Buildings, BN-pairs, Hecke algebras, classical

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As in the case of GL(n + 1) and An , study of these buildings will yield an indirect proof that Cn really is the signed permutation group. Four affine families 27 The oriflamme Coxeter system Dn has generators which we write as s1 , s2 , s3 , . . , sn−3 , sn−2 , sn , sn with data 3 = m(s1 , s2 ) = m(s2 , s3 ) = . . = m(sn−3 , sn−2 ) and 3 = m(sn−2 , sn ) = m(sn−2 , sn ) and 2 = m(sn , sn ) (that is, they commute) and all other pairs commute. Thus, unlike An and Cn , the element sn−2 has non-trivial relations with three other generators, and concomitantly the Coxeter diagram has a branch.

Sn ∅ . Then C = Co , C1 , . . , Cn is a gallery from C to a chamber Cn having x as face. In effect, the induction hypothesis is that λ and λo agree on all vertices of Co , C1 , . . , Cn−1 . We may as well consider only the case that x is the unique vertex of Cn not shared with Cn−1 , since otherwise we are already done, by induction. Let F = Cn−1 ∩ Cn . Then λ(x) must be a singleton set disjoint from λ(F ), and λo (x) must be a singleton set disjoint from λo (F ). Since, by induction, λo (F ) = λ(F ), it must be that λo (x) = λ(x).

Further: • For a chamber D in A and a face x of C, dX (x, D) = dA (x, D) • When restricted to any other apartment B containing C, ρ gives an isomorphism ρ|B : B → A which is the identity map on the overlap A ∩ B. • Let C be another chamber in A, and let B be an apartment containing both C, C . Then when restricted to B, ρA,C is equal to ρA,C . • This ρ = ρA,C is the unique chamber map X → A which fixes C pointwise and so that for any face x of C and any chamber D in X dX (x, D) = dX (x, ρD) Remarks: The retraction constructed in the proposition is the canonical retraction of X to A centered at C.

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