BRUHAT-TITS THEORY AND BUILDINGS 1

Jiu-Kang Yu, May 7-9, University of Ottawa

HISTORY AND LITERATURE

Prehistory.

— O. Goldman and N. Iwahori: The theory of p-adic norms, Acta Math. 109 (1963), 41pages.

— N. Iwahori and H. Matsumoto: On some Bruhat decomposition and the structure of theHecke ring of p-adic Chevalley groups, Publ. IHES. bf 25 (1965), 44 pages.

— H. Hijikata’s Mimeographed notes at Yale University (1964).

The beginning. Proceedings of the Boulder conference (1965) “Algebraic groups and discon-tinuous subgroups”.

p-adic Groups BY

FRANGOIS BRUHAT

1. Bounded subgroups. If G is a real connected Lie group, then the following two statements are well known:

(1) Any compact subgroup of G is contained in a maximal compact subgroup of G.

(2) Two maximal compact subgroups are conjugate by an inner automorphism. Now let P be the quotient field of a complete discrete valuation ring 0. Let

p be the maximal ideal of 0, and let n be a generator of p, and K be the residue field of 0 by p, i.e., p = 071, K = O/p. P is locally compact for the topology induced by the valuation if and only if K is a finite field.

Let G be a linear algebraic group defined over P, realized in GL(V) where V is a vector space defined over P. Let G, be the group of P-rational points of G. GP can be considered as a subset of GL(n, P), and also a subset of the ambient space P”’ of GL(r2, P). With the topology induced by P”‘, GP is a topological group. If P is locally compact, then G, is locally compact.

Let K be a subgroup of G,, then the following three statements are equivalent : (i) There exists a locally faithful matricial rational representation p of G defined

over P s.t. the coordinates of the elements of p(K) are bounded, (ii) For any matricial rational representation, the coordinates of the elements of

p(K) are bounded. (iii) For any rational linear representation p of G in a vector space I/ over P,

there exists a lattice L in V s.t. p(k)L = L for any k in K. If K satisfies one of the above conditions, K is called a bounded subgroup

of GP. The condition (iii) implies that any bounded subgroup is contained in an

open and bounded subgroup. On the other hand, the open and bounded sub- groups of G are related with the structure of G as group scheme over the ring 0: let P[G] the afhne algebra of G. The product in G gives a structure of coalgebra on P[G], i.e., a linear map d : PEG] -+ P[G] BP P[G] which is defined by the condition :

d(f) = Cfr off’-f(xY) = CfKNfl’(Y)*

Now, an &structure for G is an O-subalgebra of finite type &[G] such that P[G] = Pd[G] and d(&G]) c &[G] a0 &[G].

EXAMPLE. Let G = GL(V), let L a lattice in VP, (gij) the matrix of g E G with respect to some basis of L. Then the algebra C&[GL] = O[g,,, (det(gJ)- ‘1 is an

1Notes for a mini-course in a Fields institute workshop. These are slighted edited and expanded version of slicesused in the lectures.

P-ADIC GROUPS 69

This conjecture has been firstly proved by N. Iwahori and H. Matsumoto in the split case [7] (for the O-structure of $3 Example 2) then by H. Hijikata for quasi-split case [S] and classical cases [6].

REMARK. A triple of groups (G, B, N) is called a “Tits system” if it satisfies the following :

(0) G is generated by B and N, B n N is a normal subgroup of N, IV= N/B n N is called the Weyl group of the Tits system.

(1) Wis generated by involutive elements r E I, r2 = 1, (2) rBr-’ # B for any rE1, (3) rBw c BrwB u BwB for any r E I and any w E IV. If (G, B, N) is a Tits system, then: (I) w + BwB gives a bijective map from W to B/G/B. (2) For any subset J of I, let IV” be the group generated by r (r E J), then

G = BVV’B is a group. G, = NG(GJ). G, is conjugate to G, if and only if K = J. Any subgroup of G containing B has the form of GJ.

As a consequence if the Conjecture II is true, G, has at least r + 1 classes of maximal bounded subgroups where r = d&(G). Actually mhas r + 1 generators I = {WI, l l * w,, wO}, and a maximal bounded subgroup Ki (i = 0, 1, . . l r) is given by Ki = BW,_,,&

CONJECTURE (II) (iv). Any maximal bounded subgroup of GP contains a con- jugate of B.

REMARK. (iv) is known for classical groups modulo some exception in the case of characteristic of K = 2. [5].

Added in November 1965. During the conference, considerable progress was made towards an affirmative solution of the conjectures above. It also appeared that the properties thus established have interesting applications ; for instance, they provide a simplified approach to Kneser’s theorem on H’ of simply con- nected groups over the p-adics. A joint paper on this subject is in preparation, by F. Bruhat and J. Tits.

These results were exposed orally by J. Tits at the conference. The precise form on which they are given in the mimeographed notes of his talk must however be somewhat modified; in particular, it is not true that minimal k- parahoric subgroups of a group G-as defined in these notes-are conjugate by elements of G, . In fact, the notion of k-parahoric subgroup given there does not appear to be “the good one” when G does not split over an unramified extension of k.

On the other hand, the methods sketched there turn out to give further results. For instance, it can be shown that the Conjecture (II) (iv) above is essentially a consequence of the other parts of that conjecture and, in particular, is true in the split case.

BIBLIOGRAPHY

1. F. Bruhat, Distributions SW url groupe localemen t compact et applications tious des groupes p-adiques. Bull. Sot. Math. France 89 (1961), 43-75.

d l’dude des represert ta-

Announcements by Bruhat-Tits.

— Groupes algebriques simples sur un corps local, Proceedings of a Conference on LocalFields (Driebergen, 1966), 14 pages.

— BN-paires de type affine et donnees radicielles, C.R. Acad. Sci. Paris 263 (1966), 4 pages.

— Groupes simples residuellement deployes sur un corps local, ibid., 3 pages.

— Groupes algebriques simples sur un corps local, ibid., 4 pages.

— Groupes algebriques simples sur un corps local: cohomologie galoisienne, decompositionsd’Iwasawa et de Cartan, ibid., 3 pages.

Tits’ summary. It is in the Corvallis proceedings (1977). A must read.

The canon.

BT1 Groupes reductifs sur un corps local I, Publ. Math. I.H.E.S. 41 (1972), 247 pages.

BT2 Groupes reductifs sur un corps local II, Publ. Math. I.H.E.S. 60 (1984), 180 pages.

BT3 Schemas en groupes et immeubles des groupes classiques sur un corps local, Bulletin Soc.Math. France 112 (1984), 43 pages.

BT4 Schemas en groupes et immeubles des groupes classiques sur un corps local II: groupesunitaires, Bulletin Soc. Math. France 115 (1987), 55 pages.

BT5 Groupes algebriques sur un corps local, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987),28 pages.

Remark. (i) Officially, BT1, BT2, BT5 are labelled as Chapter I, Chapter II and Chapter III.(ii) Somehow BT3, BT4, BT5 are relatively unknown to users of Bruhat-Tits theory. Some ofthe results there have been rediscovered years later.

Some comments.

— The writing took over 20 years (1965–1987).

— The canon all together is over 550 pages.

— The formulation evolved during the period.

— Tits’ summary, which is the portal for most users of Bruhat-Tits theory, was written waybefore most of the canon.

— BT1-5 always tried to pursue maximal generalities. Some of these are not necessary forrepresentation theory. But it is amazing that essentially everything in their theory becamequite useful to certain part of mathematics later.

Some work on Bruhat-Tits theory itself.

— I.G. Macdonald: Spherical functions on a group of p-adic type (1971).

— G. Rousseau’s thesis (1977), unpublished.

— P. Garret: Buildings and classical groups (1997), about 400 pages.

— W.T. Gan and J.-K. Yu: Schemas en groupes et immeubles des groupes exceptionnels surun corps local, Premiere partie: Le groupe G2, Bulletin SMF, 55 pages.

— W.T. Gan and J.-K. Yu: Schemas en groupes et immeubles des groupes exceptionnels surun corps local, Deuxieme partie: Les groupes F4 et E6, Bulletin SMF, 43 pages.

— J.-L. Kim and A. Moy: Involutions, classical groups and buildings, J. Algebra (2000), 17pages.

— E. Landvogt, A compactification of the Bruhat-Tits building, Springer LNM (1995), 156pages.

— E. Landvogt, Some functorial properties of the Bruhat-Tits building, J. Reine Angew.Math. 518 (2000), 29 pages.

— A. Moy: Displacement functions on the Bruhat-Tits building, The mathematical legacy ofHarish-Chandra (Baltimore 1998), 17 pages.

— G. Prasad: Galois-fixed points in the Bruhat-Tits building of a reductive group, BulletinSMF.

— PY G. Prasad and J.-K. Yu, On finite group actions on reductive groups and buildings,Invent. Math. (2002).

— J.-K. Yu: Smooth models associated to concave functions in Bruhat-Tits theory, preprint(2003), 34 pages.

Remark. Several articles by G. Prasad and S. Raghunathan were written before BT2. Theyworked out many results and formulas independently. Therefore, these articles can be used asalternative references to some extent. Their Annals article about central extensions (which isthe precursor of Moy-Prasad theory) is particularly useful.

Applications to representation theory. (The list is short and incomplete, concentrating on morerecent ones).

— J.D. Adler: Refined anisotropic K -types and supercuspidal representations, Pacific J. Math.185 (1998), 32 pages.

— J.D. Adler and S. DeBacker: Some applications of Bruhat-Tits theory to harmonic analysison the Lie algebra of a reductive p-adic group, Michigan Math. J. 50 (2002), 19 pages.

— D. Barbasch and A. Moy: A new proof of the Howe conjecture, Journal of the AMS 13(2000), 12 pages.

— S. DeBacker: Some applications of Bruhat-Tits theory to Harmonic analysis on a reductivep-adic group, Michigan Math. J. (2002), 21 pages.

