Complete Reducibility in Euclidean Twin Buildings
Abstract
In [Ser04], J.P. Serre defined completely reducible subcomplexes of spherical buildings in order to study subgroups of reductive algebraic groups. This paper begins the exploration of how one may use a similar notion of completely reducible subcomplexes of twin buildings to study subgroups of algebraic groups over a ring of Laurent polynomials and KacMoody groups. In this paper we explore the definitions of convexity and complete reducibility in twin buildings and some implications of the two in the Euclidean case.
1 Introduction
Buildings were introduced by J. Tits as a geometric tool for studying certain algebraic groups over a field. A building can be thought of as a simplicial complex which is obtained by gluing together subcomplexes called apartments, which are made up of chambers (the simplices of maximal dimension) satisfying certain axioms. The apartments of a building are all isomorphic to a Coxeter complex. For example, consider the reflection group . The elements of act on the plane and we can consider the set of hyperplanes corresponding to the reflections. By cutting the unit circle by these hyperplanes we get a decomposition of the circle into simplices, and this simplicial complex is a spherical Coxeter complex. If then the simplicial complex will be a hexagon.
We can construct a building associated to for a field as follows. Let be a field and let be the abstract simplicial complex with vertices being the nonzero proper subspaces of , and with the maximal simplices being the chains of such subspaces. Then is a building and any basis of yields an apartment. This apartment consists of the vertices which correspond to subspaces spanned by proper nonempty subsets of the basis, and the simplices correspond to chains of these subspaces. For example, if and is any basis for , then we get an apartment of . The vertices correpond to the six proper nonempty subsets and the onedimensional simplices correspond to chains of these subsets, hence we have the hexagon mentioned above. Since the Coxeter complex is spherical, this is called a spherical building.
In spherical Coxeter complexes there is a bounded distance between any two points so there is a natural idea of opposite vertices and hence opposite chambers, which leads to many interesting properties of spherical buildings. In buildings of nonspherical type (e.g. Euclidean buildings), there is no bound on the distance between any two vertices so there is no notion of opposition.
Twin buildings were introduced by M. Ronan and J. Tits as a tool for studying groups of KacMoody type. They arise from these groups much like spherical buildings arise from algebraic groups and they extend to nonspherical buildings some of the ideas of spherical buildings, such as opposition. A twin building consists of a pair of buildings of the same type with an opposition relation between the chambers of the two components.
One consequence of the existence of opposites in spherical buildings is that one can use properties of the building to study completely reducible subgroups of a group which acts on a spherical building. In [Ser04], J.P. Serre gives a definition for a completely reducible subgroup of a reductive algebraic group which generalizes the definition of a completely reducible representation and uses the existence of opposite simplices in the corresponding spherical buildings. His definition in terms of opposite simplices can be extended to a definition of complete reducibility in twin buildings.
Recall that if is a representation of a group then is completely reducible if and only if for every proper invariant subspace of there is a proper invariant subspace such that . Since vertices in the spherical building associated to correspond to subspaces of and opposite vertices correspond to complementary subspaces this can be rephrased in terms of the building as follows.
For a vector space over a field , the group acts on a spherical building, call it . For a subgroup of , let be the set of points of which are fixed by the action of , then is completely reducible if and only if every vertex of has an opposite vertex in . This definition has an analogue in terms of parabolic subgroups containing since the simplices fixed by correspond to the parabolic subgroups containing . Serre then extends the idea of complete reducibility to subgroups of any group which acts on a spherical building, specifically reductive algebraic groups.
The points fixed by form a convex subcomplex and the definition of complete reducibility can be applied to an arbitrary convex subcomplex of a spherical building. A convex subcomplex is completely reducible if and only if every simplex of has an opposite in .
In [Cap09], P. E. Caprace introduces the definition of completely reducible subgroups of a group with a twin pair: a subgroup of is completely reducible if is bounded and if given a parabolic subgroup of finite type which contains , then there is a parabolic subgroup opposite which is of finite type and contains .
A group with a twin pair gives rise to a twin building (see [AB08] Chapter 8 for details) in such a way that the parabolic subgroups of correspond to the simplices (or equivalently, residues) of . Then the above definition of complete reducibility is equivalent to requiring that for every simplex (residue) in the fixed point subcomplex of in , there is an opposite simplex (residue) in the fixed point subcomplex of .
The points fixed by form a convex subcomplex of and we can extend this definition of complete reducibility to any convex subcomplex of a twin building such that is not empty and every simplex of has an opposite simplex in .
Convexity in a single building is more understood than convexity in twin buildings. P. Abramenko and K.S. Brown give a definition of convexity for chamber subcomplexes of a twin building in [AB08] and Abramenko explores general convex subcomplexes in twin buildings in [Abr96] but leaves several questions. Completely reducible subcomplexes are not always chamber subcomplexes so it is important to develop an understanding of general convex subcomplexes of twin buildings.
A subcomplex of a twin building is convex if and only if its intersection with any twin apartment is convex, so it suffices to study convexity in a twin apartment. A useful tool for studying apartments has been the Tits cone, which was introduced to study Coxeter complexes geometrically. The Tits cone is a (possibly infinite) hyperplane arrangement of a subset of a real vector space and the chambers in an apartment correspond to simplicial cones defined by hyperplanes. In nonspherical buildings the Tits cone is a convex subset of the vector space, so we can take the union of this subset with its negative and obtain a good representation of a twin apartment called the twin Tits cone.
The definition of convexity in the vector space agrees with the definition of convexity in a building, but since the twin Tits cone is strictly contained in the vector space we need a slightly modified definition of convexity. We can define convexity in the twin Tits cone, , as follows: if is a subset of and are points in , then is convex if and only if the geodesic is contained in . This leads to the following result about convexity in twin apartments.
Theorem.
Let be a pair of nonempty subcomplexes of a twin apartment such that and each contain a spherical simplex. Then the following are equivalent:

