arXiv:1606.05638v1 [] 17 Jun 2016

A LIGHTCONE EMBEDDING OF THE TWIN BUILDING OF A
HYPERBOLIC KAC–MOODY GROUP
LISA CARBONE, ALEX FEINGOLD, AND WALTER FREYN

Abstract. The twin building of a Kac–Moody group G encodes the parabolic subgroup structure of G and admits a natural G–action. When G is a complex Kac–Moody group of hyperbolic
type, we construct an embedding of the twin building of G into the lightcone of the compact real
form of the corresponding Kac–Moody algebra. When G has rank 2, we construct an embedding
of the spherical building at infinity into the set of rays on the boundary of the lightcone.

Contents
1.

Introduction

2

2.

Kac–Moody algebras and Kac–Moody groups

4

2.1.

Kac–Moody algebras

4

2.2.

Kac–Moody groups

6

2.3. BN –pair and Tits building of a minimal Kac–Moody group

7

2.4.

9

Compact real forms of Kac–Moody algebras and groups

3.

Tits cone and lightcone of hyperbolic Kac–Moody algebras of compact type

10

4.

Group actions of the compact real form K

13

4.1.

The Adjoint action of K on k

13

4.2.

The local structure of the Adjoint action

15

4.3.

The action of the compact real form on the twin building

17

5.

Simplicial structure on Hr

18

6.

The main embedding theorem

19

7.

Cartan involutions and the embedding of the twin building

22

8.

Understanding of the Tits building in rank 2

23

9.

Embedding the spherical building at infinity in rank 2

30

Date: April 5, 2018.
This material is based upon work supported by the National Science Foundation under Grant No. 1002477.
The first author was supported in part by NSF grant number DMS–1101282. All authors wish to thank the IHÉS
for support during various visits in May 2013, 2014 and 2015. The second author wishes to thank the Max-Planck
Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany, for support during visits in
June 2013, 2014 and 2015.
1

10.

Conclusion and further directions

40

References

41

1. Introduction
Buildings were introduced by J. Tits in the 1950’s to provide a ‘geometric’ interpretation of finite
dimensional simple algebraic groups and finite groups of Lie type. Following ideas originally developed by F. Klein in his Erlanger program, Tits aimed at understanding a large class of groups,
containing simple algebraic groups and finite groups of Lie type, as the automorphism groups of
carefully constructed geometric objects called ‘buildings’. It turns out that the buildings of Tits
correspond to groups which admit an additional structure, called a BN –pair or equivalently
a Tits system. This structure also gives rise to a Bruhat decomposition for the corresponding
group. Tits’s approach was the geometric counterpart to the ‘algebraic’ construction of these
groups by C. Chevalley using automorphisms of Lie algebras [Che55].
In general, the building B of a group G with a BN –pair is an abstract simplicial complex
constructed from group theoretical data. Namely, simplices in B are in bijection with the union
of all cosets G/P , where P runs through a set of representatives of the conjugacy classes of
parabolic subgroups. These simplices satisfy boundary relations which can be phrased in terms
of inclusions of these cosets.
For Kac–Moody groups, the closest infinite dimensional analogue of simple algebraic groups,
the structure of buildings and BN –pairs is richer. As Kac–Moody groups have two conjugacy
classes of Borel subgroups, they admit the definition of two ‘opposite’ BN –pairs which together
form a ‘twin BN –pair’. Consequently the geometry associated to a Kac–Moody group G naturally consists of two related components. This object is called a ‘twin building’ B = B + ∪ B −
associated to a twin BN –pair, (B + , B − , N ), where the subgroups B + and B − are the standard
Borel subgroups constructed from the positive and negative roots of the Kac–Moody algebra
respectively. Thus by construction the building is related to the combinatorial structure of its
Kac-Moody groups.
Kac-Moody algebras and groups fall naturally into three types, finite type, affine type and indefinite type. While Kac-Moody groups of finite type (simple Lie groups) and of affine type
are well-understood, there are far fewer results known for the indefinite type. The most important subclass of indefinite type is the hyperbolic type, studied since the 1980s by various
authors [LM79, Fei80, FF83, KMW88, KM95] but certainly of interest to physicists [Jul85,
DHN02, DH09, Wes01].
While the algebraic properties of the hyperbolic Kac-Moody groups and algebras attracted some
attention, there have been only a few mathematical results concerning their geometry, for example, the study of homogeneous or symmetric spaces associated to them, and the study of
‘polar actions’ or isoparametric submanifolds. These classes of objects are very well understood
for finite dimensional Lie groups and for affine Kac-Moody groups. It is hoped that the understanding of this geometry, called Kac-Moody geometry, will shed new light on algebraic and
structural properties of these groups. As a first step towards the goal of understanding the geometry of hyperbolic Kac-Moody algebras and groups, we show in this work that for hyperbolic
2

Kac–Moody groups G over C, the associated Tits building is not only an abstract simplicial
complex admitting an action of G, but that it admits a natural embedding inside the compact
real form k of the Kac–Moody algebra g. As a consequence, the structure of the Tits building
and of the compact real form of the Kac–Moody algebra are closely related.
Our results generalize work of Quillen and Mitchell in the finite dimensional case, and Kramer
and Freyn in the affine case. Mitchell, in a paper based on ideas of Quillen, used embeddings of
spherical buildings associated to simple (real or complex) Lie groups into the tangent space of
the associated noncompact real symmetric space [Mit88]. In particular, the topological building
of a real noncompact Lie group G with maximal compact subgroup K is canonically identified
with a space homeomorphic to the unit sphere in the tangent space of the noncompact real
symmetric space G/K. For example, the topological building of type A1 , isomorphic to S 1 , can
be embedded into the unit circle in the tangent space of H2 = SL2 (R)/SO(2) which is R2 .
(1)

Kramer gave a topological construction of the complex twin building of type An and an equivariant embedding of this building into the associated affine Kac–Moody algebra g [Kra02].
Freyn gave a 2–parameter family ϕ`,r of equivariant embeddings of affine ‘twin cities’ B = B + ∪B −
into the ‘s-representations’ of affine Kac–Moody symmetric spaces [Fre09, Fre11, Fre12b, Fre13c,
Fre13b]. A ‘twin city’ is the natural completion of a twin building; affine twin cities correspond
to completions of affine Kac–Moody groups in a similar way as twin buildings correspond to
minimal Kac–Moody groups. Twin cities carry a natural topology that is derived from the
topology on the corresponding Kac–Moody group.
Denoting by g = k⊕p the Cartan decomposition, and restricting Freyn’s result to non-completed
affine Kac-Moody algebras, yields an identification of the twin building with the intersection p`,r
of the sphere of squared length ` ∈ R with horospheres parametrized by rd = ±r 6= 0, where
rd is the real coefficient of the derivation d. The positive and negative components of the twin
building, B + and B − , are immersed into the two sheets of p`,r described by rd < 0 respectively
rd > 0.
For affine Kac-Moody groups of the compact type, which are symmetric spaces called of ‘type
II’ following Helgason’s classification (see [Hel01, Fre13a]), this result includes embeddings of
complex buildings into the compact forms of affine Kac–Moody algebras; this case is the affine
counterpart to the embeddings constructed in this paper for symmetrizable hyperbolic Kac–
Moody algebras. We refer the reader to [Fre09, Fre12a, Fre11, Fre12b] for additional details.
For g a hyperbolic Kac–Moody algebra we are motivated in part by the appearance of g/k and
G/K in coset models of certain supergravity theories, where coset spaces of the split real forms
occur as parameter spaces for the scalar fields of the theory [Jul85, DHN02, Wes01].
The results in this paper can be extended to include a wider class of Kac–Moody algebras, namely
Lorentzian algebras. Nevertheless as soon as one leaves the realm of hyperbolic Kac–Moody algebras, the equivalence between the lightcone and the Tits cone breaks down. Consequently,
the geometry of the buildings and their lightcone embeddings becomes more complicated. On
the other hand interesting new phenomena occur, for example there are infinite families of rank
3 Lorentzian, non–hyperbolic Kac–Moody algebras whose buildings are isomorphic. Up to isomorphism, the same hyperbolic building will embed into each algebra in this infinite family, but
in different ways. An extensive collection of examples is being worked out by A. Tichai [Tic14].
3

The authors wish to thank Peter Abramenko for his helpful comments on an earlier draft of
the manuscript. They would also like to thank Victor Kac for helpful comments in May 2015
at IHÉS. AF would like to thank Kai-Uwe Bux, Max Horn, Tobias Hartnick, Ralf Köhl and
Peter Abramenko for helpful discussions at the June 2015 conference on “Generalizations of
Symmetric Spaces” in Israel.

2. Kac–Moody algebras and Kac–Moody groups
2.1. Kac–Moody algebras. A Kac–Moody algebra gF (A) over a field F may be constructed by
generators and relations using a collection of data which includes a matrix A = (aij )i,j∈I called
a generalized Cartan matrix satisfying the following conditions for all i, j ∈ I = {1, . . . , `}:
aij ∈ Z,

aii = 2,

aij ≤ 0 if i 6= j,

and

aij = 0 ⇐⇒ aji = 0.

A generalized Cartan matrix A is indecomposable if there is no partition of the set I = I1 ∪I2 into
non-empty subsets so that aij = 0 for i ∈ I1 and j ∈ I2 . The matrix A is called symmetrizable if
there is an invertible diagonal matrix D = diag(d1 , · · · , d` ) such that DA = (di aij ) is symmetric.
One distinguishes various types of generalized Cartan matrices:
- Finite type: A is positive-definite. In this case A is the Cartan matrix of a finite dimensional semisimple Lie algebra and det(A) > 0.
- Affine type: A is positive-semidefinite, but not positive-definite, and all minors are positive definite. In this case det(A) = 0.
- Hyperbolic type: A is neither of finite nor affine type, but every proper, indecomposable
submatrix is either of finite or of affine type. In this case det(A) < 0.
A complex Kac–Moody algebra gC (A) has at least two real forms, that is, real Lie algebras gR
such that gC (A) = C ⊗ gR . The split real form of gC (A) is gR (A) (see [BVBPBMR95]). From
this point on, all generalized Cartan matrices in this paper are assumed to be indecomposable,
symmetrizable and of hyperbolic type.
Given field F = C or F = R:
- a hyperbolic generalized Cartan matrix A = (aij )i,j∈I , and
- a vector space h over F (which will play the role of a Cartan subalgebra) with dimF (h) = `,
and basis {hi | i ∈ I},
then there is a set of simple roots Π = {αj | j ∈ I} ⊆ h∗ such that the pairing h·, ·i : h∗ × h → F
given by hα, hi = α(h) satisfies αi (hj ) = aij for all i, j ∈ I, and the hyperbolic Kac–Moody
Lie algebra g = gF (A) is generated by the elements {ei , fi , hi | i ∈ I}, subject to the relations
([Kac90, GK81]):
-

[hi , hj ] = 0,
[h, ei ] = hαi , hiei , h ∈ h
[ei , fj ] = δij hi ,
(adei )1−aij (ej ) = 0, i 6= j