— S. DeBacker: Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. 156(2002), 38 pages.

— S. DeBacker: Homogeneity results for invariant distributions of a reductive p-adic group,Ann. Sci. Ecole Norm. Sup. 35 (2002), 32 pages.

— A. Moy and G. Prasad: Unrefined minimal K -types for p-adic groups, Inv. Math. 116,393–408 (1994)

— A. Moy and G. Prasad: Jacquet functors and unrefined minimal K -types, Comment.Math. Helvetici 71, 96–121 (1996)

— P. Schneider and U. Stuhler: Representation theory and sheaves on the Bruhat-Tits build-ing, Inst. Hautes Etudes Sci. Publ. Math. 85, 97–191 (1997).

— J.-K. Yu: Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14,579–622 (2001).

COMMENTS. For reasons commented above, Bruhat-Tits theory is not easy to read for manypeople. In particular, people in representation theory only want to learn it as a foundation.For that, going through 550 pages would be a lot. Tits’ summary is the portal into Bruhat-Titstheory for most people, and is highly recommended.

Macdonald’s little book is also quite accessible, but it only sketches the case of simplyconnected groups. Now Landvogt’s book is a usable replacement to many results of BT1 andBT2. However, it doesn’t treat groups associated to concave functions which have becomeincreasingly useful now (this theory is not covered in Tits’ summary or Macdonald’s bookeither). My article on smooth models can replace most of the algebro-geometric part of BT2,at least in the case of discrete valuations.

Garret’s book describes the buildings of classical groups as a simplicial complex, in a ratherconcrete way. However, it is not clear that the building in his book is the building describedin Bruhat-Tits theory. In fact, Bruhat and Tits realized this kind of matching is not obvious atall, so they wrote BT3 and BT4 to match their buildings and the theory of Goldman-Iwahori.Today it is fairly easy to match the building in Garret’s book with that in Bruhat-Tits theory. Ageneral strategy was developed in the two articles of Gan and myself, and then was applied toexceptional groups. I sketched the case of classical groups in a lecture at Banff. Notes of thatlecture can be found on my webpage http://www.math.purdue.edu/˜jyu .

In addition, I will give a few suggestions, which are of course very subjective.

About learning to use Bruhat-Tits

— Don’t try to swallow all proofs.— Learn a bit about symmetric spaces.— Learn a bit about spherical buildings of a reductive group.— Learn the case of GLn really well.— Learn a bit about affine root systems.— Learn a bit about BN-pairs (aka Tits systems).— Draw/play with the 2-dimensional apartments.— Understand the case of split/quasi-split groups.— (Learn the theory of schemes).

Remark. It may look like I am recommending a lot stuffs not directly related. You may suspectthat this is overloading on top of something which is already complicated. However, most usersof Bruhat-Tits theory want to learn the facts and to get the feeling, not going through detailsand proofs. It is best to achieve this through analogies. We only need a little bit of everythingfrom above and I believe that in the end you will find the experience rewarding.

A few things that I do not recommend.

— Don’t try to picture a building of dimension > 1.— It is not necessary to learn a lot about (poly-)simplicial complexes.— It is not necessary to learn a lot about general buildings.— Don’t think of Bruhat-Tits theory as just a case of BN-pairs (even though the building

can be described using a BN-pair, the B and the N are not easy to specify; moreover,there are many features in Bruhat-Tits theory that are not in the theory of BN-pairs).

— Although one can say a lot by just talking about chambers (aka alcoves) and apartments,don’t be afraid of talking about the whole building.

ABOUT THIS MINI-COURSE

Tits’ summary in the Corvallis proceedings has been the portal to Bruhat-Tits theory formost people, and it will continue to be the best user guide. It is very well-written and veryprecise. It does a very good job in hiding the technicalities and in describing everything interms of elegant abstract characterizations.

The goal of my lectures is to provide more hints and help to people who want to useBruhat-Tits theory. I try to be as orthgonal to Tits’ summary as possible. Therefore, I do notgive a systematical account of the theory itself. In the first lecture, I provide some backgroundmaterials that are probably most useful if you learn them before starting to read Tits’ article.In the second lecture, I give diversed complements to Tits’ article. From my experience, theseshould be helpful to someone who is currently studying Tits’ article. In particular, I try toexplain how to go between Tits’ article and [BT1-5]. For the third lecture, I discuss morerecent developments in representation theory that go beyond Tits’ article.

SYMMETRIC SPACES OF REAL REDUCTIVE GROUPS

References.

— T.A. Springer, Reductive groups, the Corvallis proceedings (A. Borel and W. Casselman,Eds.), v.1., 3–27 (1977).

— W.T. Gan and J.-K. Yu: Schemas en groupes et immeubles des groupes exceptionnels surun corps local, Premiere partie: Le groupe G2, Bulletin SMF, 55 pages.

Notation.

— G: a connected reductive group over R;— K : a maximal compact subgroup of G(R);— g := Lie G.

The symmetric space S = S(G) of G (or of G(R) is essentially G(R)/K . But we want totalk about this more intrinsically without singling out K .

If G is semisimple, we can say that S is the set of maximal compact subgroups of G(R)

(since all of them are G(R)-conjugate to K ).But I prefer to use the bijections

{maximal compact subgroups of G(R)}

↔ {Cartan involutions on G(R)}

↔ {Cartan involutions on g}

to say that S is the set of Cartan involutions on g.

Recall. A Cartan involution on g is an order 2 automorphism of g such that if k and p are the(−1)- and (+1)-eigenspaces of g under θ , then k + i p is a compact form of g [and compactnessof a real form can be detected by the positivity of the Killing form].

Remark. We will deal with all reductive groups later.

Example. Let G = SLn. Then the standard Cartan involution is X 7→ −t X on g = sln, and

g 7→ (t g)−1 on G(R).

More intrinsically, think of G = SL(V). A maximal compact subgroup of G(R) is SO(q)

for a positive definite quadratic form q on V . The Cartan involution is again g 7→ (t g)−1, andnow t g is the adjoint of g relative to the symmetric bilinear form 〈−, −〉 = 〈−, −〉q associatedto q:

〈g.v, w〉 = 〈v, t g.w〉.

Of course,√

q is then a Euclidean norm on V . Therefore,

S(SL(V)) is the space of norms on V (up to constant multiples).

In general, we write Sred(G) for the space of Cartan involutions on g (or on G).

Example. Let G = Gdm be an m-dimensional split torus. Then Sred(G) consists of one element,

which is X 7→ −X on Lie G and g 7→ g−1 on G(R). On the other hand, S(G) = (R×

>0)m

'

(R×/{±1})m.

If G is anisotropic, then G(R) is compact and Sred(G) consists of only one point, which isX 7→ X on Lie G and g 7→ g on G(R).

More generally, if G is anisotropic modulo its center, then Sred(G) consists of one pointonly.

Almost every statement about S(G) that has a counterpart in Bruhat-Tits theory can bededuced from the following (together with a few standard facts):

Theorem Let G ⊂ H be connected reductive groups over R. Then

(i) If θ ′∈ Sred(H) stabilizes G(R), then θ ′

|G(R) ∈ Sred(G).(ii) For any θ ∈ Sred(G), the set

{θ ′∈ Sred(H) : θ ′

|G(R) = θ}

is non-empty, and is permuted transitively by the real points of ZH (G), the centralizer ofG in H .

Apartments. For each maximal R-split torus S of G, let Ared(S) = {θ ∈ Sred(G) : θ(S(R)) ⊂

S(R)}. We call Ared(S) the apartment of S. We notice that θ ∈ Ared(S) if and only if θ extendsthe only Cartan involution of S(R).

Example. Let G = GL(V). A maximal R-split torus S of G amounts to a decomposition of Vinto 1-dimensional subspaces: V = Rv1 ⊕ · · · ⊕ Rvn.

The quadratic form q lies in Ared(S) if and only if the form is diagonalized w.r.t. the basisv1, . . . , vn:

q(x1v1 + · · · + xnvn) = a1x21 + · · · + anx2

n, ai > 0.

Remarks.

— If we give Sred the structure of a Riemannian symmetric space, then the apartments arethe maximal flat subspaces.

— All apartments are conjugate by G(R). Since all the S’s are conjugate.

— Each point lies in an apartment. The unique Cartan involution of S(R) always extends toa Cartan involution of G(R) by (ii) of the theorem. Therefore, at least one Cartan invo-lution lies on an apartment. It follows that any Cartan involution lies on an apartment.

— S(R) acts on Ared(S) transitively. If θ1 and θ2 both extend the only Cartan involution of S,then θ2 = z.θ1 are conjugate by some z of Z(R) by (ii) of the theorem, where Z = ZG(S).We can write z = sk, where z ∈ S(R) and k is in the maximal compact subgroup ofZ(R) (Cartan decomposition). Since k ∈ Kθ1, we have Kθ2 = zKθ1z

−1= sKθ1s

−1. Henceθ2 = s.θ1.

— The dimension of Ared(S) is rankR G − rankR Z(G). Let θ ∈ Ared(S). The normalizer ofKθ in G(R) is CG(R).Kθ , where CG is the center of G. In particular, the stabilizer of θ inS(R) is S(R) ∩ CG(R).Kθ .

— Any two points x, y ∈ Sred(G) lie on some apartment Ared(S). Suppose that x ∈ Ared(S).Then we have the Cartan decomposition G(R) = Kx S(R)Kx. Suppose that Ky = gKxg−1

with g = ksk′, where k, k′∈ Kx and s ∈ S. Then it is clear that Ky = (ks)Kx(ks)−1 and

y lies on the apartment associated to (ks)S(ks)−1= kSk−1. It is also clear that x lies

in kSk−1 also since θx(kSk−1) = θx(k)θx(S)θx(k)−1= kSk−1. Thus x, y both lie on

Ared(kSk−1).

