By Erasmus Landvogt

The objective of this paintings is the definition of the polyhedral compactification of the Bruhat-Tits development of a reductive workforce over a neighborhood box. moreover, an particular description of the boundary is given. with a view to make this paintings as self-contained as attainable and in addition obtainable to non-experts in Bruhat-Tits concept, the development of the Bruhat-Tits development itself is given completely.

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Extra resources for A Compactification of the Bruhat-Tits Building

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I) follows immediately from the definition. The inclusion F~ C_ l w ( L • clear. 18). Ad (iii): Obviously, L ~ = L2 9t and w(1-~ 2/ E 22Z. By using (ii) we obtain both equalities. Ad (iv): Let x + t y E L ~ with x , y E L2. Hence 2 x + a y = 0 and therefore x + ty = x(1 - 2ta-1). 16) (i) it follows F~a = 22~. 17) (i) and (ii) we obtain w(x + )~y) = inf(w(x) + w(),)w(y)) • 22Z for suitable x E L2 and y E L ~ since w(x) E 2 ~ and w(y) E F~a = 2~. So A 9~ 22Z, from which the remaining equality follows.

3 Corollary of Proposition 9 the length of/3 + 2~. a is given by -n(/3, a) where ( n ( - , - ) ) denotes the Cartan matrix of ~. Hence the length is _ 1. On the other hand, in the quasi-split case the arising irreducible components are of type An, D,~ or E6 (see the classification). , see [Bou 1] VI Appendix) one obtains the validity of the assertion. 7. Let a E ~ and let a E E with cr(a) = a. If there is a root 5 E fi~ such that 5 ~ a(5), 5 + a ( 5 ) ~ a a n d a - h E ~, t h e n a - 5 - a ( 5 ) E ~.

Then: (/) / a ( 2 a ) = 2fa(a), if2a E ~. #i) If b E 9 with a + b E 02, then ffl(a) § f•(b) >_ f a ( a § b). (iii) Ia(a) + la(-a) >__0. (iv) For a finite set of roots {ai} in q~, we have f f l ( ~ ai) <_ 2 ffl(a~). i 4 P r o o f . The assertions (i), (ii) and (iii) follow immediately from the definition. 5. one can show that (iv) follows from (i) - (iii). 1a,fa(a), respectively. Let a, b E 9 with b ~ - N + a . 17)). depends on the choice of the ordering of (a, b). Of course, %,b The following proposition gives us one fundamental property which will be needed for the examination of the geometry of the Bruhat-Tits building.

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