[h, fi ] = −hαi , hifi , h ∈ h,

and
and

(adfi )1−aij (fj ) = 0, i 6= j,
4

where adx (y) = [x, y]. For each i ∈ I, let sli2 be the Lie subalgebra (isomorphic to sl2 (F)) with
basis {ei , fi , hi }, so that g is generated by these subalgebras. The abelian Lie subalgebra h with
basis {hi | i ∈ I} is called the standard Cartan subalgebra of g.
The algebra g = g(A) is infinite dimensional since A is not positive definite, and it admits an
invariant symmetric bilinear form ( , ) which is unique up to a global scaling factor ([Kac90],
section II), and which extends the form on h given by 2(hi , hj )/(hj , hj ) = αi (hj ) = aij . The
nondegeneracy of the pairing h·, ·i between h∗ and h determines a corresponding form on h∗ .
This means that (αi , αj ) = (hi , hj ) and
aji (αi , αi )/2 = (αj , αi ) = (αi , αj ) = aij (αj , αj )/2
so that we may take the diagonal matrix D = diag(d1 , · · · , d` ) with di = 2/(αi , αi ) and the
symmetric matrix DA = (di aij ) = (2aij /(αi , αi )). The standard way to choose the global
scaling factor is so that the longest square length of any simple root is 2. The adjoint action of
h on g is diagonalizable, and the simultaneous nonzero eigenspaces for that action,
gα = {x ∈ g | [h, x] = α(h)x, h ∈ h}
for α 6= 0 are called root spaces.
The root system of g is the set Φ = {α ∈ h∗ | α 6= 0, gα 6= 0}, and the Z-span of Φ, called the
root lattice of g, is denoted by Q. From the relations defining g we see that for each i ∈ I,
gαi = Cei and g−αi = Cfi , so that ±αi ∈ Φ. In fact, we have Q = Zα1 ⊕ · · · ⊕ Zα` is a free
P
Z-module. Every α ∈ Φ can be written uniquely as α = `i=1 ki αi where either all ki ≥ 0, in
which case α is called positive, or all ki ≤ 0, in which case α is called negative. The set of all
positive roots is denoted Φ+ , and the set of all negative roots is denoted by Φ− . Any root is
either positive or negative.
For each simple root αi , i ∈ I = {1, . . . , `}, we define the simple root reflection
wi (αj ) := αj − αj (hi )αi .

(1)

The wi generate a group W = W (A) of orthogonal transformations of h∗ , called the Weyl
group of A. The non-degenerate pairing between h and h∗ gives a corresponding action of W as
orthogonal transformations on h. A root α ∈ Φ is called a real root if there exists w ∈ W such
that wα is a simple root. A root α which is not real is called imaginary. We denote by Φre the
set of all real roots and Φim the set of all imaginary roots.
The multibracket, [ei1 , ei2 , · · · , ein ] = adei1 adei2 · · · adein−1 ein is in the root space gα for α =
Pn
+
j=1 αij ∈ Φ , while a similar multibracket with each eij replaced by fij is in g−α . Therefore,
g has a root space decomposition ([Kac90], Theorem 1.2)
M
g=h⊕
gα = h ⊕ g+ ⊕ g−
α∈Φ

where
g+ =

M

g− =

gα ,

M

gα .

−

+

α∈Φ

α∈Φ

The standard positive Borel subalgebra b ≡ b+ is defined by b+ = h ⊕ g+ and the standard
negative Borel subalgebra by b− = h ⊕ g− .
5

2.2. Kac–Moody groups. There are various ways to define abstract Kac-Moody groups (see
for example [KP85, Tit87, Rém02]). The main point of the abstract approach is to give a flexible
definition of Kac-Moody groups, allowing the construction of groups whose Adjoint action is not
faithful. There are indeed important examples of that kind, the smallest one being the finite
type Kac-Moody group SL(2, C), where the two matrices ±Id both act as the identity operator
in the Adjoint representation. Hence the Adjoint representation of this Kac-Moody group is
actually the group P SL(2, C), and that is what we get by using the definition of the adjoint
Kac-Moody group given in equation (2) for the Cartan matrix A = [2] of type A1 .
By the definition of the abstract Kac-Moody group, there is a surjective group homomorphism:
Ad : G −→ Gad from an abstract Kac-Moody group onto the Adjoint Kac-Moody group whose
kernel is exactly the center of G (see [Rém02, 9.6.2]). Since, as we will see in subsection 2.3, for
the subgroups B ± and N defined there, we have B ± ∩ N is abelian, and the center of G is the
kernel of the action of G on the twin building (see [Cap09, Lemma 1.7.]). Hence, without loss of
generality, to understand the action on twin buildings we can work with the Adjoint Kac-Moody
group. Our references for this section are [Kum02, KP85, MP95].
Let g be a symmetrizable Kac–Moody algebra over C, L be a complex vector space and let
π : g −→ End(L) be any integrable representation, so that all π(ei ) and π(fi ) are locally
nilpotent on L and the linear operators
χπαi (t) = exp(π(tei ))

and

χπ−αi (t) = exp(π(tfi )), for t ∈ C,

are well-defined in GL(L). Let the minimal Kac–Moody group associated to such a representation
π be the group generated by these operators,
Gπ = Gπ (C) = hχπαi (t), χπ−αi (t) | i ∈ I, t ∈ Ci ≤ GL(L) .
In fact, for any x ∈ gα , α ∈ Φre the operator π(x) is locally nilpotent on L, so χπx = exp(π(x))
is well-defined and these give the real root groups Uαπ . Now apply the above to the adjoint
representation
ad : g −→ End(g),
which is integrable. This gives the minimal adjoint Kac–Moody group
(2)

ad
Gad = Gad (C) = hχad
αi (t), χ−αi (t) | i ∈ I, t ∈ Ci ≤ GL(g)

and for all x ∈ gα , α ∈ Φre we have well-defined operators χad
x = exp(adx ) ∈ GL(g) giving
the real root groups Uαad . These generators act on g as Lie algebra automorphisms, so we have
Gad ≤ Aut(g).
Since adei and adfi are locally nilpotent, g is the direct sum of finite dimensional sl2 (C)-modules
for each of the subalgebras sli2 . For fixed i ∈ I, on each such summand the exponentials above
generate the group SLi2 isomorphic to SL2 (C), so G is also generated by the subgroups SLi2 ,
i ∈ I.
The operators π(h) ∈ End(L) for h ∈ h are semisimple so they can also be exponentiated to
give a commutative group of operators TCπ = TCπ (G) = {χπh (t) = exp(π(th)) | h ∈ h, t ∈ C} ≤ Gπ
which is called the standard maximal torus of Gπ . We also define the standard Borel subgroups
(B π )± = TCπ hUαπ | α ∈ (Φre )± i and the normalizer of TCπ denoted by NCπ .
It is well known that the operators
w
eiad = exp(adei )exp(ad−fi )exp(adei ) = exp(ad−fi )exp(adei )exp(ad−fi ) , 1 ≤ i ≤ `,
6

f ad in Gad such that the restriction of w
in Gad , generate a subgroup W
eiad to the standard Cartan
ad
subalgebra h equals the simple Weyl group reflection wi and w
ei (ei ) = −fi . It means that W is
ad
ad
f
f
a homomorphic image of W . Note that W is a subgroup of NCad and that NCad /TCad ∼
= W.
2.3. BN –pair and Tits building of a minimal Kac–Moody group. Our references for
this section are [AB08, Ron89]. A Kac–Moody group G over C has subgroups B ± ⊆ G and
N ⊆ G, with twin BN –pair (B + , B − , N ) satisfying the Tits axioms:
T 1: G = hB ± , N i, H = B ± ∩ N  N , W = N/H.
T 2: Define S by the condition W = hSi and for 1 ≤ i ≤ `, wi ∈ S we have wi2 = 1.
T 3: For wi ∈ S and w ∈ W we have
wi B ± w ⊆ B ± wi wB ± ∪ B ± wB ± .
T 4: For wi ∈ S we have
wi B ± wi−1 6⊆ B ± .
For a hyperbolic Kac-Moody group GC (A) we have TC = H. Thus the group T (G) = N ∩B ± is a
normal subgroup of N . The group W = NG (T (G))/T (G) is isomorphic to our earlier definition
in section 2.1 of the Weyl group W as a group of orthogonal transformations of h∗ given by
formula (1).
We identify W (non-canonically) with a subset (not a subgroup) of N which contains exactly
one representative for every element of W . By abuse of notation, this set of representatives will
also be called W . We have the (positive and negative) standard Borel subgroups, corresponding
to the standard Borel subalgebras, B ± = T (G)U ± where U + is generated by all positive real
root groups and U − is generated by all negative real root groups. The BN –pairs (B + , N ) and
(B − , N ) have Birkhoff and Bruhat decompositions:
a
a
G =
B ± wB ∓ =
B ± wB ± .
w∈W

w∈W

A proper subgroup P ± of G is called parabolic when it contains a conjugate of a Borel subgroup
B ± , and it is called positive or negative, depending on the sign. For each subset J ⊂ I define
the subgroup WJ = hwj | j ∈ Ji of W and the corresponding subgroups of G,
a
PJ± =
B ± wB ± .
w∈WJ
±
Note that PI± = G is not parabolic, P∅± = B ± , and we write Pi± = P{i}
. For J ( I we call PJ± a
standard parabolic subgroup, and these form a complete set of representatives of the conjugacy
classes of parabolic subgroups, so there are 2 · (2` − 1) conjugacy classes of parabolic subgroups.
A parabolic subgroup P ± is called maximal if there is no parabolic subgroup P 0± such that
±
±
P ± ( P 0± . For each i ∈ I, P[i]
= PI\{i}
is a maximal standard parabolic subgroup, so there are
2` conjugacy classes of maximal parabolic subgroups, ` positive and ` negative.

A Tits building consists of a simplicial complex B together with a collection of subcomplexes A,
each of which is called an apartment, such that
(1) each apartment is a Coxeter complex,
7

(2) each pair of chambers, i.e. simplices of maximal dimension in B, is contained in a
common apartment,
(3) for two apartments A and A0 there is an isomorphism ϕ : A → A0 , fixing the intersection
A ∩ A0 .
A twin building B consists of a disjoint union B + ∪ B − of buildings, which are ‘twinned’. This
twinning can be described in various ways, for example via twin apartments. We will not go
into details here, but direct the interested reader to [AR98].
From now on let B denote the twin building associated to a Kac-Moody group G. The simplices
of B are in bijection with parabolic subgroups in such a way that simplices in B + correspond to
positive parabolic subgroups and simplices in B − correspond to negative parabolic subgroups.
The vertices (0-simplices) of the twin building B are in bijection with maximal parabolic subgroups in G. Chambers are in bijection with positive and negative Borel subgroups. We will
denote these simplices by their corresponding parabolic subgroups. Since parabolic subgroups
are self-normalizing, the simplicies in the building can be equivalently indexed by the coset
spaces G/PJ± , where {PJ± | J ( I} is a complete set of representatives of the conjugacy classes
of parabolic subgroups.
The incidence relation on the set of vertices is given by intersections of parabolic subgroups as
follows. For 0 ≤ r ≤ ` − 1 the r + 1 vertices P[i±1 ] , . . . , P[i±r+1 ] span an r-simplex if and only if
the intersection P[i±1 ] ∩ · · · ∩ P[i±r+1 ] = P[i±1 ,...,ir+1 ] is a parabolic subgroup. Hence B is a simplicial
complex of dimension dim(B) = ` − 1. We note that for Kac–Moody groups, the intersection
of a positive parabolic subgroup with a negative parabolic subgroup never contains a Borel
subgroup; hence the associated geometry consists of two buildings B + and B − , corresponding to
the two opposite BN –pairs (B + , N ) and (B − , N ), yielding a twin building B = B + t B − . In B ±
±
±
the (` − 1)-simplex P[1,...,`]
= P[I]
= P∅± = B ± is called the fundamental chamber of B ± . Each
±
fundamental chamber has boundary consisting of the simplices ∆±
J = PJ , and has closure
[
∆± =
PJ± .
J(I

Using the
that the simplices in each building are in bijection with the union of the coset
S property
±
spaces J(I G/PJ , we describe the buildings B + and B − associated to a Kac–Moody group G
as follows:

(3)
B ± := G/B ± × ∆± / ∼
±
±
±
±
The equivalence relation ∼ is defined by (f B ± , ∆±
J ) ∼ (gB , ∆J 0 ) in (G/B , ∆ ) if and only if
±
±
±
±
∆J = ∆J 0 (so J = J 0 ) and f −1 gPJ ⊂ PJ .