Remark. In fact, the above statement is equivalent to the Cartan decomposition. Notice thatwhen G = GL(V), the above statement is the following familiar fact in linear algebra: any twopositive definite symmetric real matrices can be diagonalized simultanenously.

— Kx permutes apartments containing x transitively. Suppose that x ∈ Ared(S) and x ∈

Ared(S′). We may assume that S′= gSg−1. The points of Sred(G) on Ared(S′) are precisely

gS(R)g−1.(gx) = gS(R).x. Therefore, there exists s ∈ S(R) such that x = gs.x. Letk = gs, then we have k ∈ Kx and kSk−1

= S′. Therefore, Ared(S′) = k.Ared(S).

— Another description of the apartments containing x. let g = k ⊕ p be the Cartan decom-position relative to θx. Then there is a bijection between the apartments containing xand the maximal abelian sub-algebras of p. The bijection is Ared(S) 7→ Lie S. Then theprevious statement is the well-known fact that Kx permutes the abelian sub-algebras of p.

In general, let V be the maximal vector subgroup in the center of G(R), and we defineS = Sred × V . We can put a natural action of G(R) on S to reinstate the property that thestabilizer of points on S are precisely the maximal compact subgroups.

We also define the extended apartment A(S) as Ared(S) × V(G). In particular, dim A(S) =

dim S. See the appendix of Gan-Yu for more details.

The point of these discussions is: the symmetric space has many properties analogous tothose the Bruhat-Tits building. But they are a lot easier to derive from basic facts in Lie groups.

The analogous results in Bruhat-Tits theory are often related to analogous structure theoryand analogous decompositions. But the arguments are often more convoluted. Therefore,learning the real case helps us to get a picture, and pinpoint some basic ingredients.

THE BUILDING OF A p-ADIC GLn

References.

— Goldman-Iwahori.

— BT1, 10.2, Note ajoutee sur epreuves.

— Tits’ summary, 2.9.

— BT3: this is the definitive treatment.— Gan-Yu, the G2 article.

Motivation. For a real vector space V ,

S(GL(V)) is the space of Euclidean norms on V .

Notation. Let K be a non-archimedean local field, and V a finite-dimensional vector spaceover K .

Definition. A (classical) norm on V is a function || − ||α : V −→ R≥0 satisfying:

• ||x + y||α ≤ max{||x||α, ||y||α)}, for all x, y ∈ V;• ||λx||α = |λ| · ||x||α, for λ ∈ K and x ∈ V;• ||x||α = 0 if and only if x = 0.

Definition. An (additive) norm on V is a function α : V −→ R ∪ {∞} satisfying:

• α(x + y) ≥ min{α(x), α(y)}, for all x, y ∈ V;• α(λx) = ord(λ) + α(x), for λ ∈ K and x ∈ V;• α(x) = ∞ if and only if x = 0.

Clearly, || − ||α is a classical norm ⇐⇒ α(−) = log || − ||α is an additive norm. From nowon, “norm” means “additive norm”.

Definition. The building of GL(V), B := B(GL(V)), is the set of all norms on V .

We let GL(V) act on B by

g.α = α ◦ g−1 : v 7→ α(g−1.v).

It is clear that g.α is also a norm on V .

Later, we will put a metric on B so that B becomes a topological space.

Examples.

(i) Say V = K .v is 1-dimensional. Pick c ∈ R and put α(λ.v) = ord(λ)+c. Then α is a normon V .

Clearly, all norms on V = K .v is of this form.

(ii) If V = V1 ⊕ V2, and αi is a norm on Vi . Put α(v1 ⊕ v2) = min(α1(v1), α2(v2), then α is anorm on V .

(iii) Pick a basis v1, . . . , vn of V , and real numbers c1, . . . , cn. Put

α(∑

λi vi

)= min

{ord(λi ) + ci

}.

Then α is a norm on V .

Definition. We say that {v1, . . . , vn} is a splitting basis for α if (iii) holds.

Fact 1. Any norm on V admits a splitting basis.

Compare. Any quadratic form on a real vector space can be diagonalized.

Fact 2. Any two norms on V has a common splitting basis.

Compare. Any two positive definite quadratic forms on a real vector space can be diagonalizedsimultaneously.

We now interprete Fact 2 in a special case. In the description of a splitting norm, if ci = 0for all i , then

α(v) = max{m : v ∈ πmL},

where L is the lattice O〈v1, . . . , vn〉, O = OK , π is a prime in O.Such a norm is a called a hyperspecial norm, or a hyperspecial point on B. Now Fact 2 for

hyperspecial points means the following:

Let L , M be lattices in V . Then there is an O-basis {v1, . . . , vn} of L and integerse1 ≤ · · · ≤ en such that πe1v1, . . . , π

envn is a basis of M .

This is the fundamental theorem of finitely generated modules over a PID/DVR (theoremof elementary divisors). It is also equivalent to the Cartan decomposition for GLn(K ):

GLn(K ) = GLn(O).A. GLn(O),

where A consists of diagonal matrices diag(πe1, . . . , πen) with e1 ≤ · · · ≤ en.

The metric. We now use Fact 2 to put two metrics on B as follows: if α, β are two norms withcommon splitting basis {v1, . . . , vn}, we put

d(α, β) =

(∑i

(α(vi ) − β(vi )

)2)1/2

,

d′(α, β) = max{|α(vi ) − β(vi )| : i

}.

Challenge. Show that d is well-defined (independent of the splitting basis; I don’t know anysimple proof of this). Show that d and d′ satisfy the triangle inequality. It is easy to show: d′ iswell-defined, and if d is also well-defined, d and d′ define the same topology on B.

Observation. If {v1, . . . , vn} is a basis of V , c ∈ Rn, and αc is the norm αc(vi ) = ci . Thenc 7→ αc is an isometry Rn

→ B for the metric d.

Definition. The image of this isometry is called the apartment associated to the split torus Sfor which {v1, . . . , vn} is an eigen-basis for all g ∈ S(K ). We denote this apartment by A(S).

Interpretation of Fact 1. Every point of B lies on an apartment.

Interpretation of Fact 2. Every two points lie on a common apartment.

Observation. S 7→ A(S) is a bijection.

Observation. The action of S(K ) on B stabilizes A(S). But it doesn’t act on A(S) transitively.For any x ∈ A(S), S(K ).x looks like a set of lattice points on the euclidean space A(S).

Observation. A(S) is isometric to a Euclidean space. But there is no natural base point. A(S) isan affine space, not a vector space. The space of translations can be identified with X∗(S)⊗Z R.

Observation. Let N be the normalizer of S. Then the elements of N(K ), represented by ma-trices w.r.t. the basis {v1, . . . , vn} are simply the group of monomial matrices. The quotientN(K )/S(K ) is the Weyl group of (G, S). The action of N(K ) on B also stabilizes A(S), andthere the action is given by affine transformations.

Simplicial structutre. The topological space B carry a canonical poly-simplicial structure. Forsimplicity I want to get rid of the prefix “poly”. So I define α ∼ β if α(v) = β(v) + c for somec ∈ R and put Bred = B/ ∼. This is the reduced building of GL(V), or the building of PGL(V).Now Bred is truely a simplicial complex.

The vertices are the hypespecial points, which now correspond to lattices modulo the equiv-alence L ∼ πnL for all n ∈ Z. So they are really lattice classes.

Now k+1 vertices form a k-simplex if and only if there are lattices L0, . . . , Lk representingthe corresponding lattice classes, such that

L0 ) L1 ) · · · ⊂) Lk ) π L0.

Now we have seen a few essential feautures of the Bruhat-Tits building associated to a p-adicreductive group, namely

• B(G) is a complete metric space (with an affine structure);• B(G) is a (poly-)simplicial complex;• G(K ) acts isometrically on B(G) by (poly-)simplicial automorphisms;• B(G) has a collection of distinguished subsets, known as apartments, which are indexed

by the maximal split tori of G. Each apartment is isometric to a Euclidean space.

Remark. If you are familiar with the work of Bushnell-Kutzko, you would recognize a sim-plex as a lattice chain. It is possible to rephrase the notion of “norm” in a lattice-theoreticallanguage. Bruhat-Tits did this and called the notion “a graded lattice chain”. The same no-tion has been rediscovered again and again. For example, they correspond to “filtrations” inMoy-Prasad theory, and “lattice sequences” in Bushnell-Kutzko’s work on semisimple types ofGLn.

Therefore, we review here the translation between the language of norms, and that offiltration/lattice sequences/graded lattice chains. We will adopt the terminology of “filtration”.

Definition. A filtration on V is a decreasing family of lattices in V , indexed by real numbers:{Lr }r ∈R, such that Lr +1 = π Lr .

Observation. There is a bijection

{norms on V} ↔ {filtrations on V}.

The construction is: α 7→ {Lr }r ∈R such that Lr = {v ∈ V : α(x) ≥ r }.

CHARACTERIZATION OF THE APARTMENT. 2

32 J. TITS

afine space A = A(G, S, K) under V on which N(K) operates. a system (D,f =

@af(G s9 a f B f f o a ne uric ions on A and a mapping a w X, of Qaf onto a set of

subgroups of G(K), such that the relation I. l(4) holds for s E N(K), that the vector parts v(a) of the firnctions a E Qaf are the elements of CD, and that, for a E (D, the groups Xa with v(a) = aform afiltration of UJK).

We first proceed with the construction of the space A; the set @, and the Xa’s will be defined in ssl.6 and 1.4. The relations (5) show us the way. The group X*(Z) of K-rational characters of 2 can be identified with a subgroup of finite index of X*. Let Y: Z(K) -+ V be the homomorphism defined by

(1) iC(m> = - o(~(z)) for z E Z(K) and x E Xz’(Z),

and let 2, denote the kernel of Y. Then, il = Z(K)/& is a free abelian group of rank dim S = dim V. The quotient r = N(K)/& is an extension of the finite group U @ by /I. Therefore, there is an affine space A (= A(G, S, K)) under V and an extension of Y to a homomorphism, which we shall also denote by Y, of N in the group of affine transformations of A. If G is semisimple, the system (A, Y) is canonical, that is, unique up to unique isomorphism. Otherwise, it is only unique up to isomor- phism, but one can, following G. Rousseau [19], “canonify” it as follows: calling 9G” the derived group of Go and Si the maximal split torus of the center of Go, one takes for A the direct product of A(5?G”, Go n S, K) (which is cano- nical) and X,(&) @ R .The affine space A is called the apartment of S (relative to G and K). The group N(K) operates on A through @.