is convex in , i.e. closed under projections.

is an intersection of twin roots.

Let be the union of the cells corresponding to in the twin Tits cone . Then is convex in .
Euclidean buildings have the unique property that there is an associated spherical building at infinity and in [Ron03], M. Ronan shows that for a twin Euclidean building there are subbuildings of the corresponding buildings at infinity which are naturally twinned. Our main result allows us to only consider the subcomplexes of the spherical buildings at infinity to determine if a subcomplex is completely reducible.
Theorem (Main Theorem).
Let be a Euclidean twin building and a convex subcomplex of . Let be the set of interior points in the buildings at infinity as in Section 4.2 and the subcomplex of corresponding to . Then is a completely reducible subcomplex of if and only if every simplex of maximal dimension in has an interior opposite in .
We also show that we only need to consider the set of vertices at infinity in our study of complete reducibility.
Theorem.
A convex subcomplex is completely reducible if and only if every vertex in has an interior opposite in .
As an example for how this can be applied to a group with a twin pair, Let be a field, , , and . Then has a twin pair and an associated Euclidean twin building. Let be the geometric realization of this twin building and let be a subgroup of with fixed point complex with non empty for each . Then we have the following consequences of the preceding theorem.
Proposition.
The subgroup is completely reducible if and only if every invariant submodule of which is a direct summand of has a invariant complement.
Proposition.
Let and let be a completely reducible subgroup of . Then where each is a invariant submodule such that is irreducible in .
2 Background
We assume the reader has a basic knowledge of buildings and we will briefly discuss the definition and some results that are useful here. The definitions and results in this chapter can also be found in [AB08].
Let be a Coxeter system.
Definition 1.
A building of type is a pair consisting of a nonempty set , of elements called chamber, and a map called the Weyldistance function, such that for all , the following conditions hold:

if and only if .

If and satisfies then is or . If in addition , then where is the length function on with respect to .

