Arithmetic Groups by James E. Humphreys (auth.)

By James E. Humphreys (auth.)

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N' = NH' H' c B n N' N/N n H' ((BNI) then is = W (BN2) assume , so H' and the being henceforth : /'~ Bn n~N (G,B,N'~R') B n N' is clear, H' isomorphic normalized Moreover, and We R, H' = N/H = , canonically corresponds Proof. N' by is N to N/H = W, is a s'aturated and H', N'-invariant, = B n N'. remaining so and BN-axioms R' BN-pair H' hence Therefore if is lies normal in little loss H'; W'= N'/H'=NH'/H ' are obvious automatic). (with with of g e n e r a l i t y ) that in 47 (G,B,N,R) defined is s a t u r a t e d .

Therefore recover is an isomorphism On t h e of trace group y(z) O. ,Yn_l(Z)) allows us to it is immediate that B groups. Ok in Bn l(a) ' ' ' ' - = set the ÷ of Lie ~n- 1 = (Zl-Z2, all matrices algebra of diag the (Zl, .... ,Zn_l-Zn). Since n Z z- = 0, it is c l e a r that y i=l 1 t o p o l o g i c a l groups, ~n-I having We can d e f i n e and exp : ~ maps b e t w e e n in the o b v i o u s + A , (~>0)n-] which and are is an i s o m o r p h i s m the usual ~n-i (additive) topology. way h o m o m o r p h i s m s inverse of log to each other, The d i a g r a m : A ÷ and s i m i l a r l y then c o m m u t e s : z~ 39 A * .

B, Gj = G K. so Theorem M ~ Gj in r e d u c e d whence the lattice to the l a t t i c e M ~ B, BwB, N G ( B ) = B). (e) that is i s o m o r p h i c union forces to (in p a r t i c u l a r , at once c G K. J c R. ) M is the of the sets for such w). G Jw But Lemma of 2 Gj c M W (b) Use Lemma are c o n j u g a t e (c) Again (d) If so (e) iff they use L e m m a Wj c WK, (by L e m m a If 4; since 3) Gj c GK, B c Gj, GK, are c o n j u g a t e these by some subgroups of G = w ~ W. 4. each r ~ J is a p r o d u c t o£ e l e m e n t s of K, r ~ K.

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