Non-discrete affine buildings and convexity
Affine buildings are in a certain sense analogs of symmetric spaces. It is therefore natural to try to find analogs of results for symmetric spaces in the theory of buildings. In this paper we prove a version of Kostant’s convexity theorem for thick non-discrete affine buildings. Kostant proves that the image of a certain orbit of a point in the symmetric space under a projection onto a maximal flat is the convex hull of the Weyl group orbit of . We obtain the same result for a projection of a certain orbit of a point in an affine building to an apartment. The methods we use are mostly borrowed from metric geometry. Our proof makes no appeal to the automorphism group of the building. However the final result has an interesting application for groups acting nicely on non-discrete buildings, such as groups admitting a root datum with non-discrete valuation. Along the proofs we obtain that segments are contained in apartments and that certain retractions onto apartments are distance diminishing.
Kostant’s convexity theorem for symmetric spaces, proven in [Kostant], describes the image of a certain orbit under a projection on a maximal flat as a convex set. His result is a generalization of a well known theorem of Schur [Schur]. The precise geometric statement is as follows:
Let be a symmetric space and a maximal flat of . Then there is a natural action of a spherical Weyl group on with fixed point . Write for the Iwasawa projection of onto . Kostant proves that the image of the orbit of an element under the Iwasawa projection is precisely the convex hull of the Weyl group orbit . In terms of groups his result provides a criterion for the non-emptiness of intersections of certain double cosets of group elements.
Let be a non-compact semi-simple Lie-group with Iwasawa decomposition with unipotent, abelian and compact. Geometrically left-cosets of elements of correspond to points in a certain maximal flat of the symmetric space . Kostant’s theorem translates o the fact that intersections of double cosets of the form are non-empty if and only if is contained in the convex hull of the point .
Affine buildings are in a certain sense analogs of symmetric spaces. Similarly symmetric spaces are important for the classification of semisimple Lie groups, so do affine buildings play a major role in the classification of semisimple algebraic groups defined over fields with valuation. Part of the analogy in terms of geometry is as follows: Maximal flats in symmetric spaces correspond to apartments in buildings. Both of them admit an action of a spherical Weyl group.
The notion of an “Iwasawa projection” onto an apartment does make sense in a building, too. In terms of groups it is defined precisely in the same way as in the context of semi-simple Lie groups, but there is as well a definition using the geometry of an affine building. The orbit in Kostant’s result corresponds, when talking about buildings, to the preimage of under a second type of retraction onto , which we will denote by . Hence we might ask again whether the projection of this set onto is a convex hull of the Weyl group orbit of . Or, spoken in group language, whether for algebraic groups (to be precise groups with affine and split spherical BN-pair) the same criterion guarantees the intersections of double cosets to be non-empty.
In the simplicial case this question was answered in [Convexity]. The purpose of the present paper is to prove a convexity result in the spirit of Kostant’s for a class of spaces more general than simplicial affine buildings.
Generalized affine buildings
Simplicial affine buildings, which are a subclass of the geometric objects studied in this paper, were introduced by Bruhat and Tits in [BruhatTits] as spaces associated to semisimple algebraic groups defined over fields with discrete valuations.
In [TitsComo] and [BruhatTits, BruhatTits2] they were generalized allowing fields with non-discrete (non-Archimedean) valuations rather than discrete ones. The arising geometries do no longer carry a simplicial structure and are nowadays usually called non-discrete affine buildings or -buildings. In [TitsComo], -buildings were axiomatized and for sufficiently large rank classified under the name système d’appartements.
Finally, in [Bennett, BennettDiss] Bennett introduced a class of -metric spaces called affine -buildings using axioms similar to the ones in [TitsComo]. Examples of these spaces arise from simple algebraic groups defined over fields with valuations taking their values in an arbitrary ordered abelian group . Bennett was able to prove that affine -buildings again have simplicial spherical buildings at infinity and made major steps towards their classification. To be more precise a generalized affine building is a set together with a collection of maps called atlas. Each is an injective map from a (fixed) model apartment to . The images are called apartments of . As a set is the union of its apartments, which need to satisfy certain axioms in addition. Compare Definition 3.1.
For generalized affine buildings one can again define an “Iwasawa projection” onto an apartment and one can, as already mentioned above, define a second type of retraction whose preimage of corresponds precisely to the -orbit of . Therefore it is natural to ask whether a a convexity theorem exists for this more general class of affine buildings.
Throughout this text we will refer to affine -buildings as generalized affine buildings to avoid the appearance of the group in the name. Note that the class of generalized affine buildings does not only include all previously known classes of (non-discrete) affine buildings, but does also generalize leafless -trees. These trees are simply the generalized affine buildings of dimension one.
