A Compactification of the Bruhat-Tits Building by Erasmus Landvogt

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By Erasmus Landvogt

The goal of this paintings is the definition of the polyhedral compactification of the Bruhat-Tits construction of a reductive crew over a neighborhood box. additionally, an specific description of the boundary is given. which will make this paintings as self-contained as attainable and likewise available to non-experts in Bruhat-Tits concept, the development of the Bruhat-Tits construction itself is given completely.

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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) :@-@-. (relative root system: Bn-1); 2E 6 - aD4'6D4 (relative root system: F4); : N @~-I~1 (relative root system: G2); Here every I denotes a simple root in ~ and two I's are connected if and only if the root groups, associated with the corresponding roots in/~, have a non-trivial commutator.

T2 Now let ~2~Ta,a be the standard open subset of s Obviously, the generic fibre of ~ s , a equals Ws. 55 x gt_s,a with respect to ds. 10. 8) Proposition The morphism fl~ : W ~ --+ U_~ • T x Ua extends u n i q u e l y to a m o r p h i s m P r o o f . 3) it suffices to show that #r C u-,,,n(o,r • ~:(o~:) • u o , . ( o K ) . e. d ~ ( x ~ ( u ) , x _ ~ ( u ' ) ) E o K, from which w(l - u u ' ) = 0 follows. 6) we know t h a t 7~~ ~ ( ~ m / o L , ) is the canonical oK-group scheme associX OLc~ ated with ~L~ ( ~ m / L ~ ) , hence 1 - u u ' e OL~ = T~oK (Gm/oL~)(oK).

The filtrations (U~,t)ee~ and (U2~,e)~E~ of U~(K) and U2a(g) are independent of the choice of (a, a'). P r o o f . 10). [] In order to give a description of the structure of the sets of values F~ and I ' ~ , we have to go a little deeper into the valuation theory of L. 3. We let L~ ={v 9 ~=0} , L1 = {v 9 ~'=1} and nlm~x = {A 9 L 1 : w(A) = s u p { w ( x ) : x 9 L1}} . 16. (i) There exist elements t 9 L and r, s 9 L2 such that L = L2[t], t 2 - rt + s = 0 and w(at -1) < w(2) (if a 7~ 0). ~a~. ) ]> O.

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