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 an area box. furthermore, an particular description of the boundary is given. as a way 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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A canonification will be given in w The Galois group E acts on the set of all U~ (a C ~). If a E ~ and a E E, then a(U-~) = ~;~(~). Let E~ be the stabilizer of a in E and let L~ = / ; / ~ . Then ~)~ is defined over L~. 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).

16)). 1. We have: (i) There is a unique TL~-equivariant L~-group homomorphism exp~ : WL~ (0r --9 GL~ which induces the inclusion O~ --4 0 | L~ on the associated Lie algebras. (/i) If X e 0~\{0}, then u ~ e x p A u X ) is a n~-group homomo~phism ~a/L~ ~ GL~, which is TL~-equivariant for the TL~-action on ~a/L~ through a. By this assignment we obtain a bijection o] oa\{O } onto the set of all such La-group homomorphisms ~a/n~ -4 Gnu. (iii) There is a unique Lc,-linear map < , > : Oa | ~--a -+ L~ and a uniquely determined L~-group homomorphism r* : @m/Lr ''') TL~ such that expa (X) exp_~ (Y) Y X = e x p - ~ ( l + < X , Y > ) r * ( l + < X , Y >)expa(1 + < X , Y > ) for all X 6 gc, and Y 6 0-~.

The simple factor G ~ C_ G ~ x g /~ with respect to a is obviously defined over L~. 2). In the sequel we will identify ~L~ (G o) and G ~. In the same way one can show that the inclusion ~ r C_ U~ x K /~ induces a canonical isomorphism 7~L~(Lr~) --+ U~. 6). Thus we will also identify 7~L~ (0n) and U~. Therefore we get isomorphisms 9 (V,o/Lo) --+ V =o = which induce group isomorphisms z-4-~ : L,~ --+ U+a(K) (recall L~ = L - s ) . 8. (i) Let ~ a : Ua(K) --+ ~U {oc} be defined by ~p,~(u) = W(Xal(U)) and let (ii) F~ := F'~ := ~ ( U ~ ( K ) \ { 1 } ) C_ ~ .

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