Let be a thick generalized affine building, as defined in 3.1 and 3.2 with model space . There is an action of an affine Weyl group on . The stabilizer of in can naturally be identified with the spherical Weyl group . We fix a fundamental domain of the action of on and call it the fundamental Weyl chamber . Weyl chambers in are images of under the affine Weyl group and Weyl chambers in are images of Weyl chambers in . Fixing a chart of an apartment in it therefor makes sense to talk about an origin and a fundamental Weyl chamber in . We say that two Weyl chambers based at the same vertex are equivalent if they intersect in a set with nonempty relative interior. The equivalence class of a Weyl chamber based at is called the germ of at .
To state the main theorem we need to introduce two retractions onto the given apartment . The first one, denoted by , is defined with respect to the germ of the fundamental Weyl chamber in . It preserves distances to and its restriction to apartments containing the germ of at is an isomorphism onto . The inverse image of the Weyl group orbit of under corresponds precisely to the orbit in Kostant’s setting. The second retraction is sometimes (mostly when talking about algebraic groups) called “Iwasawa projection” onto . The geometric definition of is given with respect to the parallel class of the opposite of the fundamental Weyl chamber. Here two Weyl chambers are parallel if their intersection contains a Weyl chamber. We demand that the restriction of to an apartment containing a Weyl chamber parallel to is an isomorphism onto . As it turns out this leads to a well defined retraction of onto . We obtain the following theorem.
Given a vertex in one has
Restating this in terms of a group acting “nicely” on a thick generalized affine building , one obtains the following result about non-emptiness of intersections of double cosets in . Denote by the stabilizer of the origin in and assume that it is transitive on the apartments containing . Let further be the stabilizer of the equivalence class and assume that splits as , where is the group of translations in and acts simply transitive on the apartments containing at infinity. Then
For all we have
or, since , equivalently
Techniques used to prove Theorem LABEL:Thm_convexityGeneral are geometric properties of generalized affine buildings and methods borrowed from metric geometry. An important idea is inspired by and, with enough technical effort, adapted from a result of Parkinson and Ram proven in [ParkinsonRam]. They give a combinatorial proof of the existence of certain positively folded galleries. The main idea of their proof can be modified to obtain a result on retractions onto apartments of generalized affine buildings.
Outline of proof
The first problem arising concerns the two retractions and . In order to be able to define them in the general setting of the present paper, we first have to establish several structural results on the local and global behavior of generalized affine buildings. This is done in Section LABEL:Sec_local-global. The definition of the retractions can be found in Section LABEL:Sec_retractions In this context we introduce residues, which are as sets simply the collection of all germs based at the same vertex, but carry the structure of spherical buildings. One can think of residues as “tangent spaces” at points in .
Further we want to prove that both retractions do not increase distances between arbitrary points in the building. This fact is much easier to prove in the simplicial case. In order to verify that they are distance-non-increasing (see Corollary LABEL:Cor_rho-distancediminishing), we need to generalize Lemma 7.4.21 of Bruhat and Tits [BruhatTits] which is a covering property of segments in buildings. The proof of Lemma 7.4.21 given in [BruhatTits] uses compactness arguments of -metric spaces which cannot be applied in our setting. In Section LABEL:Sec_FC we prove these properties for generalized affine buildings.
We are then ready to prove Theorem LABEL:Thm_convexityGeneral. The major problem occurs in the proof of the fact that every element of the convex set has a preimage under which is contained in the set . As already mentioned earlier this is done by modifying an idea of Parkinson and Ram [ParkinsonRam]. Given an element of we define, in the proof of Proposition LABEL:Prop_preimage, a sequence of points depending on a chosen presentation of the longest element of the spherical Weyl group . This sequence of points helps to define more or less explicitly a preimage of under , which is by construction contained in . As pretty as the main idea may be as technical is the actual proof. For the convenience of the reader we therefore repeat the underlying ideas without proof in Section LABEL:Sec_ParkinsonRam.
To finish the proof of LABEL:Thm_convexityGeneral it remains to show that the image of the set is in fact contained in the convex hull of the Weyl group orbit of . This is done in Proposition LABEL:Prop_image. Methods used in the proof are borrowed from metric geometry and mimic differentiation. The ideas come from the similarity between germs of Weyl chambers and tangent vectors of curves in manifolds or metric spaces.
We have not mentioned so far that the notion of convexity used in the present paper is not the usual one where convex sets are defined to be finite intersections of half-apartments. However our notion of convexity, as defined in Definition 2.16, generalizes the metric convex hull in terms of the Euclidean metric defined in the geometric realization of simplicial buildings. An apartment of an -building is naturally equipped with a Euclidean metric. The metric convex hull of a Weyl group orbit in such an apartment defined with respect to the Euclidean distance corresponds precisely to the convex hull as it is defined in 2.16 (in case the building is equipped with the full affine Weyl group). However dealing with -metric spaces there is nothing like a Euclidean distance. We therefore define a metric on apartments differently. Using this metric, described in definition 2.13, it is no longer true that the metric convex hull of equals the convex hull in the sense of Definition 2.16. Only the weaker observation 2.17 remains.