If then for any there is a chamber such that and .
If in reduced form, then the length of is . If , then the distance from to is .
Let and let . Two chambers in are said to be equivalent if . This is an equivalence relation and the equivalence classes are called residues. A subset is a residue if it is a residue for some and is called the type of , is called the cotype and is the rank. A residue is said to be spherical if it is a residue for some such that is finite.
The above definition of a building is equivalent to the simplicial definition of a building (which is denoted by ) and the residues of correspond to the simplices of . The chambers of correspond to the residues of type which are the chambers of , the simplices of codimension 1 (also called panels) correspond to the residues of type for , and the vertices correspond to residues of rank . In the simplicial building we say that the type of a simplex is where is the type of the corresponding residue, hence the type of a simplex in is the cotype of the corresponding residue in . So the vertices of have type for (note that each chamber of contains exactly one vertex of type for each ).
For , every residues is isomorphic to a building of type and if is finite the residue and the corresponding simplex are said to be spherical.
An important property of spherical buildings is the existence of opposites. Let be an apartment of a spherical building of type . Then there is a unique element of longest length in , denoted . If are chambers of such that then we say that and are opposite. This induces an isometry on called the opposition involution which maps each chamber to its opposite in . If is the geometric realization of then the opposition involution is defined on all the simplices of , and for any simplex of the opposite of is op. Note that if is a vertex of then is the vertex which is diametrically opposite .
We will work primarily with the simplicial building and its geometric realization but the Weyl distance definition best generalizes to twin buildings.
2.1 Twin Buildings
Definition 2.
A twin building of type is a triple where and are buildings of type and is a codistance function satisfying the following conditions for each , any , and any with .

.

If such that and then .

For any there is a chamber with and .
For nonspherical buildings there is no element of maximal length so there is no notion of opposition, but in a twin building we can say two chambers are opposite if . We define the numerical codistance between chambers by . Then two chambers are opposite if and only if .
2.1.1 Projections and Convexity
Assume that is a twin building of type . It is known that if is a spherical residue of and is a chamber of then there is a unique chamber such that has maximal length in . This chamber is called the projection of onto and is denoted by proj. This chamber also satisfies the following equality for all
which gives the following analogue of the gate property:
Since residues correspond to simplices, the projection of a chamber onto a spherical simplex is the unique chamber containing with maximal codistance from .
A pair of nonempty subsets of and respectively is called convex if proj for any and any panel that meets . This is equivalent to saying that is closed under projections. Given two subsets and of , let Con denote the convex hull of and . We will explore convexity in more detail in Chapter 3.
2.1.2 Twin Apartments
Consider a pair of nonempty subsets of a twin building with an apartment of and an apartment of , then is a twin apartment of if every chamber of is opposite to exactly one chamber of . Then the opposition involution op associates to each chamber its unique opposite in . A twin apartment is the convex hull of any pair of opposite chambers contained in and such a pair of opposite chambers is called a fundamental pair of chambers for . The following lemma (5.173 in [AB08]) is useful throughout this paper.
Lemma 3.
Let be a twin apartment and let or .

op is an isomorphism.

Given and , let op. Then .

Let be any chambers in . Then , where is the distance or codistance function which makes sense for each pair of chambers.