Hence on the chambers ∆±
∅ , the equivalence relation ∼ is trivial, while on simplices in the
boundary it is nontrivial.
Let φ : G → G be an involution centralizing the Weyl group and such that φ(B ± ) = B ∓ . Then
φ induces a twin building involution as follows (for details see [DMGH09, Hor09])
(4)

∓
∓
φ(gB ± , ∆±
J ) = (φ(g)B , ∆J ) .
8

One can give a geometric realization of a building as follows. Let {e1 , · · · , e` } denote the standard
orthonormal basis of R` . Each r-simplex is identified with a copy of the standard simplex
(
)
r+1
r+1
X
X
∆r := x =
ai ei ∈ Rr+1 0 ≤ ai ≤ 1,
ai = 1
i=1

i=1

Rr+1 .

inherits the topology from
Appropriate identifications must be made among the copies of
these standard simplices in order to reflect the incidence structure among the simplices in the
building. For details see [AB08].
For the buildings associated with the hyperbolic Kac-Moody groups we wish to study, the
geometric realization of apartments in the buildings B + and B − can be chosen to be isometric
to hyperbolic spaces tesselated by the action of the hyperbolic Weyl group W .
2.4. Compact real forms of Kac–Moody algebras and groups. Let g = gC (A) be a complex Kac–Moody algebra and let h be the standard Cartan subalgebra. The Cartan involution
ω0 : g −→ g
is the automorphism of g determined by ω0 (ei ) = −fi , ω0 (fi ) = −ei and ω0 (hi ) = −hi . Composing ω0 with complex conjugation, we obtain a conjugate linear involution ω, called the
Cartan-Chevalley involution. Then k = {x ∈ g | ω(x) = x} is a Lie algebra over R called the
compact real form of g ([Kum02] p. 243).
Note that ω0 and ω both centralize the Weyl group, so they each induce twin building involutions
via formula (4).
We may give generators for the compact real form k as follows. For each j = 1, · · · , `, let gj = slj2
be the Lie subalgebra of g isomorphic to sl2 (C) with basis {ej , fj , hj }, so that g is generated by
the subalgebras gj and ω(gj ) = gj . Then the real Lie algebra of fixed points of ω on gj , kj = suj2
has basis
i
i
1
xj = (ej − fj ), yj = (ej + fj ), zj = (hj )
2
2
2
with brackets [xj , yj ] = zj , [yj , zj ] = xj , [zj , xj ] = yj , and the compact real form k is generated
by all of the subalgebras kj , j = 1, · · · , ` (see Proposition 1 in [Ber85]). A Cartan subalgebra in
the compact real form k is an abelian subalgebra whose complexification is a Cartan subalgebra
in g. The standard Cartan subalgebra t = h ∩ k in k has real basis {zj | 1 ≤ j ≤ `}.
Let G = GC (A) be the complex adjoint Kac–Moody group associated to g. The involution ω
of g lifts to a unique involution of G, also denoted by ω, exchanging positive and negative real
root groups since ω(gα ) = g−α . We have the following more general lemma about the action of
f ad .
any Lie algebra automorphism, which we will apply to ω as well as to w
eiad ∈ W
Lemma 2.1. For any α ∈ Φre , eα ∈ gα and any φ ∈ Aut(g), we have φ ◦ exp(adeα ) ◦ φ−1 =
ad for 1 ≤ i ≤ `.
ad and w
exp(adφ(eα ) ) so that, in particular, ωUαad ω −1 = U−α
eiad Uαadi (w
eiad )−1 = U−α
i
Proof. For any x ∈ g, since φ is a Lie algebra automorphism, we have


X 1
X 1
φ(exp(adeα )x) = φ 
(adeα )k (x) =
(adφ(eα ) )k (φ(x)) = exp(adφ(eα ) )(φ(x)) .
k!
k!
k≥0

k≥0
9

ad since ω(g ) = g
The formula with φ = ω gives ωUαad ω −1 = U−α
eiad gives
α
−α , and with φ = w
ad since w
w
eiad Uαadi (w
eiad )−1 = U−α
eiad (ei ) = −fi .

i

We set K = F ixG (ω). Then K is called the unitary form or compact real form of G. We will
use the latter by analogy with the finite dimensional case, even though K is not compact. The
group K is generated by subgroups Kj such that kj = Lie(Kj ) for each j = 1, . . . , ` ([Cap09],
[Tit86], [KP85]).
For each v = a + bi ∈ C and 1 ≤ j ≤ ` we write a generator of TCad as
exp(advhj ) = exp(adahj )exp(adbihj ) = exp(adahj )exp(adb2zj ) .
This gives the decomposition TCad = TRad T where
(5)

TRad = hexp(adahj ) | a ∈ R, 1 ≤ j ≤ `i

is the split real torus and T = hexp(adbzj ) | b ∈ R, 1 ≤ j ≤ `i is the compact real torus.
f ad ≤ K, but it is not so obvious that these
It is clear from the two expressions for w
eiad that W
operators can be expressed as a single exponential w
eiad = exp(adπ xi ). This is a special case of
π
the formula w
ei = exp(π(π xi )) for any integrable representation π : g → End(V ), where the
inner π is in R, proven in [FF16] and first found in certain important cases by [DH09].
Proposition 2.2. All Cartan subalgebras of k are conjugate under the action of K.
This result follows from [Cap09], Proposition 8.1 (iii). See also [KW92], Corollary 5.33 and [KP87],
Proposition 3.5.
3. Tits cone and lightcone of hyperbolic Kac–Moody algebras of compact type
Let g = gC (A) be a Kac–Moody algebra and let h be its standard Cartan subalgebra. The split
real form gR = gR (A) of g contains the R–subalgebra hR = h ∩ gR which is just the R–span of
{hi | i ∈ I}. We call hR the standard Cartan subalgebra of gR .
The Weyl group W has been defined above as the group of orthogonal transformations of h∗
generated by the simple reflections. The isomorphism between h∗ and h gives the action of
W on h, where the formula for simple reflections is just wi (hj ) = hj − αj (hi )hi . This same
formula restricted to hR gives the action of W and it also determines the action of W on t by
wi (zj ) = zj − αj (hi )zi . These operators are orthogonal with respect to the restriction of the
bilinear form (·, ·) to hR and to t, and therefore W preserves each surface of constant square
length, (hR )r = {x ∈ hR | (x, x) = r} and tr = {x ∈ t | (x, x) = r}.
Since we are assuming that the Cartan matrix A is hyperbolic, the form (·, ·) is Lorentzian on
hR and on t. The surface where r = 0 is called the nullcone, each surface where r < 0 is called
timelike, and each surface where r > 0 is called spacelike. The set of all timelike points has two
connected components, one called forward and denoted T L+ and the other called backward and
denoted T L− . We have T L− = −T L+ .
Each of these components is preserved by the linear action of W , which acts consistently on
rays, Rayx = {rx | 0 < r ∈ R} since w(rx) = rw(x). A fundamental domain for the action of
10

W on each of the timelike components is defined by
C ± = {x ∈ T L± | αi (x) ≥ 0, 1 ≤ i ≤ `}.
The union
X± =

[

w(C ± )

w∈W

is called the positive (respectively negative) Tits cone and X = X + ∪ X − is called the Tits cone.
Clearly, we have X − = −X + . For g hyperbolic, X ± ⊇ T L± , since it is possible that C ± , and
therefore X ± , contains rays on the nullcone. This happens for the rank 3 hyperbolic Cartan
matrices




2 −2 0
2 −2 −2
−2 2 −1
−2 2 −2
and
0 −1 2
−2 −2 2
corresponding to the hyperbolic Kac–Moody algebra F [FF83] whose Weyl group is the hyperbolic triangle group T (2, 3, ∞), and the “ideal” hyperbolic Kac–Moody algebra I whose Weyl
group is the hyperbolic triangle group T (∞, ∞, ∞), respectively. For A strictly hyperbolic, that
is, whose principal minors are of finite type, we have X ± = T L± . Such an example is E10 .
We have the following description of the closure of the Tits cone (see [Kac90], 5.10.2)
X = X + ∪ X − = {h ∈ hR | (h, h) ≤ 0}.
We introduce the notations
LhR = {h ∈ hR | (h, h) ≤ 0},
L0hR = {h ∈ hR | (h, h) < 0} and
∂LhR = {h ∈ hR | (h, h) = 0}.
Proposition 3.1. Let g be a complex Kac–Moody algebra, k its compact real form and gR its
split real form. Let h, hR , t be the standard Cartan subalgebras of g, gR , k, respectively. Then
hR ∼
= R`−1,1 has signature (` − 1, 1) and
t = ihR ,
where i2 = −1 and ihR = {ix | x ∈ hR }. The signature of t = ihR ∼
= R1,`−1 is (1, ` − 1).
We use a sign convention on k adopted from the theory of finite dimensional Riemannian symmetric spaces [Hel01] and set
(6)

(·, ·)k = −(·, ·)|k .