The paper is organized as follows
In section 2 the building block of a generalized affine building, the so-called model space of an apartment, is defined. Generalized affine buildings are then defined in Section 3, where we also describe their local and global structure and prove preliminary results which are necessary for the definition of retractions.
These retractions are then defined in Section LABEL:Sec_retractions. Local covering properties (generalizing a Lemma by Bruhat and Tits), which are used to prove that the retractions of the previous chapter are distance diminishing, are investigated in Section LABEL:Sec_FC.
The following Section LABEL:Sec_ParkinsonRam might be skipped. Here we recall the convexity theorem proven in the setting of simplicial affine buildings and explain a geometrical construction of certain positively folded paths. This is done in order to make the technical proof of Theorem LABEL:Thm_convexityGeneral, given in Section LABEL:Sec_convexityThm and relying on the ideas of Section LABEL:Sec_ParkinsonRam, more approachable.
An application to groups acting nicely enough on thick affine buildings is then given in Section LABEL:Sec_application. This application is similar to the one obtained in the simplicial case and the direct analog of the result by Kostant on non-emptiness of intersections of double cosets.
Finally an open problem is discussed in Section LABEL:Sec_looseEnds.
The author would like to thank Linus Kramer for many helpful discussions and encouragement. We also thank James Parkinson for the reference to [ParkinsonRam]. The author was partially supported by the Studienstiftung des deutschen Volkes and the SFB 478 "Geometrische Strukturen in der Mathematik" while working on this topic. This work is part of the author’s doctoral thesis at the Universität Münster
2. The model space
Geometric realizations of simplicial affine buildings are metric spaces “covered by” Coxeter complexes which are isomorphic to a tiled . The basic idea of the generalization is to substitute the real numbers by a totally ordered abelian group .
2.1. Definition and basic properties
Let be a (not necessarily crystallographic) spherical root system and a subfield of containing the set of co-roots evaluated on roots . Assume that is a totally ordered abelian group admitting an -module structure. The space
is the model space of a generalized affine building of type .
We omit in the notation, since we can always choose to be . If is crystallographic then is always a valid choice. If there is no doubt which root system and which we are referring to, we will abbreviate by .
A fixed basis of the root system provides natural coordinates for the model space . The space of formal sums
is canonically isomorphic to and the evaluation of co-roots on roots can be extended linearly to its elements.
An action of the spherical Weyl group on is defined as follows. Let , and , let be the linear extension of
to , where the reflection is defined by
We call the fixed point set of a hyperplane or wall and we denoted it by or , since .
A basis of determines a set of positive roots . The subset
of is the fundamental Weyl chamber with respect to , and is denoted by .
Given a non-trivial group of translations of which is normalized by we define the affine Weyl group with respect to to be the semi direct product . In case we call it the full affine Weyl group and write . Elements of can be identified with points in by assigning to the image of the origin under . Given we write for the translation defined by .
The actions of and on induce an action of , respectively , on .
In order to emphasize the freedom of choice for the translation part of the affine Weyl group, the model space with affine Weyl group is referred to as .
An element of which can be written as for some nontrivial and with is called (affine) reflection. A hyperplane in is the fixed point set of an affine reflection . It is called special with respect to if .
Note that for any affine reflection there exists and such that the reflection is given by the following formula
Further, easy calculations imply that
The fixed point set of is given by
As in the classical case each hyperplane defines a positive and a negative half-apartment
A vertex is called special if for each there exists a special hyperplane parallel to containing . Hence is the intersection of the maximal possible number of special hyperplanes.
Note that the translates of by are a subset of the set of special vertices.
A Weyl chamber in is an image of the fundamental Weyl chamber under the full affine Weyl group . If Weyl chambers are simplicial cones in the usual sense. Therefore Weyl chambers and their faces are called Weyl simplices. The faces of co-dimension one are referred to as panels.
Note that a Weyl chamber contains exactly one vertex which is the intersection of all bounding hyperplanes of . We call it base point of and say is based at .
The following proposition is used to introduce a second type of coordinates on .
[Bennett, Prop. 2.1] Given and . Then there exist a unique and a unique such that
The value of is . Furthermore .
Define and consider . Then
and is contained in . It remains to prove uniqueness. Let and be such that Then and . Therefore we have
and conclude that and . ∎
Let be a root system of rank with basis . A point is uniquely determined by the values , which will be called hyperplane coordinates of with respect to .
2.2. The metric structure of
The remainder of this section is used to define a -invariant -valued metric on the model space of a generalized affine building and to discuss its properties.