1.3. Remark. Since V = Hom(X*, R) = Hom(X’:(Z), R), the groups Hom(X*, r) and Hom(X*(Z), r) are lattices in Vand one has

(I) Hom(X’$ Z’) c 21(2(K)) = /1 c Hom(X*(Z), Z’).

If G is connected and split, both inclusions are equalities, but in general they can be proper. Suppose for instance that G = RLIK Mult, where L is a separable extension of K of degree n, and let ri be the value group of L. The group X*(Z) is generated by the norm homomorphism NL,K, hence has index n in X*. On the other end, /I is readily seen to be equal to n*Hom(X*(Z), &). In particular, the first (resp. the second) inclusion (1) is an equality if and only if the extension L/K is unramified (resp. totally ramified). A semisimple example is provided by G = SUs with split- ting field L; exactly the same conclusions as above hold with n = 2 (indeed, in that case Z = RL,K Mult). One can prove that thefirst inclusion (1) is an equality when- ever G splits over an unramlJied extension of K.

1.4. Filtration of the groups U,(K). Let a E @ and u E U,(K) - {l}. It is known (cf. [3, $51) that the intersection U_,uU-, n N consists of a single element m(u) whose image in 0 w is the reflection ra associated with a, from which follows that r(u) = y(m(u)) is an affine reflection whose vector part is ra. Let a(a, u) denote the affine function on A whose vector part is a and whose vanishing hyperplane is the fixed point set of r(u) and let @’ be the set of all affine functions whose vector part belongs to CD. For a E cli’, we set X, = {u E U,(K) 1 u = 1 or a(a, u) 2 a>. The following results are fundamental.

1.4.1. For every a as above, X, is a group. 1.4.2. If a, p E CD’, the commutator group (Xa, Xp) is contained in the group gen-

erated by all XPa+,p for .p, q E N* and pa + qfi E @‘.

This is one of the questions I got asked most often regarding Tits’ article. What does thisparagraph mean? It is not difficult, but not entirely obvious either. The detail can be found inLandvogt’s book.

Explanation. The principle is this: suppose that you have an extension of groups

1 → 3 → N → W → 1,

where 3 is a free abelian group of finite rank, normal in N, and W is finite. Let V = 3 ⊗Z R.Then we can pull out the above diagram with 3 ↪→ V and get

1 → V → N ′→ W → 1.

If you represent the first extension by a class in H2(W, 3), the second extension is representedby the image of that class in H2(W, V). But H2(W, V) = 0, so the second extension is trivial.

Therefore, N ′' W n V . The isomorphism may not be unique, but the obstruction to

uniqueness lies in H1(W, V), which is again 0, so this isomorphism is actually unique.We notice that the action map a : W → Autgroup(V) is induced from W → Autgroup(3) =

GLZ(3), hence factors through GLR(V). Thus it is obvious that there exists a pair (A, f ) suchthat A affine space under V (i.e. a principal homogeneous space of V), f : N → Autaffine(A)

such that f (λ) is translation by λ for all λ ∈ 3, and d(

f (n))

= a(n) for all n ∈ N, where n isthe image of n in W.

Now assume that VW= 0. Suppose that both the pairs (A, f ) and (A′, f ′) have the above

property. We claim that there is a unique isomorphism (A, f ) ' (A′, f ′). Indeed, a simpleanalysis shows that the obstruction to uniqueness lies in VW

= H0(W, V) = 0.

2This is the beginning of the 2nd lecture, which is the most fragmented one, offering various complements toTits’ article.

This, when applied to the context in [Tits, 1.2], shows that for semisimple G, A(G, S, K )

is uniquely characterized as an affine space (up to a unique isomorphism). For reductive G, thereduced apartment has the same uniqueness.

EXTENDED BUILDING VERSUS REDUCED BUILDING.

[Tits] deals exclusively with the extended building (aka enlarged building, reductive build-ing). But throughout [BT], “building” usually mean the reduced building (aka semi-simplebuilding).

We recall that the reduced building is really canonically defined. The extended buildingis canonical in the sense that we can “canonify” the definition. But this is somewhat artificialand hence the behavior is not ideal. The main reason to favor the extended building is this:when the center of G is of split rank > 0, the stabilizer of points on Bred is no longer a compactsubgroup of G(K ).

We now recall that the construction of Bext from Bred. It is completely analogous to goingfrom Sred to S = Sext. We put

G(K )1= {g ∈ G(K ) : ordχ(g) = 0 ∀χ ∈ HomK (G, Gm)}.

Then G(K )/G(K )1 is a finite generated abelian group, and there is an isomorphism(G(K )/G(K )1

)⊗Z R ' X∗(Z) ⊗Z R,

where Z is the center of G. We let G(K ) acts on the vector space X∗(Z) ⊗Z R by translationsvia the above isomorphism, and define Bext as the product of two G(K )-sets: Bext = Bred ×

(X∗(Z) ⊗Z R).There is another way of dealing with this, and this viewpoint is also often used in [BT]:

instead of using G(K ) and Bext, we use G(K )1 and Bred.

In fact, since most results in [BT] are stated using Bred, it is often necessary to do the above.Notice that this means that the results in [BT] often actually apply to G(K )1 instead of G(K ).Therefore, one gets decomposition theorems etc. for G(K )1. Then one does a little bit morework to get the results for G(K ).

THE MAXIMAL BOUNDED SUBGROUPS. I want to bring attention to the fact that “max-imal bounded subgroups”, “stabilizer of vertices”, and “maximal parahoric subgroups” are alldifferent concepts. Although Tits’ article is rather clear about this, this is still a common mis-conception (probably because people tend to extrapolate the easier case of BN-pairs). A fewexamples should impress you about this. For simplicity, assume that K is a locally compactnon-archimedean field.

It is a consequence of Bruhat-Tits fixed-point theorem that every maximal compact sub-group of G(K ) is the stabilizer of some point on the building of G. But not all such stabilizersare maximal compact subgroups.

If x is a vertex (for the canonical polysimplicial structure on B), then the stabilizer G(K )x

of x is a maximal compact subgroup. If G is semi-simple and simply connected, this givesprecisely all the maximal compact subgroup.

This fails for non-simply connected groups. A simple and instructional example is G =

PGL2(K ). Recall that the lattice classes represented by the lattices L = O.e1 + O.e2, M =

O.e1 + πO.e2 corresponds to vertices x, y on the building which are connected by a 1-simplex

xy. For most point z on the interior of xy, G(K )z is G(K )x∩ G(K )y

=

{(a πbc d

)}, and is

equal to the Iwahori subgroup I associated to xy.

But their is one exception: when z is the midpoint of xy, G(K )z contains an extra element

which is represented by(

0 1π 0

). Notice that this switches x and y and hence fixes z. In this

case, [G(K )z : I] = 2 and G(K )z is another maximal compact subgroup.

Another Example. Let G = SO(2n), n ≥ 4. For simplicity say G is split. Write G = SO(V), andlet e1, . . . , en, f1, . . . , fn be a Witt basis of V (so the quadratic form is q(

∑xi ei +

∑yi fi ) =∑

xi yi ).For 0 ≤ i ≤ n, let L i be the lattice with basis

πe1, . . . , πei , ei +1, . . . , en, f1, . . . , fn.

According to Bruhat’s Boulder conference article (1965), each G(K ) ∩ GL(L i ) is a maximalcompact subgroup of G(K ), and there are exactly n + 1 conjugacy classes of maximal compactsubgroups. There are n + 1 vertices on a chamber. Are the maximal compacts simply thestabilizer of vertices? This seems reasonable. But if you look at the local Dynkin diagram(whose vertices correspond to vertices of a chamber) of this group, you may be puzzled: howcan we assign these lattices to the vertices on the local Dynkin diagram reasonably?

The answer is: no, there are two maximal compacts (up to conjugacy) which are are stabi-lizer of vertices. There are 4 “terminal vertices” on the local Dynkin diagrams, let’s call themx, y, z, w so that x, y is on one end and z, w on the other end. Then the stabilizer of x is con-jugate to that of y, and the stabilizer of z is conjugate to that of w. Moreover, the stabilizer ofthe midpoint of xy (resp. zw) is also a maximal compact. This accounts for the n + 1 maximalcompacts.

THE PARAHORIC SUBGROUPS.

It is quite interesting to notice what Tits said about Iwahori subgroups and parahoric sub-groups in his summary.

50 J. TITS

(1) P(k bi%) = inf{h) -f, w(x-j) +f, w(q)) -8 1 1 s i 5 n} i=l

(one can then choose the basis (bj) so thatf = 0 or 3&i)). The norm p is hyper- special if L/K is unramified and if there is a basis (bj) of E such that (1) holds for f=O.

3. Stabilizers and centralizers. From now on, G is assumed to be connected. 3.1. Notations; a BNpair. For every algebraic extension Kl of K with finite

ramification index and every subset Sa of the building a(G, Kl), we denote by G(Kl)O the group of all elements of G(K,) fixing Q pointwise. If Q is reduced to a point X, we also write G(K# for G(K#. Note that if F is a facet of @G, Kl) and if x is a point of F “in general position”, one has G(K#’ = G(K#. The stabilizers G(K)r of special (resp. hyperspecial) points x E &Y are called special (resp. hyper-

special) subgroups of G(K).

We recall that if G is semisimple and simply connected, the group @ =

NJww) coincides with the Weyl group W of the affine root system &. As be- fore, we set g = @G, K).