Note that in the affine case (·, ·)k , this sign convention naturally occurs in the loop group realizations [PS86, Fre09]. With respect to (·, ·)k , the Cartan subalgebra t has Lorentzian signature
(` − 1, 1).
Lemma 3.2. The invariant symmetric bilinear form (·, ·)k on k is Lorentzian with signature
(∞, 1).
11

Proof. As a vector space, a complex hyperbolic Kac–Moody algebra g has a basis consisting of
the Cartan subalgebra generators, hi , i = 1, . . . , `, and certain multibrackets [ei1 , ei2 , · · · , ein ]
and [fi1 , fi2 , · · · , fin ]. The action of ω on such multibrackets is simply ω([ei1 , ei2 , · · · , ein ]) =
(−1)n [fi1 , fi2 , · · · , fin ] and ω(hi ) = −hi so an R-basis for the compact real form k consists of
the compact Cartan basis elements zi = 2i hi , i = 1, . . . , `, and the elements obtained from basis
multibrackets above
i
1
([ei1 , ei2 , · · · , ein ] + [fi1 , fi2 , · · · , fin ]) and ([ei1 , ei2 , · · · , ein ] − [fi1 , fi2 , · · · , fin ]) for n even,
2
2
1
i
([ei1 , ei2 , · · · , ein ] − [fi1 , fi2 , · · · , fin ]) and ([ei1 , ei2 , · · · , ein ] + [fi1 , fi2 , · · · , fin ]) for n odd.
2
2
In particular, for n = 1 these include the elements xi = 12 (ei − fi ) and yi = 2i (ei + fi ).
Following our sign convention, the Cartan subalgebra t spanned by the elements zi , i = 1, . . . , `,
has signature (` − 1, 1). Using our sign convention and the characterization of the ad–invariant
scalar product in [Kac90], Equation 2.2.1, applied to the basis vectors yi and zi , we find that
the bilinear form on each subalgebra ki = sui2 is positive definite.
Furthermore the ad–invariant bilinear form has the following properties on root spaces (see [Kac90],
Sections 2.1 and 2.2). The root spaces gα and g−α , which are interchanged by ω, have dual bases
{ejα | 1 ≤ j ≤ dim(gα )} and {fαj = ej−α | 1 ≤ j ≤ dim(gα )}, respectively
such that
(fαj , em
β ) = δj,m δα,β .
As a consequence, for positive roots α, the basis elements of k, xjα =
j
i j
2 (eα + fα ) satisfy

1 j
2 (eα

− fαj ) and yαj =

1
j
m
j
m
(xjα , xm
β ) = − δj,m δα,β = (yα , yβ ) and (xα , yβ ) = 0.
2
Hence it follows from (6) that
1
j
m
j
m
(xjα , xm
β )k = δj,m δα,β = (yα , yβ )k and (xα , yβ )k = 0.
2
The Cartan subalgebra has signature (` − 1, 1) and for each positive root α the subspaces
hxjα | 1 ≤ j ≤ dim(gα )i and hyαj | 1 ≤ j ≤ dim(gα )i
in k each have positive signature and are orthogonal to each other. For different α these spaces
are orthogonal to each other and to the Cartan subalgebra t. Thus k has Lorentzian signature
(∞, 1).

This could also be deduced from [RC89], Section 2.1.
Remark 3.3. Note that the bilinear form (·, ·) on gR is indefinite with signature (∞, ∞) because,
while the split Cartan hR has the signature (`−1, 1), each pair of dual root vectors {ejα , fαj } forms
a hyperbolic plane, and for different positive α and distinct j these planes are orthogonal.
12

We recall that the ad–invariant bilinear form on hR extends C–bilinearly to the complexification
hC . In particular, for x ∈ hR we have
(ix, ix) = −(x, x) = (x, x)k

(7)

so the map ϕ : hR −→ t given by ϕ(x) = ix is an isometry. In addition, for w ∈ W we have
ϕ(wx) = wϕ(x), hence ϕ is W -equivariant.
We introduce notations for certain subsets in k:
Lk = {x ∈ k | (x, x)k ≤ 0},

Lt = Lk ∩ t = ϕ(LhR ),

L0k = {x ∈ k | (x, x)k < 0},

L0t = L0k ∩ t = ϕ(L0hR ),

∂Lk = {x ∈ k | (x, x)k = 0},

∂Lt = ∂Lk ∩ t = ϕ(∂LhR ).

The ones inside t are related by the W -invariant isometry ϕ to the corresponding subsets defined
earlier in hR . Furthermore, we have
ϕ(X ± ) = Xt± ,

ϕ(C ± ) = Ct± ,

ϕ(T L± ) = T L±
t ,

±

±

ϕ(X ) = X t ,

ϕ(X) = X t

which correspond to those subsets defined earlier in hR . We call Xt the Tits cone of t and note
that X t = Lt .
Remark 3.4. We will refer to Lk as the lightcone of k and Lt as the lightcone of t.
4. Group actions of the compact real form K
4.1. The Adjoint action of K on k. Recall that K denotes the compact form of the complex
adjoint Kac–Moody group G. We set
[
H=
ktk −1 = {kxk −1 ∈ k | k ∈ K, x ∈ t} .
k∈K

By Proposition 2.2 all Cartan subalgebras of k are K–conjugate, thus the definition of H is
independent of the choice of t.
Proposition 4.1. Let t be any Cartan subalgebra of k and let z = ih ∈ t for h ∈ hR satisfying
α(h) 6= 0 for all α ∈ Φ+ . Then the subspace [k, z] of k has basis
[
{xjα | 1 ≤ j ≤ dim(gα )} ∪ {yαj | 1 ≤ j ≤ dim(gα )} .
α∈Φ+

Proof. For α ∈ Φ+ and 1 ≤ j ≤ dim(gα ) we have
i
2

1
2

i
2

[xjα , z] = −[ih, (ejα − fαj )] = − [h, (ejα − fαj ))] = − α(h)(ejα + fαj ) = −α(h)yαj ,
i
2

1
2

1
2

[yαj , z] = −[ih, (ejα + fαj )] = [h, (ejα + fαj ))] = α(h)(ejα − fαj ) = α(h)xjα
and [t, z] = 0, so no basis vectors of t are in [k, z].



By analogy with the finite dimensional and affine cases, the following proposition shows that in
the hyperbolic case, the K–orbits, K · z = {kzk −1 ∈ k | k ∈ K} for each z ∈ t, intersect each
Cartan subalgebra orthogonally. For z ∈ t let Tz (K · z) be the tangent space of the submanifold
K · z at the point z.
13

Proposition 4.2. Let t be any Cartan subalgebra of k. The orbits K · z for z ∈ t are orthogonal
to t with respect to the ad–invariant bilinear form (·, ·)k , that is, (Tz (K · z), t)k = 0.
Proof. For z ∈ t we have
Tz (K · z) = [k, z] .
By definition, for w ∈ Tz (K · z), there is some y ∈ k such that w = [y, z]. Since the form (·, ·)k
is ad–invariant, for z 0 ∈ t we obtain
(w, z 0 )k = ([y, z], z 0 )k = (y, [z, z 0 ])k = (y, 0)k = 0.

Let X ⊂ V be a subset of a real vector space V equipped with a bilinear form (·, ·). For any
real number r we define a ‘sphere’ of radius r in X by
Xr := {x ∈ X | (x, x) = r} .
If (·, ·) is Lorentzian and r < 0 then Xr is a two-sheeted hyperboloid of constant curvature
κ = − r12 whose connected components we call Xr+ and Xr− . In particular we will use tr ⊂
L0t ⊂ t, kr ⊂ L0k ⊂ k and Hr ⊂ H and note that k0 = ∂Lk and t0 = ∂Lt . For ` > 2 we find
∼ − ∼ `−1 , (` − 1)–dimensional hyperbolic space.
t+
−1 = t−1 = H
Definition 4.3 (Lightlike closure). Let V be a possibly infinite dimensional real vector space
equipped with a Lorentzian form (·, ·) and let ∂LV = {x ∈ V | (x, x) = 0} denote its nullcone.
The boundary at infinity B ∞ (∂LV ) of the nullcone consists of all rays Rayx for 0 6= x ∈ ∂LV . If
dim(V ) = 2 it consists of four points. If dim(V ) > 2 it consists of two components, corresponding to the future timelike boundary B ∞ (∂LV )+ and the past timelike boundary B ∞ (∂LV )− , each
of which can be identified with a sphere of dimension ` − 2 We define the lightlike closure of V
by
Vb = V ∪ B ∞ (∂LV ) .
For each r < 0, define B ∞ (Vr± ), the boundary at infinity of Vr± , to be equivalence classes of
geodesic rays, where two rays are equivalent if the distance between them is finite at all points.
For details, see [Ebe96]. Note that for each r < 0, B ∞ (Vr± ) can be identified with B ∞ (∂LV )± .
Using this observation we define the lightlike closure of Vr± to be
(8)

Vbr± = Vr± ∪ B ∞ (∂LV )± .

Since the bilinear form (· , ·)k is ad–invariant, the Adjoint action of K on k preserves the surfaces
Hr ⊂ H. For k ∈ K we have k · tr = (k · t)r . There is an induced action of K on B ∞ (∂Lt ) and
for each r < 0 on B ∞ (∂Ltr )± , as well as on bt and on bt±
r .
Hence, for each r < 0, the Adjoint action of K on k induces a well–defined action on the surface:
K : Hr −→ Hr ,

k · x = Adk (x) = kxk −1

br.
as well as on the lightlike closure H
Since all Cartan subalgebras are conjugate, we can define the following surjective map
ψ(k, x) = kxk −1 .

ψ : K × tr −→ Hr ,
14

Note that the choice of the standard Cartan subalgebra t in the definition of ψ does not restrict
the generality.
Let T = exp(t) = T (Gad ) ∩ K be the torus associated to t and let
N = NK (T ) = NGad (T (Gad )) ∩ K
be the normalizer of T in K. For k ∈ K, t ∈ T and u ∈ t we have:
Adkt (u) = ktut−1 k −1 = kuk −1 = Adk (u).
Thus the Adjoint action on t is T –invariant and factors to the quotient space K/T yielding a
surjective map
ψ : K/T × tr −→ Hr ,
ψ(kT, u) = kuk −1 .
For any Cartan subalgebra t0 = ktk −1 of k, Lt0 coincides with the closure of the Tits cone of t0 ,
X t0 . Hence X t0 is the closure of the cone {su ∈ t0 | s > 0, u ∈ t0r }, for any fixed r < 0, which
includes the boundary ∂Lt0 . We distinguish two cases:
(1) When every proper Cartan submatrix of A is of finite type (i.e. A is (` − 1)-spherical or
strictly hyperbolic) then we have Xt0 = L0t0 and there is a bijection between each surface
(t0r )± for r < 0 and the set of rays in L0t0 . In this case we do not need to consider the
b r in order to embed the building in it.
lightlike closure H
(2) When A contains an affine Cartan submatrix the fundamental chambers C ± contain rays
that accumulate at rays on the lightcone ∂Lt0 . So the corresponding points on the surface
(t0−1 )± accumulate at points on the boundary of the (`−1)–dimensional hyperbolic space
H`−1 . Therefore, in these cases we do have to consider the lightlike closure in order to
embed the building. In the example of F, the surfaces (t0−1 )± ∼
= H2 are isometric to
the Poincaré disk, and the tesselation by the hyperbolic Weyl group T (2, 3, ∞) includes
chambers which have an ideal vertex on the boundary. In such an example, the building
would have 0-simplices corresponding to those ideal vertices, so to achieve our goal of
embedding the twin building of a hyperbolic algebra inside the lightcone of k, we must
br.
use H
Since W = N/T ≤ K/T , the restriction to tr of the Adjoint action of N on kr coincides with
the Weyl group action of W on tr .
Restriction of the second coordinate of the domain of ψ to either fundamental domain Cr± for
the action of W on tr gives the surjective maps
(9)

ψ ± : K/T × Cr± −→ H±
r ,

ψ ± (kT, u) = kuk −1 .