Let be a totally ordered abelian group and let be a set. A metric on with values in , short a -valued metric, is a map such that for all the following axioms are satisfied
if and only if
the triangle inequality holds.
The pair is a -metric space.
An isometric embedding of a -metric space into another is a map such that for all and in one has . Such a map is necessarily injective, but need not be onto. If it is onto we call it an isometry or an isomorphism of -metric spaces.
Let be as in 2.5. The distance of points and in is given by
If is contained in the fundamental Weyl chamber the distance equals , where is half the sum of the positive co-roots.
Let be a basis of . Choosing equal to and identifying the co-roots with in the definition of the metric in [BennettDiss] we precisely obtain the metric defined in 2.13.
The latter generalizes the chamber distance (or length of a translation) in a Coxeter Complex. This distance is defined as follows: Let be a vertex in a Euclidean Coxeter complex. Let the length of a translation be the number of hyperplanes crossed by a minimal gallery from to . This number is given by the formula . The fact that is the direct generalization of a combinatorial length function justified, at least in my opinion, to make a specific choice of the appearing in the definition of the metric as written in [BennettDiss].
The distance defined in 2.13 is a -invariant -valued metric on .
By definition and if and only if , since otherwise, by Corollary 2.2 in [Bennett], one of the terms would be strictly positive. It remains to prove that :
Hence is a metric. We prove -invariance: Let be a translation in . Then
Therefore is translation invariant. With we have
The second last equation holds since permutes the roots in . Therefore is invariant and -invariance follows. ∎
Recall the definition of the hyperplane coordinates of with respect to a basis introduced in Corollary 2.10.
Fix a basis of and let be an element of . The distance is uniquely determined by the hyperplane-coordinates of . With we have
Assume first that . Then has hyperplane coordinates defined in Corollary 2.10, with for all . Hence, using , we have
If is not then -invariance of implies that , where is the unique element of contained in and . Further and the assertion follows. ∎
2.3. Convexity and parallelism
As in the classical case, one can define convexity.
A subset of is called -convex if it is the intersection of finitely many special half-apartments. The -convex hull of a subset of is the intersection of all special half-apartments containing .
Note that Weyl chambers and hyperplanes are -convex, as well as finite intersections of -convex sets. Special hyperplanes and Weyl chambers are also -convex.
[BennettDiss, Prop.2.13] For any two special vertices in the model space the segment is the same as the -convex hull .
Two subsets of a -metric space are at bounded distance if there exists such that for all there exists such that for . Subsets of a metric space are parallel if they are at bounded distance.
Note that parallelism is an equivalence relation. One can prove
[Bennett, Section 2.4] Let equipped with the full affine Weyl group . Then the following is true
Two hyperplanes or Weyl simplices are parallel if and only if they are translates of each other by elements of .
For any two parallel Weyl chambers and there exists a Weyl chamber contained in and parallel to both.
Moreover is parallel to for all where is the translation in by .
3. The definition of generalized affine buildings
Throughout the following let be as defined in 2.5.
Let be a set and a collection of injective charts . We call the images of charts apartments of and we define Weyl chambers, hyperplanes, half-apartments, special vertices, … of to be images of such in under any . The set is a (generalized) affine building with atlas (or apartment system) if the following conditions are satisfied
For all and the concatenation is contained in .
Given two charts with . Then is a closed convex subset of . There exists with .
For any two points in there is an apartment containing both.
Axioms imply the existence of a -distance on , that is a function satisfying all conditions of Definition 2.11 but the triangle inequality. Define the distance of points in to be the distance of their preimages under a chart of an apartment containing both.
Given Weyl chambers and in there exist sub-Weyl chambers in and such that .
For any apartment and all there exists a retraction such that does not increase distances and .
Let and be charts such that the associated apartments pairwise intersect in half-apartments. Then .
By the distance function on is well defined and satisfies the triangle inequality.
The dimension of the building is , where .
Tits defined his “système d’appartements” in [TitsComo] by giving five axioms. The first four are the same as above. The fifth axiom originally reads different from ours but was later replaced with as presented in the definition above. In fact if axiom follows from . But in the general case this additional axiom is necessary as illustrated with an example given on p. 563 in [Bennett]. However in [Bennett] axiom is mostly used to avoid pathological cases and to guarantee the existence of the panel and wall trees. One can find a short history of Tits’ axioms in [Ronan]. Equivalent sets of axioms are discussed by Parreau in [Parreau].
Let be a generalized affine building with model space and apartment system . We call thick with respect to if for any special hyperplane of there exist apartments and , with such that and is one of the two half-apartments of (or ) determined by . Furthermore apartments do not branch at non-special hyperplanes.
If in the previous definition then is a building branching everwhere.
Two affine buildings of the same type are isomorphic if there exist maps , , further maps , and an automorphism of such that
and the following diagram commutes for all with