3.1.1. Suppose that p = W. Then G( K)F = G(K)x for every facet F of 9Y and every x E F. Furthermore, if C is a chamber of A = A(G, S), the pair (G(K)C, N(K))

is a BN-pair (or Tits system: cf. [5], [23]) in G(K) with Weyl group W. In that case, the groups G(K)x for x E g are called the parahoric subgroups of G(K) (cf. [S]), but we shall avoid using that terminology here in order not to prejudge of its most suitable extension to the nonsimply connected case. An alternative con- struction of the building a starting from the above BN-pair (which can be defined independently of the building, as we shall see) and using the parahoric subgroups defined by means of that BN-pair is given in [S, 521.

Let Q be a nonempty subset of the apartment A whose projection on the building of the semisimple part of G (cf. last paragraph of 2.1) is bounded. For any root a E @, let a(a, 0) denote the smallest affine root whose vector part is a and which is positive on Q. Let @’ be the set of all nondivisible roots-i.e., all roots a E @ such that *a 4 @-and let 0’+ (resp. @‘-) be the set of all nondivisible roots which are positive (resp. negative) with respect to a basis of 0, arbitrarily chosen. Set

NW = N(K) n G(K)” and let ZC and Xa be defined as in 831.2 and 1.4. Then one has the following group-theoretical description of G(K)$ (cf. C&6.4.9, 6.4.48, 7.4.41) :

If Xk(Q) denotes the group generated by all Xaca sa) with a E Qft-, the product

maPPiw l-I aa’+ xc&, 0) --+ X’(Q) is bijective for evey ordering of the factors of

the product and one has G(K)Q = X-(Q) l X+(Q) l N(K)? If 0 contains an open

subset of A, the product mapping naEQ/ Xnce, Q) x 2, -+ G(K)* is btjective for every ordering of the factors of the product.

3.2. Maximal bounded subgroup. For every nonempty subset Q of a, G(K)O is a bounded subgroup of G(K) (cf. $2.2). If the residue field K is finite, G(K)Q is even compact and, in what follows, “maximal bounded” can be replaced by “maximal compact”.

From 2.3.1, one easily deduces that:

REDUCTIVE GROUPS OVER LOCAL FIELDS 51

every bounded subgroup of G(K) is contained in a maximal one and every maximal

bounded subgroup is the stabilizer G(K)% of a point x of 9?.

It is now clear that if x belongs to a facet of minimal dimension of g, G(K)% is a

maximal bounded subgroup of G(K), in particular, special subgroups are maximal bounded subgroups. From 3.1.1, it follows that the above two statements give a complete description of the maximal bounded subgroups in the simply connected case :

if G is semisimple and simply connected, the maximal bounded subgroups of G(K) are

precisely the stabilizers of the vertices of the building g; they form n rzl (& + 1) con- jugacy classes, where II, l l l , I, denote the relative ranks of the quasi-simple factors of G.

For an analysis of the nonsimply connected case, cf. [S, 3.3.51. 3.3. Various decompositions. Let C be a chamber of A = A(G, S). We identify A

with the vector space Yin such a way that 0 becomes a special point contained in the closure of C; in particular, G(K)0 is a special subgroup of G(K). Set D = RT l C

(a “vector chamber”) and B = G(K)C; if K is finite or, more generally, if G is re- sidually quasi-split, and if G is simply connected, B is an Iwahori subgroup of G(K) (cf. $3.7). Let U+ be the group generated by all Ua for which ale--and hence aID-

is positive and let Y be the “intersection of Y and @“, that is, the group of all translations of A contained in @; thus, Y is the image of Z(K) by the homomor- phism Y of $1.2. Set Y+ = Y 0 D (closure of D) and Z(K)+ = y-1( Y+), a subsemi- group of Z(K).

3.3.1. Bruhat decomposition. One has G(K) = BN(K)B and the mapping BnB w

v(n) (n E N(K)) is a bijection of the set {BgB 1 g E G(K)} onto r. If n E N(K) and y(n) = w, we also write BnB = BwB, as usual. If K = Z$ the

cardinality qW of BwB/B (used for instance in [1]) is given by the following formula in terms of the integers d(v) of 5 1.8 : set w = rl* +wo, where (rl, 9 -0, rl) is a reduced word in the Coxeter group wand we(C) = C, and let vi be the vertex of A repre- senting ri; then qW = qd with d = Cf=i d(vi). In particular, we have another inter- pretation of d(v): qd(y) = q,+) where r(v) denotes the fundamental reflection cor- responding to the vertex v of A.

More generally, for any 13, the quotient BwB/B has a natural structure of “per- fect variety” over K, in the sense of Serre [Publ. Math. I.H.E.S. 7 (1960), 1.41, and, as such, it is isomorphic to a K-vector space of dimension CfK1 d(v,), with the above notations.

3.3.2. Iwasawu decomposition. One has G(K) = G(K)OZ(K)U+(K) and the map- ping G(K)OzU+(K) I+ Y(Z) (z E Z(K)) is a bijection of {G(K)OgU+(K)( g E G(K)} onto Y.

3.3.3. Cartun decomposition. One has G(K) = G(K)OZ(K)G(K)o and the map- ping G(K)OzG(K)O I--, Y(Z) (z E Z(K)+) is a bijection of {G(K)OgG(K)Olg E G(K)}

onto Y+. In particular, we see that if K is finite, the convolution algebra of all functions

G(K) w C with compact support which are bi-invariant under G(K)0 (Hecke al- gebra) has a canonical basis indexed by Y+. That algebra is commutative.

For the proofs and some generalizations of the above results, cf. [S, $41. 3.4. Some group schemes. The results of this section and the next are special

cases of results which will be established in [9].

54 J. TITS

and if & is the vertex marked with a *. Indeed, it is readily verified that in all other cases, QF contains all nonmultipliable relative roots of G, and the assertion follows from [4, 2.23 and 4.31. In the exceptional case above, @sd is a special orthogonal group, hence not simply connected. Using the fact that in a simply connected group the derived group of the centralizer of a torus is also simply connected, one easily deduces from the preceding result the following more general one. Let us say that a special vertex of the absolute local Dynkin diagram Lill is good if it is not the vertex * of a connected component of type (1) of that diagram. Then ifG is semisimple and

simply connected and tf utrEIF O(v) contains a good special vertex out of each

connected component of Al, the derived group of GFd is simply connected.

3.6. Fixedpoints of groups of units of tori. Let M be a subgroup of the group of units SC = (8 E S(K) I 4x(s)) = 0 f or all x E X*} of S. We wish to find under which condition the apartment A = A(G, S) is the full fixed point set 99M of M in A?. From the properties of the building recalled in $2.2, one deduces that A = &?M if and only if, for every facet F of A of codimension one, the only chambers con- taining Fin their closure and fixed by M are the two chambers of A with those pro- perties. By 3.5.4, that means that the image i@ of M in S(K) has only two fixed points in the spherical building of @d over K. If a is any one of the two nondivi- sible roots in GF, that condition amounts to a(@) # (1). Thus, we conclude that:

3.6.1. A necessary and suflcient condition for A to be the fullfixed point set of A4 in g is that a(M) q~ 1 + p for every relative root a E @.

In particular, iflv has at least four elements (resp. if K N F2) A is always (resp. never) the full

Jixedpoint set of S, in B.

The preceding discussion also gives information on the fixed point set of the group of units S1 c of a nonsplit torus Si which becomes maximal split over an unramified Galois extension K1 of K: one applies 3.6.1 to the action of S1,, on a (G, K1) and one goes down to 9 by Galois descent, using 2.6.1. In that way, one gets the following result for instance:

If S1 is an anisotropic torus which becomes maximal split over an unramiJied Galois

extension of K, then S,(K) has a uniquefixedpoint in the building a.

By contrast, it is easily shown that if S1 is a maximal torus of G = SL2 whose splitting field is ramified, then S1(K) necessarily fixes a chamber of 9 and possibily more than one2; for a similar torus S1 in PGL2, S(K) may have a single fixed point in g and may have more than one.

3.7. Iwahori subgroups; volume of maximal compact subgroups. In this section, we suppose G residually quasi-split; remember that that is no assumption if the residue field Kis finite (1.10.3).

To every chamber C of the building g, we associate as follows a subgroup Iw(C) of G(K), called the Iwahori subgroup corresponding to C: if c; denotes the neutral component of the algebraic group G, (cf. 3.4), Iw(C) is the inverse image in %(o) = G(K)C of the group c:(E) under the reduction homomorphism SC(o) -+ G,(g). Clearly, all Iwahori subgroups of G(K) are conjugate. From 3.5.2, it follows that cl is a solvable group, hence is the semidirect product of a torus T by a uni-

2 This answers a question of G. Lusztig.

Tits was indeed very careful in stating everything precisely. But still, in the literature peoplesometimes quote these results incorrectly.

We saw that Tits was undecided about what parahoric subgroups should be (though he wasfirm about the case of Iwahori subgroups). Today there should be no ambiguity. In [BT2],Bruhat-Tits defined parahoric subgroups in the same way Tits defined the Iwahori subgroups.Namely, the definition has to involve the smooth group schemes constructed by Bruhat-Tits.

Therefore, the parahoric subgroups are not the stabilizer/fixer of a facet on the building.However, we caution you that in the literature, some people use conventions inconsistent withBruhat-Tits.

Bruhat-Tits’ choice has several advantages:

— There is a bijection between facets and parahoric subgroups, F 7→ G(K )F . The bijectionis order reversing: G(K )F ⊂ G(K )F ′ ⇐⇒ F ′

⊂ F . [This is just like the case of aBN-pair].

— The parahoric subgroups contained in a fixed parahoric subgroups G(K )F are in bijectionwith the parabolic subgroups of G modπ , where G is the smooth group scheme associatedto G(K )F . In other words, they are in bijection with facets on the associated sphereicalbuilding.