4.2. The local structure of the Adjoint action. In this section we describe the geometry
‘close’ to a fixed Cartan subalgebra in the compact real form k. We show that for i ∈ {1, . . . , `},
the Adjoint action of the fundamental SU (2)i –subgroup of the compact real form K, is a rotation
of the standard Cartan subalgebra around the hyperplane
Lt,i := {z ∈ t | αi (z) = 0}
fixed by the generator wi of W .
The standard Cartan subalgebra is given by t = Rz1 + · · · + Rz` . For 1 ≤ i ≤ ` and s, t ∈ R, we
have exp(sxi + tyi ) ∈ SU (2)i ≤ K and for z ∈ t we have [sxi + tyi , z] = −iαi (z)(txi − syi ) = 0
15

when αi (z) = 0 which means exp(sxi + tyi )(z) = z when αi (z) = 0. So Lt,i is the fixed point
set in t of exp(adsxi +tyi ). For each s, t ∈ R, exp(adsxi +tyi )t = ti (s, t) is either another Cartan
subalgebra such that t ∩ ti (s, t) = Lt,i or else t = ti (s, t). Note that with v = ( 2s + ti2 ) ∈ C we
have sxi + tyi = vei − vfi , so
exp(adsxi +tyi ) = exp(advei −vfi ) ∈ (su2 )i
gives another parameterization of {ti (s, t) | s, t ∈ R} = {exp(advei −vfi )t | v ∈ C}. For each
1 ≤ i ≤ ` the family of distinct Cartan subalgebras obtained this way can be parameterized by a
2–sphere with antipodes identified, as we will show using the first part of the following theorem.
Theorem 4.4. ( [FF16]) For any s, t ∈ R such that 0 < r2 = s2 + t2 and for any z ∈ t, we have
(1) exp(adsxi +tyi )z = z − iαi (z)(cos(r) − 1)zi − iαi (z)

sin(r)
(txi − syi ),
r

t sin(r)
t
zi − 2 (cos(r) − 1)(txi − syi ),
r
r
s sin(r)
s
(3) exp(adsxi +tyi )yi = yi +
zi − 2 (cos(r) − 1)(txi − syi ).
r
r

(2) exp(adsxi +tyi )xi = xi −

Proof. We prove only the first relation since the others are analogous. We have
(adsxi +tyi )1 z
(adsxi +tyi )2 z
(adsxi +tyi )3 z
(adsxi +tyi )4 z

=
=
=
=

[sxi + tyi , z]
[sxi + tyi , −iαi (z)(txi − syi )]
[sxi + tyi , αi (z)(s2 + t2 )zi ]
[sxi + tyi , iαi (z)(s2 + t2 )(txi − syi )]

=
=
=
=

−iαi (z)(txi − syi ) ,
iαi (z)(s2 + t2 )zi ,
iαi (z)(s2 + t2 )(txi − syi ) ,
−iαi (z)(s2 + t2 )2 zi .

Using r2 = s2 + t2 6= 0, it is clear that for n ≥ 1 we have
(adsxi +tyi )2n z = −iαi (z)(−1)n (r2 )n zi
and for n ≥ 0 we have
(adsxi +tyi )2n+1 z = −iαi (z)(−1)n (r2 )n (txi − syi )
so we get
exp(adsxi +tyi )z = z − iαi (z)

∞
X
(−1)n r2n

(2n)!

n=1

zi − iαi (z)

∞
X
(−1)n r2n
n=0

= z − iαi (z)(cos(r) − 1)zi − iαi (z)

(2n + 1)!

(txi − syi )

sin(r)
(txi − syi ).
r


Corollary 4.5. ( [FF16]) For each 1 ≤ i ≤ `, the family of distinct Cartan subalgebras
{ti (s, t) | s, t ∈ R}, including t, is parametrized by the unit hemisphere

where r =

√

{(sin(r)sin(ψ), −sin(r)cos(ψ), cos(r)) ∈ R3 | 0 ≤ r < π, 0 ≤ ψ < π}
s2 + t2 ≥ 0 and ψ is defined when r > 0 by
sin(ψ) =

t
r

and
16

cos(ψ) =

s
.
r

Proof. The Cartan subalgebra ti (s, t) is spanned by Lt,i and
exp(adsxi +tyi )zi = sin(r)(sin(ψ)xi − cos(ψ)yi ) + cos(r)zi
from theorem 4.4 (1) and the fact that αi (zi ) 6= 0. Since for 0 ≤ r < π and 0 ≤ ψ < π no two
vectors of the above form are colinear, all of the corresponding subspaces ti (s, t) are distinct.
But for antipodal points on the unit sphere, (r, ψ) and (π−r, ψ+π), the corresponding subspaces
are the same.

4.3. The action of the compact real form on the twin building. To describe the action
of K on the twin building, we use the Iwasawa decomposition [DMGH09] which yields
G = KAU ± ,
where G denotes a complex Kac–Moody group, K denotes the compact real form of G,
A∼
= Rrank(G) is an abelian subgroup and U ± is the group generated by all positive (respectively
negative) real root groups. Recall that t denotes a Cartan subalgebra in k, and T= exp(t) its
torus. Using TC (G) = T A, the decomposition B ± = TC (G)U ± and the Iwasawa decomposition,
we have
∼ K/T .
(10)
G/B ± =
Recalling the description of the building from equation (3) in section 2.3

(11)
B ± = G/B ± × ∆± / ∼ ,
we obtain an equivalent new description by equation (10):

(12)
B ± = K/T × ∆± / ∼ .
In the description of the twin building B = B + ∪ B− given by equation (11), the natural action
of G via left multiplication is apparent, while in the description of B as in equation (12) the
G-symmetry is broken to K–symmetry. Note that in equation (11) the cosets of the two opposite
buildings B + and B − are defined with respect to different subgroups B + and B − respectively.
Hence there are subgroups of G which act differently on B + and B − . For example, for any
g ∈ G the coset gB + ∈ G/B + is fixed by all elements in the subgroup Bg+ = gB + g −1 , but that
subgroup acts transitively on {f B − ∈ G/B − | g −1 f ∈ B + B − } which certainly contains gB − .
But the description of B ± in equation (12) shows that the action of K is the same in both.
In particular this means that the group K does not act transitively on any apartment system.
More precisely, for A ∈ B ± an apartment, define the orbit AK (A) := K · A. Then for each
chamber c ∈ B ± there is exactly one apartment Ac ∈ AK (A) such that c ∈ Ac .
An apartment A is called ω–stable iff ω(A) = A. If A is ω–stable, the set AK (A) contains all
ω–stable apartments.
±
±
±
±
Let c = (f B ± , ∆±
∅ ) be any chamber in B = (G/B × ∆ ) / ∼, and let Cham(B ) denote the
set of all chambers of the building B ± . For i ∈ I, the i-wall of c is (f B ± , ∆±
i ) and the i–residue
±
±
±
±
±
of c is defined to be Ri (c) = {d = (gB , ∆∅ ) ∈ Cham(B ) | (f B , ∆i ) ∼ (gB ± , ∆±
i )}. Then
1
1
∼
b
we have Ri (c) = P (C). Identifying P (C) with the Riemann sphere C = C ∪ ∞, the action
b by Möbius
of the subgroup SU (2)i on Ri (c) can be identified with the action of SU (2) on C
transformations. Additional details may be found in [AB08], Chapter 6.
17

5. Simplicial structure on Hr
In this section we define for each r < 0 a simplicial structure on the set Hr ⊂ kr . We begin
−
in the standard Cartan subalgebra with tr ⊂ t. Recall, that tr = t+
r ∪ tr is (up to rescaling)
isometric to a pair of hyperbolic spaces (for ` > 2) which are both tesselated by the action of
the Weyl group W . We call an element x ∈ tr singular if it is fixed by a Weyl group element
w 6= 1 and call x regular otherwise. Let tsin
be the set of all singular elements in tr and let treg
r
r
± and (treg )± . Then we have
be the set of all regular elements, and similarly we have sets (tsin
)
r
r
the decomposition
reg
tr = tsin
r ∪ tr .
reg
We denote by Comp±
r the set of connected components of tr . In each sheet one connected
±
component coincides with the fundamental domain C . For each sign ±, the Weyl group acts
simply transitively on that set Comp±
r . We use the Weyl group action to index the elements
of Comp±
as
follows:
Let
1
∈
W
denote
the identity element. In each sheet t±
r
r we index the
±
fundamental chamber C with 1. Then we index the connected component c± = wC ± ∈ Comp±
r
by w yielding
reg ±
reg ±
Comp±
r = {(tr )w | w ∈ W } ⊂ (tr ) .
±

reg ±
±
Let (treg
r )w denote the closure of the component (tr )w and let U denote the union
[ reg ±
U± =
(tr )w
w∈W

so

U±

covers

t±
r .

For any minimal subset J ∈ W such that the intersection
\ reg ±
SJ± =
(tr )w ⊂ t±
r
w∈J

is nonempty, we identify the interior of

SJ±

with a simplex of dimension ` − |J|.

±
For example, each connected component in Comp±
r is an open subset of tr , hence of dimension
` − 1. It follows that it is identified with a simplex of dimension ` − 1.

It is straightforward to check that S ± = {SJ± | J ⊂ W, SJ± 6= ∅} is a Coxeter complex [AB08] for
the Weyl group W . That is, S ± admits a W –action which is simply transitively on simplices
of maximal dimension. Thus for any two simplices of maximal dimension, there is a chain of
maximal dimensional simplices such that two consecutive ones share a common boundary. Since
our compact real Kac–Moody algebra k has rank `, all of its Cartan subalgebras, ktk −1 , for
k ∈ K, have dimension ` and each surface (ktk −1 )r of radius r < 0 has dimension ` − 1. Thus
simplices of maximal dimension have dimension ` − 1 and their boundaries have dimension ` − 2.
We may now extend this simplicial structure to all of Hr as follows. A similar simplicial structure may be defined on any Cartan subalgebra in Hr . We must check that the Weyl group
tesselations on different Cartan subalgebras fit together in a well–defined way. Cartan subalgebras intersect exactly in hyperplanes fixed by Weyl group elements (as in section 4.2), hence
they intersect in singular elements. Thus each simplex of maximal dimension lies in exactly one
Cartan subalgebra. Thus the simplicial structures on different Cartan subalgebras fit together,
leading to a tesselation of Hr into simplices.
18

Each simplex of codimension 1 lies in the intersection of a fixed point hyperplane Lktk−1 ,i with
the surface (ktk −1 )r of radius r < 0 as calculated in Section 4.2, while simplices of codimension
n lie in the intersection of n such hyperplanes with the surface. Restriction of the Adjoint action
to a single Cartan subalgebra coincides with the Weyl group action. This follows easily from
the fact that the Weyl group W = N/T is a quotient of the normalizer of T . Hence simplicies
of lower dimension are fixed by nontrivial subgroups of W .
6. The main embedding theorem
An ‘embedding’ of the twin building B = B + ∪ B − of the Kac-Moody group G with identity
element e into the compact real form k is a bijective simplicial map from B onto the simplicial
b r for each r < 0 defined in section 5.
structure defined on H
Theorem 6.1. Let A be a symmetrizable hyperbolic generalized Cartan matrix, g its complex
Kac–Moody algebra and G its complex Kac–Moody group and let r < 0. Let K be the compact
real form of G and let k be its Lie algebra. Let B = B + ∪ B − be the geometric realization
(section 2.3) of the twin building of G over C.
b r ⊂ bk, that is, the following diagram
(1) There is a K–equivariant embedding Ψr : B ,→ H
commutes:
B


K

Ψr

br
H

AdK

/B


Ψr

br
/H

where AdK denotes the Adjoint action and K ⊂ G acts on B by left multiplication.
±
±
(2) When A is strictly hyperbolic the K–equivariant restrictions Ψ±
r : B ,→ Hr are bijective,
±
±
±
b are injective.
otherwise, Ψr : B ,→ H
r
b
(3) There is a W –equivariant embedding of the fundamental apartment of B ± into bt±
r ⊂ tr .
(4) The embedding of the twin building is a simplicial complex isomorphism with respect to
b r discussed in section 5.
the simplicial structure on H
Remark 6.2. The image Im (Ψr ) is contained in the interior of the lightcone if A is strictly
hyperbolic. Examples of this type are the rank 2 hyperbolic Kac-Moody algebras (see section 8).
Otherwise it contains points on the boundary of the lightcone. Examples of this type are the
algebras F and I.
Proof. To construct the embedding of the twin building we use the description given in equation 12

B ± = K/T × ∆± / ∼ .
Our construction is in three steps. We first define the embedding of the fundamental chamber.
Then we make use of the K-action on the building and the Adjoint action of K on the Lie
algebra. In a third step we establish the properties claimed in the theorem.
±
Part 1: We choose the fundamental chambers in the buildings B ± to be c± = (1T, ∆±
∅) ∈ B .
±
The ` walls of the fundamental chamber correspond to the simplices (1T, ∆i ) for 1 ≤ i ≤ `,
±
its vertices correspond to the simplices (1T, ∆±
[i] ). Recall that [i] = I\{i}. Two pairs (f T, ∆i )
19

and (gT, ∆±
i ) describe the same simplex if any only if f Ki = gKi where Ki = K ∩ Pi =
±
K ∩(B tBwi B); similarly for any subset J ( I two J-cells (f T, ∆±
J ) and (f T, ∆J ) are equivalent
if and only f KJ = gKJ for KJ = K ∩ PJ .
+
For r < 0 we choose a connected component (treg
r )1 where 1 ∈ W denotes the identity element
and we define

(13)

−
reg +
(treg
r )1 := −(tr )1 .