is convex in .
2.1.3 Twin Roots
Given a twin apartment of a twin building , the pair with a root of for is a twin root if op, where .
Consider a pair of adjacent chambers and let be the root of containing but not . Let and (note that and are adjacent chambers of ) and let be the root of containing but not . Then is a twin root of and is the convex hull of and . The following lemma (5.198 in [AB08]) is very useful. Denote by the set of apartments of which contain .
Lemma 4.
Let be a twin root, and let be a panel in which contains exactly one chamber of for . Then there is a bijection that assigns to each the convex hull of and .
Given a simplex in a twin apartment we say that is a boundary simplex of a twin root if there are chambers and having as a face such that and . Then the above lemma says that if is a codimension 1 boundary simplex of a twin root and if is any chamber not in which has as a face, then there is a twin apartment containing and .
2.2 Simplicial Approach
Let be a twin building, for , let be the simplicial building associated to , and . Let , the geometric realization of , and .
These are three equivalent views towards twin buildings and we will use the notations interchangeably throughout this paper.
2.2.1 Sign Sequences
Let be a Coxeter complex and let be the complete set of walls of . Each wall defines a pair of roots of . Each simplex of is either in , or . We can assign a sign where if and only if . The support of is the intersection of walls such that (note that has the same dimension as its support, Proposition 3.99 in [AB08]). The sign sequence is defined as .
Let be a twin apartment with geometric realization . A twin wall is a pair of walls in and respectively such that op. If is the sign of a simplex with respect to the wall then op.
3 Convexity
Convex subcomplexes of a single apartment are well understood and there are several equivalent definitions including being an intersection of roots, closed under products/projections, and closed under straight line segments in the corresponding Tits cone. A subcomplex of building is convex if its intersection with every apartment is convex in the apartment.
Convex subcomplexes of twin buildings are not as well understood. There is one main definition in the literature to date, namely: a subcomplex of a twin building is convex if it is closed under projections (within each building and between the two buildings). Proposition 5.193 of [AB08] says that if the subcomplex contains a chamber then being closed under projections is equivalent to the subcomplex being an intersection of roots. We show that this is also true if the subcomplex does not necessarily contain any chambers but does contain a sufficient number of spherical simplices.
3.1 Projections
Definition 5.
Given simplices and of a building the product, , is defined as the simplex with sign sequence given by
where ranges over the set of walls in an apartment containing and . This product is also called the projection of onto and denoted proj.
Definition 6.
Given a twin building let be a spherical simplex, be any simplex and be any chamber containing . Then proj is the unique chamber having as a face which has maximal codistance to and proj where ranges over all chambers having as a face.
We can also characterize the projection of onto in a twin building in terms of sign sequences. We will need the following lemma. This is Proposition 4 in [Abr96] and the proof uses the metric approach. We restate it in terms of simplices and give a simplicial proof. Note that is the link of which is the simplical building of the corresponding residue of .
Lemma 7.
Let be a twin apartment and let and be simplices with spherical. Let be the corresponding apartment in the link of . Then
.
Proof.
By definition, where runs over all chambers having as a face and is a chamber such that and where runs over all chambers having as a face and is a chamber such that . Note that .
Let be a chamber having as a face and let and . Then is maximal among distances with and is minimal among distances with . Hence is maximal in the link of . So is opposite in . Therefore, . ∎
Proposition 8.
Let be a twin apartment. Given simplices and with spherical the sign sequence of proj is
where ranges over the twin walls of .
Proof.
Since the walls of correspond bijectively to the walls of containing , the opposition involution op negates only the signs corresponding to the walls containing . Therefore,
∎
3.2 Twin Tits cone
Let be a twin apartment of type , where is infinite and irreducible. The chambers of correspond to simplicial cones in a real vector space and the union of these cells is called the Tits cone of as in section 2.6 of [AB08]. The subset of is a convex subset of and since is infinite . Let . So is a Tits cone representation of and . We define the twin Tits cone as . Let be a chamber of , and abusing notation also the corresponding simplicial cone in . The simplicial cone corresponds to the chamber of which is opposite . Then in , and two chambers, and , in are said to be opposite in if .
Example 9.
Let . The Tits cone corresponding to is the open upper half plane of plus the origin and the twin Tits cone is not inculding the ponits for as in Figure 1.
Proposition 10.
Two chambers and are opposite in if and only if their corresponding chambers in are opposite.
Proof.
Let be a fundamental pair of , and abusing notation, also the fundamental pair of the twin Tits cone . Assume that and are opposite in , hence . From Lemma 5.173 in [AB08], we get
Let . Hence in the Tits cone and so and are opposite in .
Conversely, suppose in . Let , then in , so . Then we have that hence
So . ∎
Since the twin Tits cone is not convex in , a slightly different definition of a convex subset of is needed.
Definition 11.
A union of cells contained in is convex in if for any two points , , where is the straight line connecting and in .
Example 12.
For with twin Tits cone . The shaded region minus the dotted line in Figure 2is convex in .
3.3 Convexity in a Twin Apartment
Given a simplicial complex of finite dimension, we say that is a chamber complex if all maximal simplices have the same dimension and can be connected by a gallery. Any building, and any apartment in a building is a chamber complex. Also, any convex subcomplex of an apartment is a chamber complex (though the chambers of may not be chambers of )(Proposition 3.136 of [AB08]).
We need the following lemma which guarantees a certain number of spherical simplices given at least one of maximal dimension.
Lemma 13.
Let be a Coxeter complex and let be a convex subcomplex of which contains at least one spherical simplex. Then every chamber is spherical.
Proof.
Since is convex and contains a spherical simplex , it must contain a spherical chamber which has as a face. Now assume that there is a chamber which is not spherical and consider proj. Since and both have maximal dimension in and is closed under projections, we must have that has maximal dimension in , hence . Consider the sign sequence definition of projection:
Then is spherical if and only if for finitely many . Since is spherical, for finitely many , hence for finitely many , so is spherical which is a contradiction, so does not have maximal dimension in .∎
This brings us to our main result, giving several equivalent definitions of convexity in a twin apartment.
Theorem 14.
Let be a pair of nonempty subcomplexes of a twin apartment such that and each contain a spherical simplex. Then the following are equivalent:

