as follows. Take an isomorphism \( \phi : E \rightarrow {E}^{\pr... | Yes |
Lemma 11.64. Let \( \mathfrak{C} \) be a sector in an apartment \( E \), and let \( C \) be a chamber in \( X \) . Then we can find a subsector \( {\mathfrak{C}}^{\prime } \subseteq \mathfrak{C} \) and a chamber \( {C}_{0} \) in \( E \) such that the retraction \( \rho = {\rho }_{E,{C}_{0}} : X \rightarrow E \) has the... | Proof. Note first that there is a bounded subset \( Z \) of \( E \) such that for any choice of \( {C}_{0} \), we will have \( \rho \left( C\right) \subseteq Z \) . In fact, let \( z \) be any point of \( E \) and let \( Y \) be any ball in \( X \) centered at \( z \) and containing \( C \) ; then we can take \( Z = Y ... | Yes |
Lemma 11.72. Given a point \( x \) and a ray \( \mathfrak{s} \), there is a unique ray \( \mathfrak{r} \) that is based at \( x \) and parallel to \( \mathfrak{s} \). | Proof. To prove existence, let \( E \) be an apartment containing \( \mathfrak{s} \). Let \( \mathfrak{C} \) be a sector in \( E \), based at the origin of \( \mathfrak{s} \), such that the closure of \( \mathfrak{C} \) contains \( \mathfrak{s} \). By Theorem 11.63 we can find a subsector \( {\mathfrak{C}}^{\prime } \)... | Yes |
If \( F \) is an ideal simplex and \( x \) is an arbitrary point of \( X \) , then there is a conical cell \( \mathfrak{A} \) based at \( x \) such that \( F = {\mathfrak{A}}_{\infty } \) . Consequently, there is a 1-1 correspondence between the set of ideal simplices of \( X \) and the set of conical cells based at an... | The proof is similar to that of Lemma 11.72: Write \( F = {\mathfrak{B}}_{\infty } \) for some conical cell \( \mathfrak{B} \), let \( E \) be an apartment containing \( \mathfrak{B} \), and choose a sector \( \mathfrak{C} \) in \( E \) (based at the cone point of \( \mathfrak{B} \) ) such that \( \mathfrak{B} \) is a ... | Yes |
Lemma 11.76. The ideal simplices partition \( {X}_{\infty } \) . | Proof. Any open ray \( \left( {x, e}\right) \) is contained in an apartment \( E \) . It is therefore contained in some conical cell \( \mathfrak{A} \) in \( E \) based at \( x \), whence \( e \in {\mathfrak{A}}_{\infty } \) . This shows that \( {X}_{\infty } \) is the union of the ideal simplices. Suppose now that we ... | Yes |
Lemma 11.77. Two sectors of \( X \) have the same face at infinity if and only if they have a common subsector. | Proof. It is obvious that a sector has the same face at infinity as any subsector, whence the \ | No |
Theorem 11.79. The poset \( {\Delta }_{\infty } \) is a spherical building. Its apartments are in \( 1 - 1 \) correspondence with those of \( X \) . | Proof. The proof of Lemma 11.76 showed that any two elements \( F,{F}^{\prime } \) of \( {\Delta }_{\infty } \) are contained in an apartment \( {\sum }_{\infty }^{\prime } \) . Since the latter is a simplicial complex and is closed under passage to faces, it follows that (a) any two elements of \( {\Delta }_{\infty } ... | Yes |
Proposition 11.87. Let \( \phi : E \rightarrow {E}^{\prime } \) be a type-preserving isomorphism between apartments \( E = \left| \sum \right| \) and \( {E}^{\prime } = \left| {\sum }^{\prime }\right| \) of \( X \) . Then the induced isomorphism \( {\phi }_{\infty } : {\sum }_{\infty } \rightarrow {\sum }_{\infty }^{\p... | Proof. Assume first that \( E \) and \( {E}^{\prime } \) have a common sector \( \mathfrak{C} \) and that \( \phi \) is the isomorphism that fixes \( E \cap {E}^{\prime } \) pointwise. Then \( {\phi }_{\infty } \) fixes \( {\mathfrak{C}}_{\infty } \) and all its faces, so it is type-preserving. For arbitrary \( E,{E}^{... | Yes |