If A is strictly hyperbolic the fundamental domain is contained in the interior of tr ; hence its
closure is contained in tr . On the other hand if A is not strictly hyperbolic then it is unbounded
and its closure is contained in btr but not in tr . We identify the boundary hyperplanes Lt,i of the
fundamental chamber with the generators wi , 1 ≤ i ≤ `, of the Weyl group W . We also identify
intersections of hyperplanes Lt,i1 ∩ · · · ∩ Lt,in for distinct indices {i1 , . . . , in } ⊂ I with the subset
of generators {wi1 , . . . , win }. Vertices correspond to the intersection of (` − 1) hyperplanes and
hence to subsets [i] = I\{i}. For a subset J ( I we define the boundary components
\
Lt,j .
CJ± = C ± ∩
j∈J
±
±
We define Ψr (1T, ∆±
[i] ) = C[i] and extend this map to ∆ by mapping a point x in the geometric
realization of the simplex (1T, ∆±
J ) with normalized barycentric coordinates x = [λ1 : · · · : λ` ]
to the point with the same normalized hyperbolic barycentric coordinates.

Recall the definition of normalized hyperbolic barycentric coordinates: Let ∆ be a simplex in ndimensional hyperbolic space with vertices (v0 , . . . , vn ). Some vertices may be on the boundary
of hyperbolic space. We denote by V = V (∆) the volume of ∆. For any point p ∈ ∆ we can
define n+1 simplices ∆[i] for 0 ≤ i ≤ n, spanned by the (n+1)-tuple of vertices (v0 , . . . , p, . . . , vn ),
where the vertex vi has been replaced by p. Let Vi = V (∆[i] ) denote the volume of ∆[i] . Then
the normalized hyperbolic barycentric coordinates of p are given by


V0
Vn
: ··· :
.
V
V
This yields a simplicial map Ψr : (1T, ∆± ) −→ C ± .
Part 2: Let x± ∈ ∆± . We extend the map defined in Part 1 to a map
br
Ψr : B ,→ H
by defining


Ψr (kT, x± ) = Adk Ψr (1T, x± ) .
We have to check that Ψr is well-defined. Assume we have two equivalent elements (k1 T, x±
1)∼
±
±
±
±
(k2 T, x±
),
hence
x
=
x
and
assuming
x
∈
∆
for
some
subset
J
(
I,
we
have
k
K
=
k
1 J
2 KJ .
2
1
2
1
J
Then there is some l ∈ KJ such that k1 = k2 l and we have




±
±
Ψr (k1 T, x±
1 ) = Adk1 Ψr (1T, x1 ) = Adk2 l Ψr (1T, x1 ) .
±
Since x±
1 ∈ ∆J we have

Ψr (1T, x±
1)∈

\
j∈J

20

Lt,j .

±
Hence Adl Ψr (1T, x±
1 ) = Ψr (1T, x1 ). Thus, from Adk2 l = Adk2 Adl we get




±
±
Adk2 l Ψr (1T, x±
1 ) = Adk2 Ψr (1T, x2 ) = Ψr (k2 T, x2 ) .

Part 3: We have to check that Ψr satisfies the properties stated in the theorem.
Proof of (4): Remark that Ψr maps the unique simplex (kT, ∆±
J ) for J ( I spanned by vertices
±
(kT, ∆±
),
i
∈
J,
onto
the
simplex
spanned
by
Ψ
(kT,
∆
),
i
∈ J. If two simplices (k1 T, ∆±
r
J)
[i]
[i]
±
±
and (k2 T, ∆J ) share a common boundary (lT, ∆L ), for J ( L in B then by definition we have
±
±
k1 T ⊂ Adl KL and k2 T ⊂ Adl KL . But then Ψr (k1 T, ∆±
L ) = Ψr (lT, ∆L ) = Ψr (k2 T, ∆L ) is the
±
±
commonly shared boundary of the simplices Ψr (k1 T, ∆J ) and Ψr (k2 T, ∆J ). Hence Ψr preserves
the simplicial structure of B and is thus a simplicial complex isomorphism, establishing (4).
Proof of (1): Recall the left K–action on the building B ± :
K : B ± −→ B ± ,

k · (f T, ∆± ) 7→ (kf T, ∆± ) .

We need to verify that for k1 , k2 ∈ K and J ( I
±
Ψr (k1 · (k2 T, ∆±
J )) = Adk1 Ψr (k2 T, ∆J ) .

We have
±
±
Ψr (k1 · (k2 T, ∆±
J )) = Ψr (k1 k2 T, ∆J ) = Adk1 k2 Ψr (1T, ∆J ) =
±
= Adk1 (Adk2 Ψr (1T, ∆±
J )) = Adk1 (Ψr (k2 T, ∆J )) .
±
Proof of (2): Assume two simplices (k1 T, ∆±
J ) and (k2 T, ∆J ) satisfy
±
Ψr (k1 T, ∆±
J ) = Ψr (k2 T, ∆J ) .

Then we have
±
Adk1 Ψr (1T, ∆±
J ) = Adk2 Ψr (1T, ∆J )
±
which implies Adk−1 k2 Ψr (1T, ∆±
J ) = Ψr (1T, ∆J ).
1

±
As
= CJ± its stabilizer in K is KJ . Hence k1−1 k2 ∈ KJ . Thus (k1 T, ∆±
J ) ∼ (k2 T, ∆J )
which proves injectivity.

Ψr (1T, ∆±
J)

Suppose we have an arbitrary element x± ∈ H±
r . Then there is some group element k ∈ K such
±
that Adk x± ∈ t±
so
it
is
in
the
closure
of
some
chamber, (treg
r )w , uniquely labeled by an element
r
f ad ≤ K matches the action of w on t. So
w ∈ W . The action of the corresponding element w
e∈W
±
±
±
we have Adwe−1 Adk x ∈ Cr . Therefore x ∈ Adk−1 Adwe Ψr (1T, ∆± ) = Ψr (k −1 wT,
e ∆± ) . This
shows that in the case when A is strictly hyperbolic, Ψr : B ± → H±
r is surjective. Otherwise
there can be elements x± ∈ B ∞ (H±
r ) which are in Im(Ψr ), but some may not. It would be
interesting to understand precisely which points in B ∞ (H±
r ) are in Im(Ψr ). The proof of (3) is
now clear.

Remark 6.3. Based on our understanding of the examples F and E10 , we believe that the
intersection of the nullcone with Im(Ψr ) can be characterized as those rays on the nullcone which
contain roots of g. Each such ray of null vectors corresponds to a copy of an affine KM subalgebra
inside g. Each such ray should be conjugate under the action of the Weyl group W to one ray in
the closure of the fundamental chamber, so the number of W orbits is the number of ideal points
in that fundamental domain. Classifying those rays then becomes a significant problem involving
21

the arithmetic properties of W , essentially understanding its cusps as a modular group. Such
an analysis was carried out for E10 in Lemma 5.2 of [KPN12].
Thus for each r < 0 we have established an embedding Ψr of the twin building B = B + ∪ B −
into H±
r which is inside the lightcone
Lk = {x ∈ k | (x, x) ≤ 0}
of the compact real form k of a strictly hyperbolic Kac–Moody algebra g, and into its lightlike
b ± , otherwise.
closure, H
r
Remark 6.4. Recall that B ± is contractible since each apartment is contractible (see [Ron89]).
Since Ψr for r ≤ 0 is a simplicial complex isomorphism onto its image, the image Ψr (B ± ) is
also contractible.
This observation yields the corollary
b ± are contractible.
Corollary 6.5. The spaces H
r
Remark 6.6. The analogs of theorem 6.1 for finite dimensional compact Lie groups and affine
Kac–Moody groups are well-known (see [Ebe96, Mit88, Hei06, Fre12a]). It can be generalized in
these cases to s-representations and relates in this way the (twin) building to the isotropy representations of finite dimensional Riemannian symmetric spaces and affine Kac-Moody symmetric
spaces respectively. The existence of hyperbolic Kac–Moody symmetric spaces is conjectural at
this stage. We hope to take this up elsewhere.
7. Cartan involutions and the embedding of the twin building
Recall from section 2.4 that the Cartan involution is the automorphism ω0 : g −→ g determined
by ω0 (ei ) = −fi , ω0 (fi ) = −ei and ω0 (hi ) = −hi , and ω is the conjugate linear automorphism
defined as the composition of ω0 with complex conjugation. This gives
ω0 (t± ) = t∓

and

− ω(t± ) = t∓

∓
ω0 (t±
r ) = tr

and

∓
− ω(t±
r ) = tr

b∓
ω0 (bt±
r ) = tr

and

b∓
− ω(bt±
r ) = tr .

as well as
for each r ∈ R, so that
Proposition 7.1. The following diagram is commutative:
B


k

ω

Ψr
−ω

/B


Ψr

/k

Proof. By definition −ω|k = −Id. As before, we set B ± = (K/T × ∆± )/ ∼. The action of the
Cartan involution ω on B is defined by
∓
ω(kT, ∆±
J ) = (kT, ∆J ) .
22

±
Let (kT, ∆±
J ) ∈ B . Then we have
±
±
±
−ω(Ψr (kT, ∆±
J )) = −ω(Adk Ψr (1T, ∆J )) = −Adω(k) ω(Ψr (1T, ∆J )) = −Adk Ψr ((1T, ∆J )) .

On the other hand
∓
∓
±
Ψr (ω(kT, ∆±
J )) = Ψr (kT, ∆J ) = Adk Ψr ((1T, ∆J )) = −Adk Ψr ((1T, ∆J ))

where the last equality comes from Equation (13), finishing the proof.



