By James E. Humphreys (auth.)

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B r l ) = w~ = i) to obtain: Wk = B = Bw0 so (B _l )w, the last factor using 12(a): WoW BW being work Then inside collapse parentheses The p r e c e d i n g conform with wow the first from right development the c o n v e n t i o n -I B'B = wB n k terms to Bw to left. :follows Richen of Borel [5], [I]. In order to if there exists we set (= B-- 1) W THEOREM 3. G = Proof. G = ~,J BwB w~W (by Theorem l ( a ) ) = kJ B- 1 B lWB w~W ww- (by Lemma 13) = U B'wB, w~W w DEFINITION. a normal subgroup Example.

The relative COROLLARY. d~~ is closed in AK topologies (cf. preceding on d K0 coincide exercise). 24 P r o o f of Lemma. then U nJK is o p e n the r e l a t i v e idele Conversely, relative idele and w h e r e I%lv v can all we n e e d U' ~ ~ ~K0 " topology JK0 ' c JK if topology, U is o p e n U n JK0 so S includes Any n e i g h b o r h o o d includes where has the s p e c i a l U < c = y 1, v so by c U in A K, is o p e n in ~ o p e n set (v{S)} l~v[ v ~ i. requiring ~ U ~ JK form ]yvl v = 1 with in the By d e f i n i t i o n to be v e r y small we satisfy ~--7 IYvl v = I I IYvl v < 2 V V ~S set U' c A K such that to do is find an o p e n n JKO c U ~ JKO .

2), discrete so in ~ Exercise. 2 JK d K. K* x K* JK we are free to view Special Let a "strong approximation" theorem for JK it is valid. ideles c(~) = i i l~vl v (~ c JK). Since I~vl v = 1 for almost V all v, "volume") this of is actually ~. The map a finite product, c : JK ÷ ~>0 called the content is evidently (or a continuous ho- 23 momorphism (continuous JK0 Its kernel essentially is a closed group of special det, which ideles. is brought * because subgroup (There of each JK' is continuous).