is convex in , i.e. closed under projections.

is an intersection of twin roots.

Let be the union of the cells corresponding to in the twin Tits cone . Then is convex in .
Proof.
We will prove .

Let be the support of . Then is a convex subcomplex of containing at least one spherical simplex and by Lemma 13, all chambers are also spherical. Also, all chambers are spherical.
We know from Lemma 3.137 in [1] that is an intersection of roots , each defined by a boundary panel of . This boundary panel, as defined in the proof of the Lemma, is the face of exactly two spherical chambers, so it is spherical. We need to show that . We will use contradiction.
Let be a simplex of and a boundary panel in . Note that proj. So let be any chamber of having as a face and let .
Now assume that . Then op and op . Hence . Since this holds for all chambers having as a face we must have that proj which is a contradiction. Therefore, is the intersection of twin roots with and .

It is enough to show that twin roots in correspond to halfspaces in . Then an intersection of twin roots in corresponds to an intersection of halfspaces in , which is a convex set. To show this, note that roots in correspond to roots in . So for a given and corresponding it suffices to show that corresponds to . This follows from the fact that opposition is preserved:

Given with spherical, we want to show proj. We may assume and . Let be a point in the interior of and a point in the interior of , with and viewed as cells of . Let .
Let be the segment of the line starting at and having length . Let be the cell of minimal dimension containing . We claim that proj. Then since is convex, any cell meeting [x,y] in its interior is in . Hence proj is in .
To prove the claim, first note that corresponds to the cell containing a segment of starting at and having length . In the link of , the cell opposite corresponds to the cell containing an extension of starting at and having length ; call this extension . Let be the cell of minimal dimension containing . By Lemma 7, . It remains to show that . This amounts to showing that and are not separated by any hyperplane of . For any hyperplane of there are three cases to consider: , and , and .
First, assume . Then there is some positive distance between and . Since is arbitrarily small, and are not separated by . Second, assume that and . Then by definition, is on the same side of as and is on the opposite side of as op . Hence is on the same side of as and . Thirdly, assume . Then op so and are in .
∎
3.4 “Coconvexity”
In [Abr96], P. Abramenko discusses a notion of “coconvexity” which is defined as closure under projections, but only those projections between the two components of the twin building, not projections within each component. In that book P. Abramenko states without proof the following proposition which we prove here.
Proposition 15.
Let , be spherical simplices. Then the coconvex hull of and , Con, is the intersection of all twin roots containing and .
Proof.
Since Con is contained in the convex hull of and , Proposition 14 gives the inclusion . Note that twin roots of are in one to one correspondence with halfspaces of the twin Tits cone so we need to show that . Let be the intersection of hyperplanes containing and . Note that dimdim: since is contained in we know that dimdim, and by the sign sequence of proj we know that the hyperplanes containing proj are exactly those containing both and so that dimdim hence dim dim. Since contains an infinite hyperplane arrangement and is a convex subcomplex of , all the results in [[AB08], section 2.7] apply to . So for the remainder of the proof, we will be working in , so by ”chamber” we will mean chamber, etc.
Let be the intersection of halfspaces of containing but not op , which is the intersection of with the intersection of the halfspaces of containing and . Note that there is only one chamber of having as a face: any two chambers containing are separated by at least one hyperplane , one of these chambers would have to be on the same side of as op and therefore would not be in . Consider the sign sequence of : since has as a face, if then , and since all the hyperplanes containing separate from op, if then which is the same sign sequence as proj from Lemma 8. Hence, proj.
