Download A Compactification of the Bruhat-Tits Building by Erasmus Landvogt PDF

By Erasmus Landvogt

ISBN-10: 3540604278

ISBN-13: 9783540604273

The target of this paintings is the definition of the polyhedral compactification of the Bruhat-Tits development of a reductive workforce over an area box. furthermore, an specific description of the boundary is given. to be able to 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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Extra resources for A Compactification of the Bruhat-Tits Building

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Here we denote by MorK( , ) and MorL( , ) the sets of K- and L-morphisms, respectively. 5). 1) we know that the image of (~' X o~ OL)(OL) under 7l(id ) lies in (~3m/oL)(OL). 3) we OL obtain the existence and the uniqueness of the lifting, hence T = ~oK (G-~/OL)' Obviously, the norm map X L is an element of X*K(T ). Hence if t C Tb(K), then x it follows that NL(t) E Olc [] x and therefore that t E oL = T(OK). 3. We will differ from this in so far as we will not deal with the dependence of the filtrations of the groups Ua(K) on the choice of the Chevalley-Steinberg system.

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. [] 47 Now we want to examine the action of the torus on the unipotent subgroups. 20. We consider the following L2-group homomorphisms: --V y+ " H o ( L , L2) --+SU3 with (u,v) ~-~ y- : Ho(L, L2) --~SU 3 with (u,v) ~ z : T'~L(Gm/L) -"~SU3 with t ~ m : "~L2(~m/L) ---+SU3 with t 0 0 1 0 ; u --V 1 --U a ; (10 ) o) ~+ t t-it ~ 0 o (t~) -~ -t-lff " 0 0 0 ; 9 Next let ~ : SU3 --+ G a'a' be the L2-group homomorphism defined by ~,,• = 7r o ~ o y+ Finally, we let a = ~ o n ~ , ( ~ o ~): ~ : : ( ~ m / ~ ) - ~ (Ua, U-o) and for ( u , v ) E Ho(L, L2)\{(O,O)}, we let ma(U , V) : X_a(UU -1, (Va)--I)Xa(U, V)X_a(U(VCr) -1, (Va) - I ) .

1(C)M Vc')i = [0, oo], which is a closed subset. For a~lc < 0, we have f(gg(c) M vC')i =] - oc,0], which is a closed subset, too. 1(C) is dosed in an open covering of V z. 4) (ii) and the fact that R>0 is compact the assertion (iv) follows. [] Now let " W be the Weyl group of the root system ~. Then " W acts in a natural way on V and Z. So we obtain an action of ~W on V Z "W x V z --+ V ~ ( w , x + i F ) ) , - > w ~ + <~F> extending the action of " W on V. If we equip " W with the discrete topology, then this action becomes continuous.

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