Corollary 11.92. There is a 1-1 correspondence between subbuildings of \( {\Delta }_{\infty } \) and pairs \( \left( {{X}^{\prime },\mathcal{A}}\right) \), with \( {X}^{\prime } \) a subbuilding of \( X \) and \( \mathcal{A} \) a good apartment system for \( {X}^{\prime } \) . | It is given by \( {\Delta }^{\prime } \mapsto \left( {{X}^{\prime },\mathcal{A}}\right) \), where \( {\Delta }^{\prime } \) is a subbuilding of \( {\Delta }_{\infty },\mathcal{A} \) is the set of apartments \( E \) of \( X \) such that \( {E}_{\infty } \) is an apartment of \( {\Delta }^{\prime } \), and \( {X}^{\prime... | Yes |
Proposition 11.93. Let \( \mathcal{A} \) be a good system of apartments for a Euclidean building \( X \) . If \( X \) is thick, then the building at infinity \( {\Delta }_{\infty }\left( \mathcal{A}\right) \) is thick. | Proof. Any two adjacent chambers of \( {\Delta }_{\infty }\left( \mathcal{A}\right) \) can be represented by adjacent sectors \( {\mathfrak{C}}_{1},{\mathfrak{C}}_{2} \) (with a common cone point) in an apartment \( E \) in \( \mathcal{A} \) . We may take their cone point \( x \) to be a special vertex of \( E \), in w... | Yes |
Proposition 11.96. \( \mathcal{A} \) satisfies (2) if and only if \( G = \mathfrak{B}N\mathfrak{B} \) . | Proof. The group \( G \) acts transitively on \( {\mathcal{A}}_{\infty } \), and \( N \) stabilizes \( {\sum }_{\infty } \) and acts on the latter via the quotient map \( N \rightarrow W \rightarrow \bar{W} \) . So \( N \) is chamber transitive on \( {\sum }_{\infty } \) . The result now follows from Lemma 6.4. | No |
Corollary 11.98. If \( \mathcal{A} \) satisfies (2) and \( \Delta \) is thick, then \( \left( {B, N}\right) \) and \( \left( {\mathfrak{B}, N}\right) \) are BN-pairs in \( G \) | Proof. \( {\Delta }_{\infty }\left( \mathcal{A}\right) \) is thick by Proposition 11.93. Now apply Theorem \( {6.56}\left( 2\right) \) twice. | No |
Proposition 11.99. If \( \mathcal{A} \) satisfies (2), then \( G = \mathfrak{B}K \) . | Proof. Since \( G \) acts strongly transitively on \( {\Delta }_{\infty }\left( \mathcal{A}\right) \), every apartment in \( \mathcal{A} \) containing a subsector of \( \mathfrak{C} \) has the form \( {bE} \) for some \( b \in \mathfrak{B} \) ; hence, since we know that \( \mathcal{A} \) satisfies (1), \[ {X}^{\prime }... | Yes |
Proposition 11.105. Let \( X \) be the Euclidean building \( \left| {\Delta \left( {G, B}\right) }\right| \) associated to \( G = {\mathrm{{SL}}}_{n}\left( K\right) \). (1) There is a sector \( \mathfrak{C} \) in the fundamental apartment \( E = \left| \sum \right| \) such that the stabilizer \( \mathfrak{B} \) of \( {... | Sketch of proof. Identify the fundamental apartment \( \sum \) with the complex \( \sum \left( {W, V}\right) \) studied in Section 10.1.6. [Recall that we gave an explicit way of making this identification in Section 10.1.7.] Then \( E = \left| \sum \right| \) is identified with \( V \). As \ | No |
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