Remark 7.2. The choice of −ω on k comes from the identification of k with the p-component
of the Cartan decomposition of g via the relation p = ik.
We note that while Ψr is a simplicial complex isomorphism, the apartment system of B has no
direct geometric interpretation in Ψr (B). Our embedding identifies ω–stable twin apartments
in B with Cartan subalgebras in k. The other apartments are hidden because Ψr is only K–
equivariant, but not G–equivariant.
8. Understanding of the Tits building in rank 2
In this section assume that g = g(A) is a rank 2 hyperbolic Kac-Moody Lie algebra, so that


2 −a
A=
, ab > 4.
−b 2
We impose the additional conditions that a > 1 and b > 1 because otherwise the real root groups
Ui defined below may not be abelian (see [Mor88, CMS15]).
Rank 2 hyperbolic Kac-Moody algebras were studied intensively by Lepowsky and Moody [LM79],
by Feingold [Fei80] for the “Fibonacci hyperbolic” (a = b = 3), and by Kang and Melville [KM95].
In these rank 2 cases the hyperboloids tr for r 6= 0 are hyperbolas, and there is no topological distinction between “one-sheeted” for r > 0 and “two-sheeted” for r < 0, as there is in
higher rank. In this section we will try to provide a detailed construction of the twin building,
B = B + ∪ B − , whose simplicial structure is a pair of trees related by a twin building involution,
and give the apartment system and the action of the complex group GC on each tree. We will
also give an explicit description of the smaller apartment system and the action of the compact
real form K, which will provide more details of the embedding of the building into the compact
real form in this rank 2 case. In the next section we show how this allows the embedding of a
spherical building at infinity.
√
The eigenvalues of A are λ± = 2 ± ab so λ+ > 0 and λ− < 0. This means that the signature of
the bilinear form determined by A on the split real Cartan subalgebra hR is (1, 1). By Equation
(6), the bilinear form on the compact real Cartan subalgebra t = ihR also has signature (1, 1).
The Weyl group of g is the infinite dihedral group
W = hw1 , w2 | w2 = 1 = w2 i ∼
= Z/2Z ∗ Z/2Z ∼
= Z o {±1}
1

2

which acts on h ⊂ g as well as on hR ⊂ gR and on t ⊂ k. Denote the infinite cyclic subgroup of
W by
W even = {(w2 w1 )m | m ∈ Z} ∼
= Z,
which has index 2 in W , so that for i = 1, 2,
W odd = {(w2 w1 )m wi | m ∈ Z}
23

is the other coset, and we have the relations
wi (w2 w1 )m wi−1 = (w2 w1 )−m

for m ∈ Z, i = 1, 2.

With t = w2 w1 and r = wi this gives the presentation of W as the infinite dihedral group
W = ht, r | r2 = 1, rtr−1 = t−1 i. It can be useful to index the elements of W by the integers so
that W even and W odd correspond to even and odd integers, respectively (after making a definite
choice for i, say i = 2:
(
(w2 w1 )m if n = 2m,
w(n) =
(w2 w1 )m w2 if n = 2m + 1.
Then we have w1 w(n) = w(−1 − n) and w2 w(n) = w(1 − n) for n ∈ Z, and
−1

w(n)

(
(w2 w1 )−m if n = 2m,
=
w2 (w2 w1 )−m if n = 2m + 1

(
w(−n) if n = 2m,
=
w(n) if n = 2m + 1.

It is straightforward to check that for any n, k ∈ Z, we have
(
w(n + k) if n = 2m,
w(n)w(k) =
w(n − k) if n = 2m + 1.
f by the integers in the same
We will also find it useful to similarly label certain elements of W
way:
(
(w
e2 w
e1 )m if n = 2m,
w(n)
e
=
(w
e2 w
e1 )m w
e2 if n = 2m + 1.
We define the non-standard partition of the real roots of g, Φre = Φ1 ∪ Φ2 where
Φ1 = W even α1 ∪ W odd α2 = W even {α1 , −α2 }
and
Φ2 = W even α2 ∪ W odd α1 = W even {−α1 , α2 } .
See Figure 1 as an illustration for the “Fibonacci” rank 2 hyperbolic root system, and see Figure
2 for the rank 2 hyperbolic root system coming from the Cartan matrix A given at the beginning
of Section 8, with a = 2 and b = 3, which has simple roots of different lengths. The real roots
are on the red hyperbolas, and the non-standard partition is according to whether a root is on
a left branch or a right branch.
24

w 1 w 2 α1

w 2 w 1 α2
10

5

w 1 α2

w 2 α1

Φ1
-6

α1

-4

Φ2

α2
2

-2

-α2

4

6

-α1

-w2 α1

-w1 α2
-5

-10

-w2 w1 α2

-w1 w2 α1

Figure 1: The Fibonacci root system and non-standard partition of real roots
Φre = Φ1 ∪ Φ2 where Φ1 = W even {α1 , −α2 } and Φ2 = W even {−α1 , α2 }.
For i = 1, 2 recall that
Lt,i = {x ∈ t | wi (x) = x} = {x ∈ t | αi (x) = 0}
are the lines in t fixed by the simple reflections wi , respectively. Then
Lt,1 = {t(az1 + 2z2 ) ∈ t | t ∈ R} and Lt,2 = {t(2z1 + bz2 ) ∈ t | t ∈ R}.
For i = 1, 2 the line Lt,i is the intersection of the family of Cartan subalgebras,
{exp(adsxi +tyi )t = ti (s, t) | s, t ∈ R}
parametrized by the 2-sphere with antipodes identified, the complex projective space P1 (C)
(Theorem 4.4).
25

6

w1 w2 α 1

4

w1 α 2

Φ
-6

w2 w1 α 2

2

α1

w2 α 1

1

-4

-2

-α2

α2

-w2 α1
-w2 w1 α2

Φ

2

2

4

6

-α1

-2

-w1 α2

-4

-w1 w2 α1

-6

Figure 2: The root system and non-standard partition of real roots for a rank 2 hyperbolic
with unequal root lengths, a = 2, b = 3.
We also have the corresponding statements for the split real form. For i = 1, 2 recall that in the
split real Cartan, hR , we have
LhR ,i = {x ∈ hR | wi (x) = x} = {x ∈ hR | αi (x) = 0}
are the lines in hR fixed by the simple reflections wi , respectively. Then
LhR ,1 = {t(ah1 + 2h2 ) ∈ hR | t ∈ R} and LhR ,2 = {t(2h1 + bh2 ) ∈ hR | t ∈ R}.
In Figures 1 and 2 those fixed lines are the inner green lines.
For i = 1, 2 the line LhR ,i is the intersection of the family of Cartan subalgebras,
{exp(adsei +tfi )hR = hiR (s, t) | s, t ∈ R}
parametrized by pairs of antipodal points on a 1-sheeted hyperboloid.
Let (G, B ± , N ) be a BN –pair for the complex Kac–Moody group G = GC (A). The standard
parabolic subgroups PJ± for J ( {1, 2} are
P∅± = B ± ,

P1± = B ± t B ± w1 B ± ,
26

and P2± = B ± t B ± w2 B ± .

As a simplicial complex, the twin building B = B + ∪ B − of (G, B ± , N ) is a pair of homogenous
P1 (C)-trees. The vertices of B are in bijection with the conjugates of P1± and P2± in G, while the
set of edges (chambers) are in bijection with conjugates of B ± . We identify the sets of vertices
with the disjoint union of cosets
V (B ± ) ∼
= G/P ± t G/P ± .
1

2

The set of edges is given by
E(B ± ) ∼
= G/B ± .
The group G acts by left multiplication on cosets. There are natural projections on cosets
induced by the inclusion of B ± in P1± and P2± :
πi : G/B ± −→ G/Pi± , i = 1, 2 .
If vi ∈ G/Pi± is a vertex, and StB (vi ) = πi−1 (vi ) is the set of edges with origin vi , then we may
index StB (vi ) by Pi± /B ± ⊆ G/B ± , i = 1, 2.
Apartments in B ± are infinite lines, but not every infinite line in the tree is an apartment.
The fundamental apartment A± in B ± is a union of chambers which are Weyl group translates
±
of the fundamental chamber C ± = (1B ± , ∆±
∅ ), which is fixed under the action of B , which
±
contains the root group U . The boundary vertices of that fundamental chamber are denoted
±
by vi± = (1B ± , ∆±
i ) for i = 1, 2, corresponding to the maximal parabolic subgroups Pi . By
definition, the vertex vi± = (1B ± , ∆±
i ) is stabilized by the generator wi of the Weyl group W .
The apartment system A± in B ± is the set of apartments obtained from fundamental apartment
A± by the action of G. The question is how precisely can we describe that apartment system.
In t, for each r < 0 the hyperbola tr has two connected components, t±
r , which are each tessellated
by the action of W into intervals separated by vertices which are the intersections of tr with the
W -images of the two lines Lt,i fixed by wi , i = 1, 2. The fundamental domain for the W action
±
±
on t±
r is C , which corresponds to the fundamental chamber of A , and the other chambers of
±
A correspond to:
{. . . , w1 w2 w1 C ± , w1 w2 C ± , w1 C ± , C ± , w2 C ± , w2 w1 C ± , w2 w1 w2 C ± , . . . }
so that adjacent chambers in this list are adjacent in the apartment, sharing a common vertex.
For example, the common vertex v1± = w1 C ± ∩C ± corresponds to the maximal parabolic subgroup
P1± , and v2± = C ± ∩ w2 C ± corresponds to the maximal parabolic subgroup P2± . It means that
this apartment is a line in the tree, infinite in both directions. Note that according to the
geometric pictures shown in Figures 1 and 2, this left-to-right ordering is natural in A+ , but
would be reversed in A− because the reflecting mirrors of w1 and w2 are crossed in the lower
lightcone. The order would not be reversed if we think of viewing the lower lightcone from the
origin, facing south. We use the integers n ∈ Z to label each chamber in A± , as we did above
to label elements of W ,
(
(w2 w1 )m C ± if n = 2m,
C ± (n) = w(n)C ± =
(w2 w1 )m w2 C ± if n = 2m + 1
and its boundary vertices
(
(w2 w1 )m vi± if n = 2m,
vi± (n) = w(n)vi± =
(w2 w1 )m w2 vi± if n = 2m + 1
27

but notice that each vertex has two labels: v1± (2m) = v1± (2m − 1) and v2± (2m) = v2± (2m + 1).
The chambers with boundary vertices may be written as
[v1± (2m), C ± (2m), v2± (2m)]

and

[v2± (2m + 1), C ± (2m + 1), v1± (2m + 1)]

where we read the order of vertex, chamber, vertex, left to right. When w(2k + 1) acts on
one of these, it will reverse the positions of the vertices. This can be seen using the formula
w(2k + 1)w(n) = w(2k + 1 − n). We use this notation to describe a (left or right) “ray” of
chambers starting with C ± (n) and including all boundary vertices:
±
L±
ray (n) = {C (m) | m ≤ n}

±
and Rray
(n) = {C ± (m) | m ≥ n}.

Note that the action of the Weyl group generators on the chambers of A± is given by the simple
formulas
w1 C ± (n) = C ± (−1 − n) and w2 C ± (n) = C ± (1 − n)
for any n ∈ Z, so that (w2 w1 )m C ± (n) = C ± (n + 2m) for any m, n ∈ Z, and the action on rays is
±
w1 L ±
ray (n) = Rray (−1 − n)

±
and w2 L±
ray (n) = Rray (1 − n)

so for each m, n ∈ Z we have
±
(w2 w1 )m L±
ray (n) = Lray (n + 2m)

and

±
±
(w2 w1 )m Rray
(n) = Rray
(n + 2m).

