By James E. Humphreys (auth.)
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Proof. O LEMMA 7. Proof. bii iff there iff all g ¢ BwB, ¢i(g ) = ~i(ag) 3 (and the fact that (b) If_f g ¢ G, Because in ~i(n) _> 1 ~i(ag) combination ~ I. i for all g ~i a,B > 0 det c = 1 ~i(c) fo__r all (~i = s i m p l e for all U). s w-lugsw exists i. for all i. (orthonormal) e U-, Lemma di > 0 the case Then if on ~ . D form, n(e I ^ ... ^ e i) in a given Siegel to treat is multiplicative of other canonical Since there = ~i(swlugsw ) ¢i(hg) ~ ~i(hg) of triangular ~i(n) compact If exist D It suffices relatively When (g~G, b~B).
BN-pair. For In Lie t h e o r y the d i a g o n a l one-dimensional group H subgroup GL n in is GL(n,~) of s c a l a r s BN-structure. Construct j c R, are easy Bourbaki discussed in the GL(n,k). normal here. n; Exerc~e. for for Card(R) has d i m e n s i o n plays except is finite the e x a m p l e viewed to the s y m m e t r i c see for e x a m p l e Remark. clearly 1 < i < n-l) The that group, a is the let BN-pair for SL(n,k) analogous to that rank? Wj = s u b g r o u p of W generated by J. Set Gj = BWjB.
In then c l e a r l y so it c o n s i s t s is a unit more by J Ii iff closely, Uv ~ ~> (log has I Ji and log are c o n t i n u o u s ) . iff of roots of unity. UK = K n J (~), JK(~) = JK n JK (~) the r+s R>0, maps J~(~) , to a d d i t i v e ) , onto where valuations. and c o n t i n u o u s where Define l~r+slr+s ) infinite finite v. 0 I~II 1 ..... log image for all for all that 0 (multiplicative X lalv = 1 I~Iv = 1 notice and as u s u a l is a h o m o m o r p h i s m each W n K D indicates (because of u n i t y "FEe set is a root of u n i t y UK so is finite, for all v} of roots c I = c 2 = I.
Arithmetic Groups by James E. Humphreys (auth.)