— The smooth group scheme G associatd to a parahoric subgroup are connected by defini-tion. We can apply representation theory of G(O/π). Most theory (e.g. work of Lusztig)does require G modπ to be connected.

Caution. Be careful about the following statement regarding the disconnection of stabilizers.The proof can not be found in [BT] and the statement is incorrect. It is not clear to me how tomake the right statement.

REDUCTIVE GROUPS OVER LOCAL FIELDS 53

arbitrary reductive groups: if 2a is not a root, we simply take the coroot associated with a in the split subgroup of maximal rank defined in [3, $71; if 2a is a root, we define the coroot associated with a as being twice the coroot associated with 2a (& being written additively).

3.5.1. The root system of GFd with respect to s is the system (DF (cf. 1.9); in partic- ular, its Dynkin diagram is obtained from the local Dynkin diagram A(G, K) by deleting the vertices belonging to IF (cf. 1.9) and all edges containing such vertices. The coroot associated with a root a E @i is the same for @d as for G. If o= denotes the unipotent subgroup of GFd corresponding to a, the group u=(R) is nothing else but the group X, of 5 I .4, where a is the afine root vanishing on F and whose vector part is a.

Applying that to the unramified closure of K, one gets the following immediate consequence.

. 3.5.2. The index of @d over K, in the sense of [3] and [22], is obtained from the

local index of G by deleting from Al all vertices belonging to the orbits O(v) with v E IF (the notations are those of $j 1.11) and all edges containing such vertices. In particular, if G is residually quasi-split (resp. residually split), @d is quasi-split (resp. split):, When F is a chamber, then G is residually quasi-split (resp. residually split) if and only ifcf;,ed is a torus (resp. a split torus).

If G is simply connected, the group GO is connected. In general, the group of com- ponents of GB is easily computed when one knows the group Z’i = E(G, K,) (cf. §2.5), where K1 is the maximal unramified extension of K. Here, we shall give the result only in the case of a facet.

3.5.3. The group of components of GF is canonically isomorphic with the intersec- tion of the stabilizers of the orbits O(v) with v E IF in the group El. A component is defined over K zf and only if the corresponding element of EI is centralized by Gal(Ki/K). If K isf%tite, every component of GF which is defined over K has a K- rational point (by Lang’s theorem).

The groups Gpd give an insight into the geometry of the building through the following statement:

3.5.4. The link of F in B is canonically isomorphic with the spherical building of @d over K, i.e. the “building of K-parabolic subgroups” of Gyd (cJ [23,5.2]).

The groups Gr;;‘d also provide an alternative definition of the integers d(v) of $X8. Suppose F is of codimension one and let v be the complement of IF in the set of all vertices of A. Then, @d has semisimple K-rank 1 and d(v) is the dimension of its maximal unipotent subgroups, or, equivalently, the dimension of the variety G>/PF, where PF is a minimal K-parabolic subgroup of G& the neutral component of CF. This, together with 3.5.4, implies the interpretation of d(v) given in 2.4. If G is residually split, c>/& is a projective line, hence d(v) = 1; in particular, we re- cover the last statement of $1.8.

While 3.5.2 gives an easy algorithm to determine the type of Gyd, 3.5.1, applied to the unramified closure of K, actually provides the absolute isomorphism class of that group. Here is an immediate application of that. Suppose that G is quasi- simple, simply connected and residually split and that F is a special point. Then, Gyd is a simply connected quasi-simple group except if the local Dynkin diagram is the following one:

(1) m-1’ - - -- *

Example. If G = T is a torus and K = K , and x any point on the building, then a result ofKottwitz (which was made explicit by Rapoport) says that the component group is the torsionsubgroup of the group of co-invariants X∗(T)Gal(K/K ).

VALUATION OF ROOT DATUM.

A basic feature of Bruhat-Tits theory is that if Ua is a root subgroup, then Ua(K ) is filtred.For example, for split groups, Ua(K ) is (isomorphic to) the additive group of K , and is filteredby the subgroups πnO.

For Bruhat-Tits theory, it is important that you index the filtrations on Ua(K ), for varyinga’s, in a coherent way. In fact, this point is almost the whole theory. But the languages of doingthis in [Tits] and [BT] are different.

• In [Tits], Ua(K ) is filtered by the groups Xα, where α varies over affine functions on Awith vector part a. The set of such functions form a real line, of course.

• In [BT], Ua(K ) is filtered by Ua(K )ϕ,r , a family of groups indexed by the real line Rdirectly [the notation ϕ is there to indicate the choice of a “valuation of root datum”].

The approach of [Tits] is elegant, since it does not depend on any anxiliary choice. Incontrast, the filtration in [BT] depends on the choice of ϕ, “the valuation of root datum”. Theprecise definition is somewhat long, and it is difficult to show that such an object exists. In fact,a very large part of [BT] is devoted to its existence.

Once the existence is known, the whole theory of building can be developed. In fact, it thenfollows that to give a valuation of root datum is to give a point on the building. Therefore, thenotion of valuation of root datum is indeed very nature, and it can be considered as a remoteanalog of Cartan involution.

In [Tits], this filtration of Ua(K ) indexed by affine functions is always well-defined. Whatis not clear is that this filtration has many wonderful properties. If you read [BT], you don’tsee any attempt to prove these properties from the definition in [Tits]. Instead, all the effort isputting into proving the existence of a valuation of root datum. It will follow, however (sincethere is a translation between the two languages), that all the nice properties are valid for thegroups Xα.

In order to access the results in [BT] not summarized in [Tits], one needs to know thetranslation, which we give now.

First, today “root datum” usually means the gadget (consisting of a dual pair of lattices anda dual pair of root systems) classifying a split reductive group, as defined in SGA3 (see also,Springer’s Corvallis article).

However, in [BT], “root datum” is a gadget in abstract group theory. For us, the “root da-tum” for a reductive group over K is the datum

({Ua(K )}a∈8(G,S), Z(K )

), where S is a maximal

K -split torus in G, Ua the root subgroup, and Z = ZG(S).Then, the set of valuations on the root datum

({Ua(K )}a∈8(G,S), Z(K )

)is in bijection with

points on Ared = Ared(G, S, K ). Indeed, by fixing x ∈ Ared, we can put a filtration {Ua(K )x,r }r ∈Ron Ua(K ) by setting

Ua(K )x,r =

⋃dα=aα(x)≥r

Xα.

Conversely, if these filtration groups are known, then we can recover the group Xα’s by

Xα = Ua(K )x,α(x).

Notice that I have adopt the convention that a point on the building is a valuation of a rootdatum. This completes the translation.

FIGURING OUT THE FILTRATION.

As we have mentioned, [Tits] defined the filtration {Xα}α on Ua(K ) by an elegant recipe.But it is quite hard to prove the good properties of this filtration.

32 J. TITS

afine space A = A(G, S, K) under V on which N(K) operates. a system (D,f =

@af(G s9 a f B f f o a ne uric ions on A and a mapping a w X, of Qaf onto a set of

subgroups of G(K), such that the relation I. l(4) holds for s E N(K), that the vector parts v(a) of the firnctions a E Qaf are the elements of CD, and that, for a E (D, the groups Xa with v(a) = aform afiltration of UJK).

We first proceed with the construction of the space A; the set @, and the Xa’s will be defined in ssl.6 and 1.4. The relations (5) show us the way. The group X*(Z) of K-rational characters of 2 can be identified with a subgroup of finite index of X*. Let Y: Z(K) -+ V be the homomorphism defined by

(1) iC(m> = - o(~(z)) for z E Z(K) and x E Xz’(Z),

and let 2, denote the kernel of Y. Then, il = Z(K)/& is a free abelian group of rank dim S = dim V. The quotient r = N(K)/& is an extension of the finite group U @ by /I. Therefore, there is an affine space A (= A(G, S, K)) under V and an extension of Y to a homomorphism, which we shall also denote by Y, of N in the group of affine transformations of A. If G is semisimple, the system (A, Y) is canonical, that is, unique up to unique isomorphism. Otherwise, it is only unique up to isomor- phism, but one can, following G. Rousseau [19], “canonify” it as follows: calling 9G” the derived group of Go and Si the maximal split torus of the center of Go, one takes for A the direct product of A(5?G”, Go n S, K) (which is cano- nical) and X,(&) @ R .The affine space A is called the apartment of S (relative to G and K). The group N(K) operates on A through @.

1.3. Remark. Since V = Hom(X*, R) = Hom(X’:(Z), R), the groups Hom(X*, r) and Hom(X*(Z), r) are lattices in Vand one has

(I) Hom(X’$ Z’) c 21(2(K)) = /1 c Hom(X*(Z), Z’).

If G is connected and split, both inclusions are equalities, but in general they can be proper. Suppose for instance that G = RLIK Mult, where L is a separable extension of K of degree n, and let ri be the value group of L. The group X*(Z) is generated by the norm homomorphism NL,K, hence has index n in X*. On the other end, /I is readily seen to be equal to n*Hom(X*(Z), &). In particular, the first (resp. the second) inclusion (1) is an equality if and only if the extension L/K is unramified (resp. totally ramified). A semisimple example is provided by G = SUs with split- ting field L; exactly the same conclusions as above hold with n = 2 (indeed, in that case Z = RL,K Mult). One can prove that thefirst inclusion (1) is an equality when- ever G splits over an unramlJied extension of K.

1.4. Filtration of the groups U,(K). Let a E @ and u E U,(K) - {l}. It is known (cf. [3, $51) that the intersection U_,uU-, n N consists of a single element m(u) whose image in 0 w is the reflection ra associated with a, from which follows that r(u) = y(m(u)) is an affine reflection whose vector part is ra. Let a(a, u) denote the affine function on A whose vector part is a and whose vanishing hyperplane is the fixed point set of r(u) and let @’ be the set of all affine functions whose vector part belongs to CD. For a E cli’, we set X, = {u E U,(K) 1 u = 1 or a(a, u) 2 a>. The following results are fundamental.