Now let be in with distance 1 from . Let . Since , , and since is the only chamber of containing , we have strict inequality. Hence the hyperplane defined by separates op from so which is defined to be the intersection of halfspaces containing and , and is the only chamber of containing . By the above argument proj, hence .
We continue by inducting on the distance from . Assume that all chambers of distance less than from are in . Let be a chamber of of distance from . Then is adjacent to a chamber which is in and proj for some . Let be the intersection of halfspaces containing and . If, in the above proof that , we make the following identifications:
we get because .
∎
The next example shows that being closed only under projections between the two components does not guarantee convexity in each component of the twin building, leading to the conclusion that we need to require projections within each component in our definition of a convex subcomplex.
Example 16.
Consider the group with generating set . Then the hyperbolic planes, in Figure 3 form a thin twin building of type . Let be a vertex of of type and let and be walls containing . Let be a spherical simplex of of type which is in such that and op are vertices of a common chamber. Similarly, let be a spherical simplex of of type which is in such that and op are vertices of a common chamber. Let be the wall containing but not op and let be the wall containing but not op (since and have type , their links are isomorphic to a Coxeter complex of type where and which has exactly two walls).
The coconvex hull of , , and is the shaded subcomplex in the Figure 3. Since and we know that for all and similarly for all . Since the roots defined by and are nested and similarly for the roots defined by and we have that and for all . The sign sequence for with respect to these hyperplanes is and for we have . From what we just said about we know that these zeros can only be replaced with hence there is no way to get the sign sequence which is the sign sequence for both and . Therefore, neither nor is in and is not convex.
4 Twin Buildings at Infinity
4.1 A Single Building at Infinity
To every Euclidean building we can associate a spherical building by attaching a sphere at infinity to each apartment. This is achieved as follows (see chapter 11 of [AB08]).
Let be the geometric realization of a Euclidean Coxeter complex of type with the corresponding set of hyperplanes in . Let be a point of and be the set of hyperplanes through which are parallel to some element in . Then defines a decomposition of into conical cells, called conical cells based at x. If is a special vertex (every hyperplane of is parallel to a hyperplane of ) then is a subset of and is isomorphic to the set of hyperplanes corresponding to a Coxeter complex of type where is the finite reflection group consisting of the linear parts of the elements of .
Let be a cell associated to , then for any point the conical cells based at are the translates , and if is a chamber, then is called a sector. Figure 4 shows a sector based at a vertex . The bold lines in the figure, which are called rays, are also conical cells based at . The vertex is a special vertex and is not a special vertex.
Let be the geometric realization of a Euclidean building of type . Then the building at infinity, , is the collection of ends of parallel classes of rays. The simplices of correspond to parallel classes of conical cells and the chambers of correspond to parallel classes of sectors. Two conical cells are parallel if the distance between them is bounded. For sectors this implies that their intersection contains a sector. A sector is called a subsector of . Note that is a spherical building of type .
Let be a conical cell based at with direction in an apartment . Let be the cell associated to which is opposite