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Erasmus Landvogt's A Compactification of the Bruhat-Tits Building PDF

By Erasmus Landvogt

ISBN-10: 3540604278

ISBN-13: 9783540604273

The objective of this paintings is the definition of the polyhedral compactification of the Bruhat-Tits development of a reductive staff over an area box. moreover, an particular description of the boundary is given. that allows you to make this paintings as self-contained as attainable and in addition available to non-experts in Bruhat-Tits idea, the development of the Bruhat-Tits development itself is given completely.

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6)) one can show that a + 5, a + a(5) r <~. 3 Prop. 9 (iii) we get n(a, 5), n(a, a(5)) > 0 ( ( n ( - , - ) ) denotes the Caftan matrix of ~ again). If 5 + a ( 5 ) E ~, then n(5+c~(5), a) --- n(8, a ) + n ( a ( 5 ) , a) > 0. 3 Theorem 1). For 5 + ~r(5) ~ ~, we have n(5, a(5)) = 0, since obviously 5 - a(5) r ~. 3 Theorem 1 again). 4. 2 we use sign changes like X~ F-~ - X a (a E ~) to construct a Chevalley-Steinberg system out of the ChevaJley system. The Galois group E acts on the Dynkin diagram by diagram automorphisms.

Ad (i): Obviously, it suffices to show that we have w o Xlz(oK) = 0 for all X E X ~ ( T ) . Since ~(OK) is a group, we only have to show that w o X[z(or) has a l o w e r bound. Let X E X ~ ( T ) and c E K x such that c X E og[~]. Then x(t) E c - l o g for all t E ~(og). Hence w o X[Z(oK) >- -w(c). Ad (ii): Let X E X~:(T") and t E T[(K). Then X o f E X~c(T') and therefore 0 = w((x o f)(t)) = w(x(f(t))). E X~ E X k ( T ) . , see [CaFr] II w Hence for t E Tb(K), we obtain w(x(t)) = [L: K] - 1 . w(~i(t)) = O.

12. The action u of N ( K ) of g(g) on A . on A can be extended uniquely to a continuous action P r o o f . Let us consider the canonical homomorphism W --+ V>4 vW. The existence is clear by the previous remark. [] Note t h a t the construction was made without any knowledge of the (poly-) simplicial complex in A. 4)) for the case that G is quasi-split. 9)) and canonify it in this way. Up to this point, ~fi will depend on the choice of a Chevalley-Steinberg system. In chapter III and IV it turns out that ~a(og) C G(K) is essentially the stabilizer of f~ in the Bruhat-Tits building and that the special fibre ~fi = ~ • oK k gives us information about the local structure near ~ (in the topological sense) of the building.

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A Compactification of the Bruhat-Tits Building by Erasmus Landvogt


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