A Compactification of the Bruhat-Tits Building by Erasmus Landvogt

By Erasmus Landvogt

The goal of this paintings is the definition of the polyhedral compactification of the Bruhat-Tits construction of a reductive team over a neighborhood box. moreover, an particular description of the boundary is given. to be able to make this paintings as self-contained as attainable and in addition available to non-experts in Bruhat-Tits concept, the development of the Bruhat-Tits construction itself is given completely.

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First of all, we want to examine the types of the Dynkin diagrams of G which can arise. Let us consider a connected component of the Dynkin diagram of G. Then the classifcation theory (see [Ti 1] Table II) yields the following possible types of root systems: 1) (split case): X,~ where Xn is the type of a reduced, irreducible root system (hence An, Bn, C,~, Dry, E6, E7, Es,F4, G2). 34 2) (quasi-split case): The following types remain: - 2A2,~(n >_ 1) : (relative root system: BCn); - 2A2~+l(n > 1) : (relative root system: Cn+l); 2Dn(n >_4) :@-@-.

Thus show that s = U~,e. In order to prove this, we will show that --- {x~(u, v ) : (u, v) 9 Ho(L, L2) with a~(u) > e + 3' and w(v - %u"u) >_ 2e} ---- U a , ~ . Let (u,v) C Ho(L, L2) and let x - Xa(U,V). T h e n x = x ' - x t' where X' = Xa(U,,)IU~ = x a ( j ; l ( u , O ) ) E Ua(K) and x" = xo(O,v - ),u%) = z (j;l(O,v - A u % ) ) e U2 (K) The first equality follows from this. Since p~(x') = 89 = w(u) - 7 and ~2~(x") = w(v - A u % ) , it is clear that the second set is contained in the third one.

P r o o f . 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 .

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