1.4.1. For every a as above, X, is a group. 1.4.2. If a, p E CD’, the commutator group (Xa, Xp) is contained in the group gen-

erated by all XPa+,p for .p, q E N* and pa + qfi E @‘.

[Tits] also provided several examples of computing the filtration Xα. However, in myexperience this is not a very pleasant drill, in particular for the exceptional groups.

Therefore, in practice, to figure out the filtration, it is easier to follow the strategy in [BT],which is to go through two descent steps: descending from the split case to the quasi-split case,and descending from the quasi-split case through an unramified extension.

I do not have the time to go through either steps. So I will just recast the starting point: thesplit case (which was already emphasized by [Tits] in the beginning, section 1.1) in the languageof valuation of root datum. You will see that it is very simple. The quasi-split case is also fairlysimple, and can be found in either [BT2], Landvogt’s book, or Prasad-Raghunathan’s article oncentral extensions.

So let G be a Chevalley scheme over O. This means that G is a smooth group scheme overO, such that both the generic fiber G and the special fiber are split reductive groups.

Then G has a maximal split torus S over O, with respect to which there is a root decom-position of Lie G, and root subgroups Ua, a ∈ 8(G, S), S = SK . There are isomorphismsxa : Ga → Ua over O. Then we put

Ua(K )r = (xa ⊗O K )({λ ∈ K : ord(λ) ≥ r }).

NOTATION FOR THE THIRD LECTURE. 3

• K : non-archimedean local field

• K : the maximal unramified extension of K

• O, O, π , κ = O/π , κ = O/π

3This is the beginning of the third lecture, in which I discuss more recent results beyond Tits’ article.

• G: connected reductive group over K

• S: maximal K -split torus in G

• 8 = 8(G, S): relative root system

• 8aff: relative affine root system

• {Ua}a∈8: root subgroups

• Z = ZG(S): centralizer of S, also denoted by U0

A theorem of Steinberg. G ⊗K K is quasi-split. Therefore, if T is a maximal K -split torus inG ⊗K K , then ZG(T) is a torus.

Key result of Bruhat-Tits’ “Etale Descent”. There exists a torus T over K , containing S, suchthat T ⊗K K is a maximal K -split torus.

Consequence. Bruhat-Tits theory behaves very well respect to unramified extensions. For ex-ample, B(G ⊗K K )0

= B(G), where 0 = Gal(K/K ) ' Gal(κ/κ).

Convention. Today I will always assume: G is quasi-split over K and in addition, we can takeS = T . This is for simplicity of exposition. The extension to the general case is fairly easy.

BACKGROUND ABOUT INTEGRAL MODELS

Remark. For the discussion here, G can be any linear algebraic group over K .

Model. A (integral) model of G is an affine group scheme G (of finite type) over R with genericfiber

GK := G ⊗R K = G.

Concretely, you try to rewrite the defining equations and the polynomials defining multi-plication and inverse using polynomials over O. A model is such a system of O-equations.

Remark. A model carries more structure than the original algebraic groups. Examples:

• G(R): groups of integral points, i.e. solutions to the equations with coordinates in R;

• Gκ : special fiber of G or reduction of G: take the defining equations/polynomials mod π ,we get an algebraic group over κ (not necessarily smooth!)

• Lie G: Lie algebra of G, a lattice in Lie G, closed under Lie bracket.

For simplicity, assume that K = K for a while.

One may consider all subgroups of G(K ) of the form G(R) privileged. Such a subgroup isquite special, e.g. it is open and bounded (and more). Here, we call such a subgroup schematic.

If H is schematic subgroup of G(K ), there are many G’s such that G(R) = H . But one ofthem is canonical, and is characterized by either

• it is the initial object in the category of models G such that G(R) = H ; or

• it is smooth over R.

This follows from the theory of group smoothening (Neron, Raynaud, etc). This canonicalmodel will be denoted by GH .

Remark. Bruhat-Tits showed that many groups occuring in Bruhat-Tits theory are schematic.

This theory attached to schematic subgroups H extra structures which are quite non-trivial,e.g.

• the congruence subgroups

0(πn; GH ) =

{g ∈ G(R)

∣∣∣ g modπn= 1 in G(R/πn R)

},

• the special fiber of GH ,

• the Lie algebra of GH .

Remark. If K 6= K , and H is a schematic subgroup of G(K ) which is stable under Gal(K/K ),then GH , which is defined over O a priori, is actually defiend over O.

THE FILTRATION ON Ua REVISITED

We recall that Ua(K ) is filtred by the {Xα}dα=a, or by the {Ua(K )x,r }r ∈R.

Now according to my convention S = T , the meaning of “Ua” doesn’t change when yougo from K to K . So we can also talk about Ua(K )x,r etc. This group is Gal(K/K )-stable.

Fact. The group Ua(K )x,r is schematic. Let Ua,x,r be the associated smooth model over O.Then Ua,x,r (O) = Ua(K )x,r .

Fact. 0(πn; Ua,x,r ) = Ua(K )x,r +n.

Fact. The special fiber of Ua,x,r is a product of additive groups Ga.

THE FILTRATION ON Lie Ua

It is natural to consider the Lie algebra of Ua,x,r . Although this is already considered in[BT], Moy-Prasad seems to be the first to consider the whole family {Lie Ua,x,r }r ∈R together.

Fact. Let ua,x,r = Lie Ua,x,r . Then the family {ua,x,r } is a filtration on Lie Ua in the sensewe discussed in the first lecture (of the kind called “lattice sequence” by Bushnell-Kutzko;i.e. ua,x,r +1 = πua,x,r ).

Remark. The definition here is not the one given by Moy-Prasad. They made more ad hocusage of the fact that G is quasi-split and specific knowledge of the structure of Ua. There wasno direct connection to Ua,x,r . It was just an analogy.

The definition here is more natural, and it shows that ua,x,r is attached to Ua,x,r canonically.It also makes the following transparent:

Moy-Prasad isomorphism. we have

Ua(K )x,r /Ua(K )x,r + ' ua,x,r /ua,x,r +,

where

Ua(K )x,r + =

⋃s>r

Ua(K )x,s,

etc.

THE FILTRATION ON U0(K ) AND Lie U0

We recall the convention U0 = ZG(S). Also notice that U0 is the last part of Bruhat-Tits’“root datum”

((Ua)a∈8,U0)

).

It is interesting to observe that from Chapter I of [BT], it is clear that Bruhat-Tits wanted toconsider a filtration on U0(K ). But somehow this was not done in Chapter II or any subsequentchapters. Ten years later, Moy-Prasad (Inventiones 1994) and Schneider-Stuhler (Publ. IHES1997) supplied the (same) definition.

Definition. Let Z be any torus over K . Let L/K be an extension over which Z is split. DefineZ(L)0 to be the maximal bounded subgroup of Z(L) and for r > 0,

Z(L)r := {z ∈ Z(L) : ordK (χ(z − 1)) ≥ r ∀χ ∈ HomL(Z, Gm)}.

Here ordK : L×→ Q is the valuation extending the normalized valuation ordK : K × � Z. We

then define Z(K )0 to be the Iwahori subgroup of Z(K ), and for r > 0,

Z(K )r := Z(K )0 ∩ Z(L)r .

Let z = Lie Z = Lie U0 = u0. Moy-Prasad also defined

zr = u0,r = {x ∈ z : ordK (dχ(x) ≥ r ∀χ ∈ HomL(Z, Gm)}.

One can show that {Z(K )r } satisfies the conditions prescribed in Chapter I of [BT]. How-ever, there are some problems with these definitions. The most serious one is that there is noMoy-Prasad isomorphism in general. But the Moy-Prasad isomorphism is used in a fundamen-tal way in the prof of Moy-Prasad theory.

We remark that the Moy-Prasad isomorphism is valid in many cases, for example when Zbecomes an induced torus over a tamely ramified extension. For example, the first paper ofMoy-Prasad is OK since they deal with simply connected groups, and for such groups, Z is aninduced torus.

There are now two ways to resolve the situation.

(i) Debacker has given alernative arguments to the main results in Moy-Prasad theory, with-out using the Moy-Prasad isomorphism. But some features of the theory is lost [MichiganJournal, 2000].

(ii) I have given a different definition of {Z(K )r } and {zr }. For this definition, the Moy-Prasad isomorphism is valid and hence the rest of the Moy-Prasad theory can be used withoutany modification [preprint 2002].

But the new definition involves a lot more algebraic geometry, and is more difficult tocompute. I will not give it here. However, if Z becomes an induced torus over a tamelyramified extension, the two definitions agree.

For the rest of this lecture, I will use my definition, or assume that Z becomes an inducedtorus over a tamely ramified extension.

THE MOY-PRASAD FILTRATION

Definition. Let x ∈ B(G), r ≥ 0. The Moy-Prasad filtration group associated to (G, x, r ) is thesubgroup of G(K ) generated by Ua(K )x,r for all a ∈ 8 ∪ {0}, where we put U0(K )r = Z(K )r .

Similarly, gx,r is the (direct sum) of ua,x,r for all a ∈ 8 ∪ {0}, where u0,x,r = u0,r = zr .

The Moy-Prasad isomorphism. For r > 0,

G(K )x,r /G(K )x,r + ' gx,r /gx,r +.

Main theorem in Moy-Prasad theory. Let (π, V) be an irreducible admissible representation ofG(K ). The number

ρ = inf{r ∈ G(K ) : ∃x ∈ B(G) s.t. VG(K )x,r + 6= 0}

is a rational number and the infimum can be achieved for some x ∈ B(G). It is called the depthof π .

Moreover, if VG(K )x,ρ+ 6= 0, the irreducible constituents of this space as a representationof G(K )x,r /G(K )x,r + enjoy certain nice non-degeneracy characterization (which relies on theMoy-Prasad isomorphism).

The depth of a representation is preserved by Jacquet functor and parabolic induction.

