Bruhat-Tits buildings and their compactifications

Transcription

1 Bruhat-Tits buildings and their compactifications Prof. Dr. Annette Werner Goethe-Universität, Frankfurt am Main BMS Fridays January 23, / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

2 Bruhat-Tits buildings: The setting Basic data K non-archimedean local field v : K x Z discrete valuation x = c v(x) (c > 1) absolute value on K R K = {x K : x 1} valuation ring m K = {x K : x < 1} valuation ideal K is complete, the residue field R K /m K is finite Example K = Q p = completion of Q with respect to x p = p vp(x). 2 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

3 Bruhat-Tits buildings: The setting G/K semisimple algebraic group, G GL n,k closed subgroup such that the rad (G) (=biggest connected solvable normal subgroup) = 1 G = G(K) Example: SL n,k, PGL n,k, Sp 2n,K, SO n,k SL n (D), D central division algebra over K. Bruhat-Tits building B(G, K) complete metric space with continuous G action polysimplicial structure 3 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

4 Why are Bruhat-Tits buildings useful? B(G, K) is a nice space on which G acts, B(G, K) encodes information about the compact subgroups of G. B(G, K) is a non-archimedean analogue of a Riemann symmetric space G C /H of non-compact type. (G C a complex semisimple algebraic group, H a maximal compact subgroup) Buildings can be used to prove results for symmetric spaces (e.g. Kleiner-Leeb) Representation theory of G (Schneider-Stuhler) Cohomology of arithmetic groups (Borel-Serre) 4 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

5 How is B(G, K) constructed? General case: S G maximal k split torus Z N centralizer/normalizer of S in G. N = N(K), Z = Z(K) W = N/Z Weyl group, W acts on X (S) = Hom K groups (G m, S) G = PGL n,k = GL n,k center {( )} s1 S =... K x s n Z = S N = {permutation matrices} W symmetric group S n 1 W acts by permutation of the elements η i : x diag (1,..., x,..., 1) i on X (S) = n Zη i Z( n η i ). i=1 i=1 5 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

6 How is B(G, K) constructed? A = X (S) Z R X (C) Z R C = connected center of G is called an apartment A = n Rη i R( n η i ) i=1 i=1 The action of W on A can be extended to an action of N N = S W diag (s 1,..., s n ) S acts by n x i η i n ( xi v(s i ) ) η i i=1 i=1 6 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

7 How is B(G, K) constructed? Φ = Φ(S, G) root system Φ = {a i /a j : i j} with a i : diag (s 1,..., s n ) s i For a Φ we have a root group U a l : U a \{1} R, U a u U a N = {m(u)} m(u) acts on A as reflection at some affine hyperplane H a,u = {x A : a(x)+l(m) = 0} {( )} 1 x U ai /a j =... 1 x in the i th row and j th column ( ) 1 x l... = v(x) 1 with Affine hyperplanes: { n x i η i A : x i x j = k} for all i=1 i j, k Z 7 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

8 How is B(G, K) constructed? Bruhat-Tits blackbox Basicda For every x A define a subgroup P x G Define an equivalence relation on G A by (g, x) (h, y) n N : nx = y and g 1 hn P x B(G, K) = G A/ x = 0 P x = PGL n (R K ) B(PGL n, K) {non-archimedean norms on K n } modulo scaling (Goldman-Iwahori) 8 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

9 B(PGL 2, Q 2 ) 9 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

10 Apartment for PGL 3 10 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

11 Apartment for Sp 4 11 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

12 Some part of B(PGL 3, Q p ) 12 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

13 Compactifications of Bruhat-Tits buildings Borel-Serre compactification (1976): Fix some x B(G, K) and add an endpoint to every geodesic ray starting at x. ( This works in the same way for CAT (0) spaces ). Polyhedral compactification (Landvogt 1996): analogue of the maximal Satake compactification for Riemann symmetric spaces. 13 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

14 Compactifications of Bruhat-Tits buildings Concrete compactification for PGL n (A.W. 2001/2004) - A = compactification of one apartment - refine the Bruhat-Tits blackbox to define P x for all x A - B(PGL n, K) = PGL n (K)xA/ Theorem 1: The Goldman-Iwahori identification B(PGL n, K) ({non-archimedean norms on K n } modulo scaling) can be continued to a PGL n (K) equivariant homeomorphism B(PGL n, K) ({ non-archimedean seminorms on K n } modulo scaling) Theorem 2: B(PGL n, K) can be embedded in the Berkovich projective space (P n 1 ) an. 14 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

15 What are Berkovich spaces? A Berkovich space is a p adic analytic space with good topological properties. X /K smooth, projective variety (projective manifold). The set of points X (K) inherits a non-archimedean topology from K with bad topological properties (e.g. it is totally disconnected). Berkovich adds a lot of new points to fill in the gaps. 15 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

16 What are Berkovich spaces? More precisely: X = Spec B affine variety (B = K[x 1,..., x n ] a) { } multiplicative seminorms on B X an = extending K Let x X (K), i.e. x = (x 1,..., x n ) K n such that f (x) = 0 for all f a. X defines a multiplicative seminorm on B as follows: g x = g(x 1,..., x n ) K. There are many more multiplicative seminorms on B. 16 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

17 The Berkovich projective line 17 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

18 Compactification of Bruhat-Tits buildings with Berkovich spaces ( Joint work with Bertrand Rémy and Amaury Thuillier (Lyon) ) We define a continuous map as follows: ϑ : B(G, K) G an For every x B(G, K) there exists a unique K affinoid subgroup G x G an such that for all non-archimedean field extensions L/K G x (L) = Stabilizer of x in B(G L, L) ϑ(x) = Shilov boundary point given by G x (a certain multiplicative seminorm on the algebra of G). This generalizes results of Berkovich in the split case. 18 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

19 Compactification of Bruhat-Tits buildings with Berkovich spaces Choose a parabolic subgroup P in G (If G = SL n, the parabolic subgroups are precisely the stabilizers of flags in K n ) ϑ P : B(G, K) ϑ G an G an P an is a continuous, G equivariant map. It may forget some almost simple factors of G. ϑ P only depends on the type of P. Definition B P (G, K) = closure of ϑ P ( B(G, K) ) in the Berkovich space G an /P an. 19 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

20 Compactification of Bruhat-Tits buildings with Berkovich spaces Theorem B P (G, K) = B(Q ss, K) Q good parabolic Theorem Theorem Any two points x, y in Bp(G, K) are contained in one compactified apartment. (Mixed Bruhat decomposition) Let x, y Bp(G, K) with stabilizers P x, P y G. Then G = P x NP y. 20 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,

21 Compactification of Bruhat-Tits buildings with Berkovich spaces Example: 1 P Borel subgroup (If G = SL n, a Borel subgroup is the stabilizer of a maximal flag in K n ) Then all parabolic subgroups are good and B P (G, K) is Landvogt s polyhedral compactification. 2 G = PGL n P =... the stabilizer of a hyperplane 0 0 Then B P (G, K) is the seminorm compactification described above. B P (G, K) is a non-archimedean analogue of Satake s compactifications for Riemann symmetric spaces. 21 / 21 Bruhat-Tits buildings and their compactifications BMS Fridays,