I would like to mention a nice interpretation of the depth.

Theorem. Let G = GLn, or a tamely ramified torus. Let π be an irreducible smooth represen-tation of G and φ : W′

K →L G the corresponding Langlands parameter. Then

depth(π) = inf{r : φ(Pr +) = id}.

Here, P ⊂ WK ⊂ W′

K is the wild inertia group and the filtration is the upper number filtrationby ramification groups.

We suspect that this relation is valid fairly generally. Therefore, the depth should be pre-served by Langlands functoriality.

For example, it should be preserved by the Generalized Jacquet-Langlands/Deligne-Kahzdan-Vigneras correspondence between representations of GLn and those of division algebras. Re-cently, [Lansky and Raghuram 2003] proved that it is so in some cases.

MORE GROUP SCHEMES

Theorem. The groups G(K )x,r are schematic. Let Gx,r be the associated smooth model over O,then Lie Gx,r = gx,r . Moreover,

0(πn, Gx,r ) = G(K )x,r +n.

It is interesting to remark that the definition of Moy-Prasad filtrations and Schneider-Stuhlerfiltrations are rather involved. But the theory of smooth models now allow us to give a veryshort and conceptual description of the Schneider-Stuhler filtration U (e)

F associated to facet F:

Let UF be the canonical model for Stab(G, F), and let U+

F be the dilatation on UF

of Ru((UF)◦

κ

). Then U (e)

F = 0(πe; U+

F ) for all e ∈ Z≥0.

AN INTERPRETATION OF THE MOY-PRASAD FILTRATION.

Fact. The lattices {gx,r }r ∈R form a filtration on the vector space g corresponding to a norm ong, i.e. gx,r +n = πngx,r .

Therefore, it corresponds to a norm αx on g. Recall that we can regard αx as a point onB(GL(g)). Therefore, the theory of Moy-Prasad filtrations for g is a map

ι : B(G) → B(GL(g)).

Remark. Moy-Prasad also defined a filtration on g∗. From our point of view, this is simplyobtained by composing the above map with the natural isomorphism B(GL(g)) ' B(GL(g∗)).The latter map is the one induced by the identificaiton GL(g) = GL(g∗) and is describedexplicitly in [BT3].

For simplicity, assume that G is adjoint, so G ↪→ GL(g). Then the map ι : B(G) →

B(GL(g)) enjoys the following properties.

(i) it is G(K )-equivariant;(ii) it is continuous and injective;(iii) it is toral in the sense that if S is a maximal K -split torus of G, there exists a maximal K -

split torus S′ of G = GL(g) such that S ⊂ S′ and ιA(G, S) ⊂ A(G′, S′) and the restrictionι|A(G, S) → A(G′, S′) is affine;

(iv) it is compatible with base field extensions.

This is an example of descent maps between buildings, and in this case it is canonically defined.The theory of descent maps can be considered as the functoriality of the building. Below wefirst review the case of symmetric spaces.

FUNCTORIALITY OF SYMMETRIC SPACES.

Functoriality. Assume that G, H are reductive groups over R such that G ⊂ H . Suppose thatx ∈ S(G), y ∈ S(H) are such that the associated maximal compact subgroups Gx ⊂ G(R),Hy ⊂ H(R) satisfy Gx ⊂ Hy, then we have a G(R)-equivariant map S(G) → S(H), sendingg.x to g.y. It is easy to see that Hy ∩G(R) = Gx, hence this equivariant map is always injective.

However, the Cartan involution θy associated to y may not stabilize G.

Example. Let G = SL2, H = SL4, and define the inclusion ι : G ↪→ H by g 7→

(g 00 t g−1

).

The standard maximal compact subgroup K of G(R) is the stabilizer of the quadratic formrepresented by the 2 × 2 identity matrix I2.

It is easy to see that ιK stabilizes the quadratic form q represented by the 4 × 4 matrix(aI2 bI2bI2 cI2

). When q is positive definite (for example, a = c = 1 and b sufficiently small), the

stabilizer Kq of Qq in H(R) is a maximal compact subgroup such that ιK ⊂ Kq. But the Cartaninvolution θq associated to Kq does not stabilize ιG(R) as long as b 6= 0.

Definition.

(i) We say that the map S(G) → S(H), g.x 7→ g.y is a descent map if θy stabilizes G(R) (andhence θy descends to θx: θy|G(R) = θx).

(ii) We say that S(G) → S(H) is a toral map if there exists a maximal R-split torus S of G, amaximal R-split torus T of G such that x ∈ A(S), y ∈ A(T), and S ⊂ T .

Proposition (i) A descent map is a toral map.

(ii) A toral map is a descent map.

(iii) If S(G) → S(H) is a descent map then θy′ |G(R) = θx′ for all x′∈ S(G), where y′ is the

image of x′.

Base change. Denote ResC/R(G⊗C) by GC. Then there is a canonical map Sred(G) → Sred(GC).It is constructed as follows: let θ ∈ Sred(G), and let k and p be the +1 and −1 eigenspaces of θ ong = Lie G. Then a Cartan involution θC on gC = g⊗C = Lie GC is defined by θC(k+ p) = k− pfor all k ∈ k ⊗ C, p ∈ p ⊗ C.

Now the base change map Sred(G) → Sred(GC) is θ 7→ θC. It is easy to see that this map isG(R)-equivariant.

We also have an obvious inclusion V(G) ↪→ V(GC). Combining with the previous con-struction, we get a base change map

S(G) → S(GC),

which is clearly a G(R)-equivariant descent map.

Proposition The image of the base change descent map S(G) → S(GC) is S(GC)Gal(C/R).

Proposition Assume that G ⊂ H is an embedding of complex groups in the sense that there arecomplex reductive groups G′

⊂ H ′ such that G = ResC/R G′ and H = ResC/R H ′. Then anyequivariant map S(G) → S(H) determined by an inclusion Gx ⊂ Hy is a descent map.

Descent maps and base change. Let ι : G ⊂ H be an inclusion of real reductive groups.Assume x ∈ S(G), y ∈ S(H) are such that Gx ⊂ Hy and denote by ι∗ the equivariant mapS(G) → S(H), g.x 7→ g.y.

Proposition The map ι∗ is a descent map if and only if ι∗ extends to a GC(R)-equivariant mapS(GC) → S(HC).

Finite group actions.Suppose that we have a finite group F acting on H . Then G = (H F)◦ is also reductive. We

denote the inclusion G → H by ι.

Theorem Let X be the set of descent maps ι∗ : S(G) → S(H) satisfying ι∗(S(G)) ⊂ S(H)F .Then,

(i) X is non-empty.(ii) ι∗(S(G)) = S(H)F for all ι∗ ∈ X.(iii) X is a principal homogeneous space of V(G).

FUNCTORIALITY OF BUILDINGS

Unfortunately, this is still not well-understood and there is no adequate literature. The mainreference is [BT1] and

— E. Landvogt, Some functorial properties of the Bruhat-Tits building, J. Reine Angew.Math. 518 (2000), 29 pages.

Nothing is mentioned about the analogy with the case of symmetric spaces. But I thinkthat this analogy provides very good insights and motivations. The two references also talk intotally different languages. [BT1] talks about analogues of the descent maps, namely they wantto consider a compatibility condition regarding valuation of root data. In Landvogt’s article,he considers the analogoue of the toral maps.

In an article which I am guilty of not finishing, I showed that these two concepts are thesame, and so are several analogues of the above-mentioned results in the symmetric space case.The analogue of the theorem about finite group action (with a tameness assumption) is now aTheorem of Gopal Prasad and myself (Inventiones 2002).

The most important result in this direction is the main theorem of Landvogt in the abovementioned paper, which asserts the existence of descent maps compatible with unramified basechange. However, for some applications we do want to know the existence of descent mapscompatible with ramified finite extension of base field. I can show the existence if the residuecharacteristic is not 2.

Remark. Proposition 2.4.1 in [Landvogt] can be proved easily from Proposition 1.3 [Prasad-Yu]. This would save a few pages of arguments.

Remark. The last main theorem (2.7.4 and 2.7.5) in [Landvogt] appears to be incorrect in boththe statement and the proof. The case G = split SO3 and H = SL(3) in residue characteristic 2provides a simple counterexample. However, Gopal Prasad has shown me a different argumentfor 2.7.4 when the residue characteristic is 0.

Remark. It is often asked whether for an embedding G ↪→ H and descent map ι : B(G) →

B(H), we have compatibility of Moy-Prasad filtrations:

G(K )x,r = H(K )ιx,r ∩ G(K )?

Again, this is true if the residue characteristic is sufficiently large. But it fails in some cases.

CONCAVE FUNCTIONS

Definition. A function f : 8 ∪ {0} → R is called concave if f (a + b) ≤ f (a) + f (b) whenevera, b, a + b ∈ 8 ∪ {0}.

For such a function, we define G(K )x, f to be the subgroup of G(K ) generated by Ua(K )x, f (a)

for all a ∈ 8 ∪ {0}.

Example. If f (a) = r for all a, then f is concave and the associated G(K )x, f is the Moy-PrasadG(K )x,r .

These groups are used increasingly more in representation theory. Unfortunately they arenot mentioned in Tits’ article. To prove the properties of the Moy-Prasad theory, one needs touse results about Gx, f in [BT] but not in [Tits]. We will just mention two results here.

Theorem. (i) G(K )x, f is bounded and open.(ii) If f (0) > 0, then the multiplication map∏

a∈8∪{0}

Ua(K )x, f (a) → G(K )x, f

is a bijection (the map sends (ua)a to the product of the ua’s for a suitable order.

Theorem. G(K )x, f is schematic.

This result is proved in [BT2] when f (0) = 0. The proof there doesn’t seem to extend to thegeneral case. So I proved the above result in a preprint using the theory of group smoothening.It turns out that this new approach is substantially simpler, and can be used as a subtitute for alarge part of [BT2] if one only concerns the case of discrete valuations.