We can describe the action of the real root groups U±(w1 w2 )m αi , i = 1, 2, m ∈ Z, on the
fundamental apartments A± .
Using the notation î = 3 − i, we label the roots in Φi by the integers as follows:
(
(
(w2 w1 )m αi if n = 2m
w(n)αi if n = 2m
Φi (n) =
=
(w2 w1 )m w2 αî if n = 2m + 1
w(n)αî if n = 2m + 1
so that the labels of roots in both branches are the integers in order, negative to positive going
in Φ1 from top to bottom, but going in Φ2 from bottom to top:
(
n ≤ 0 if i = 1
+
Φi (n) ∈ Φ for
n ≥ 0 if i = 2.
We also have w1 Φ1 (n) = Φ2 (1 − n) and w2 Φ1 (n) = Φ2 (−1 − n).
Here is another useful application of the labeling of the real roots by Φi (n), n ∈ Z, as shown in
the paragraph above. A consistent choice of real root vectors given by
(
(
ew(n)αi if n = 2m
w(n)e
e
αi if n = 2m
eΦi (n) =
=
,
ew(n)αî if n = 2m + 1
w(n)e
e
αî if n = 2m + 1
and then we would have w
ej eα = ewj α for all real roots α and j = 1, 2. L±
ray (n) is fixed by U±Φ2 (n)
±
±
and Rray (n) is fixed by U±Φ1 (n) for all n ∈ Z, but U±Φ2 (n) Rray (n − 1) is a distinct family of
±
rays in B ± indexed by C whose intersection with L±
ray (n) is only the vertex v2 (n) if n is even,
±
v1± (n) if n is odd, and U±Φ1 (n) L±
ray (n + 1) is a distinct family of rays in B indexed by C whose
± (n) is only the vertex v ± (n) if n is even, v ± (n) if n is odd.
intersection with Rray
1
2
Proposition 8.1. We have the following results about the action of real root groups U±Φi (k) ,
k ∈ Z, on the chambers C ± (n), in the fundamental apartment A± .
28

±
(1) The chambers of L±
ray (k) = {C (n) | n ≤ k} are each fixed by U±Φ2 (k) but
± (k + 1) is a family of distinct rays in B ± indexed by C whose intersection
U±Φ2 (k) Rray
±
with L±
ray (k) is only the vertex v2 (k).
± (k) = {C ± (n) | n ≥ k} are each fixed by U
(2) The chambers of Rray
±Φ1 (k) but
±
±
U±Φ1 (k) Lray (k − 1) is a family of distinct rays in B indexed by C whose intersection
± (k) is only the vertex v ± (k).
with Rray
1

Proof. (1) We know that C ± (n) = w(n)C ± and that for real α ∈ Φ, Uα C ± = C ± when α ∈ Φ± .
So Uα C ± (n) = C ± (n) when w(n)−1 Uα w(n)C ± = C ± , that is, when Uw(n)−1 α C ± = C ± . But that
happens when w(n)−1 α ∈ Φ± . Recall that
(
(
w(−n)
if
n
=
2r,
w(n + k) if n = 2r,
w(n)−1 =
and
w(n)w(k) =
w(n) if n = 2r + 1
w(n − k) if n = 2r + 1
so for fixed k ∈ Z, any n ≤ k, and
(
±w(k)α2
α = ±Φ2 (k) =
±w(k)α1
we have
−1

w(n)

α = ±w(n)

if k = 2m
if k = 2m + 1

(
w(n)−1 w(k)α2 if k = 2m
Φ2 (k) = ±
w(n)−1 w(k)α1 if k = 2m + 1

= 2m
w(−n + k)α2 if n = 2r, k




= 2m + 1
w(−n + k)α1 if n = 2r, k
=±

k = 2m
w(n − k)α2 if n = 2r + 1,



k = 2m + 1
w(n − k)α1 if n = 2r + 1,

−1


w(−n)w(k)α2 if n = 2r, k




w(−n)w(k)α1 if n = 2r, k
=±

w(n)w(k)α2 if n = 2r + 1,



w(n)w(k)α1 if n = 2r + 1,

Φ2 (k − n)



Φ (k − n)
2
=±

Φ
1 (n − k)



Φ1 (n − k)
since

if
if
if
if

n = 2r, k = 2m
n = 2r, k = 2m + 1
n = 2r + 1, k = 2m
n = 2r + 1, k = 2m + 1

= 2m
= 2m + 1
k = 2m
k = 2m + 1

∈ ±Φ+ = Φ±

(
s ≤ 0 if i = 1
Φi (s) ∈ Φ for
s ≥ 0 if i = 2.
+

For α = ±Φ2 (k) and t ∈ C let g(t) = exp(adteα ) ∈ Uα and consider the family of rays
±
±
Uα Rray
(k + 1) = {g(t)Rray
(k + 1) | t ∈ C}

with chambers {g(t)C ± (n) | n ≥ k + 1, t ∈ C}. Two such rays are certainly distinct if their first
chambers are distinct, so suppose that g(t1 )C ± (k+1) = g(t2 )C ± (k+1), that is, g(t1 )w(k+1)C ± =
g(t2 )w(k+1)C ± . Let w = w(k+1) and g w = w−1 gw, so we have g(t1 )w C ± = g(t2 )w C ± which gives
C ± = (g(t1 )w )−1 g(t2 )w C ± = g(−t1 )w g(t2 )w C ± = g(−t1 + t2 )w C ± . Therefore, g(−t1 + t2 )w ∈ B ± .
But for any t ∈ C, we have g(t)w ∈ Uw−1 α . Since
(
w(−k − 1) if k is odd
w−1 = w(k + 1)−1 =
w(k + 1) if k is even
29

we get
w

−1

(
(
w(−k − 1)w(k)α1 if k is odd
w(−1)α1 if k is odd
α=±
=±
w(k + 1)w(k)α2 if k is even
w(1)α2 if k is even
(
(
w1 α1 if k is odd
−α1 if k is odd
=±
=±
w2 α2 if k is even
−α2 if k is even

so w−1 α ∈ −Φ± which means Uw−1 α ≤ U ∓ . Since g(−t1 + t2 )w ∈ B ± ∩ U ∓ = {1} we get
g(−t1 + t2 )w = 1 so g(−t1 + t2 ) = 1 so t1 = t2 .We have shown that this family consists of
distinct rays indexed by C.
The proof of (2) is similar.



9. Embedding the spherical building at infinity in rank 2
Let G = GC (A) be a rank 2 hyperbolic Kac–Moody group with compact real form K. Let
B = B + ∪ B − denote its twin building, whose simplicial structure must be a pair of trees. We
+ ∪ B − , by doing so for each tree B ± as
define the twin spherical building at infinity, B∞ = B∞
∞
follows. A ray in a tree is a sequence of connected vertices and edges (v1 , e1 , v2 , e2 , · · · ) with an
initial vertex, v1 , infinite in only one direction. Two rays are called equivalent if their intersection
is infinite, that is, they have a common tail. An equivalence class of rays is called an end of the
tree. A line in a tree is a sequence of connected vertices and edges infinite in both directions,
so it can be expressed as the union of two rays having a finite nonempty intersection, which can
always be taken to be exactly one vertex. So each line has two ends, the classes of those two
rays. For each apartment, X = Xray1 ∪ Xray2 in B ± there an apartment [X] = [Xray1 ] ∪ [Xray2 ]
± whose two chambers [X
in B∞
ray1 ] and [Xray2 ] uniquely determine the line X as follows.
Without loss of generality, we may assume that Xray1 ∩ Xray2 = {X0 } is exactly one vertex.
Suppose another apartment Y = Yray1 ∪ Yray2 with Yray1 ∩ Yray2 = {Y0 } also exactly one
vertex, has [Xray1 ] = [Yray1 ] and [Xray2 ] = [Yray2 ], so the intersections I1 = Xray1 ∩ Yray1 and
I2 = Xray2 ∩ Yray2 are both infinite rays. The intersection I1 ∩ I2 ⊆ Xray1 ∩ Xray2 is either one
vertex {X0 } or empty, and I1 ∩ I2 ⊆ Yray1 ∩ Yray2 is either one vertex {Y0 } or empty. So if I1 ∩ I2
is non-empty, then it is one vertex and that vertex is X0 = Y0 , so X = Y . Assume now that
I1 ∩ I2 is empty and write I1 = (v1 , e1 , v2 , e2 , · · · ) and I2 = (w1 , f1 , w2 , f2 , · · · ) so v1 is a vertex
in Xray1 ∩ Yray1 and w1 is a vertex in Xray2 ∩ Yray2 but the connected path in X from v1 to w1
goes through X0 while the connected path in Y from v1 to w1 goes through Y0 . If those two
paths were distinct, there would be a loop in the tree, so they must be identical, giving X = Y .
For any apartment X in B ± and for any vertex v in X, we may write X = X1 (v) ∪ X2 (v) where
Xi (v), i = 1, 2, are the rays starting from v and going towards the two ends of X, which have
been labeled 1 and 2. Let us suppress the ± for the moment to simplify the notation. Vertex v
determines a partition of the tree into two “rooted” trees, T reei (v), i = 1, 2, such that
T ree1 (v) ∩ T ree2 (v) = {v},

Xi (v) ⊂ T reei (v), i = 1, 2.

There can be no other vertices or edges in common between these two trees, since otherwise there
would be a loop passing through that intersection and v. Sometimes the phrase “Nile Delta”
is used to describe X1 (v) ∪ T ree2 (v) and X2 (v) ∪ T ree1 (v), since they look like a long river
30

splitting into tributaries at the point v. We may use this terminology later to help understand
the action of the group G on the tree.
Roughly speaking, a spherical building at infinity of a tree B ± consists of a ‘sufficiently large’
set of ends. The minimal spherical building at infinity may be constructed as follows. For
each sign ± let A± be the fundamental apartment of B ± with fundamental chamber C ± , and
let A± = {g · A± | g ∈ G} be the set of all G–translates of A± . Then A± is an apartment
± is an end of an apartment in A± , and Cham± denotes
system (see [AB08]). A chamber in B∞
∞
± . Of course, this building is highly degenerate. As a chamber
the set of all chambers of B∞
complex it consists of a set of points, and its apartments are exactly subsets of two distinct
points corresponding to the two ends of a line. It is straightforward to show that G acts doubly
transitively on Cham+
∞ . (See Lemma 9.1.) So in this rank 2 case, using this doubly transitive
action, the existence of a spherical BN –pair with Weyl group W∞ = {±1} ∼
= Z/2Z follows, thus
relating the spherical building at infinity to the structure of G. We denote all objects associated
to the spherical building at infinity, i.e. apartments, chambers or the Weyl group, with the
subscript ‘∞’. Our construction of the spherical BN-pair follows the approach used in [CG03]
to define spherical BN-pairs for rank 2 hyperbolic Kac-Moody groups over finite fields, which
are then used to construct lattices in trees.
± is doubly transitive.
Lemma 9.1. The action of G on B∞
±
±
Proof. For 1 ≤ i ≤ 2, let c±
i , di ∈ Cham∞ we want to show that there is g ∈ G such that
±
±
± there exist apartments A± and A± , such that c±
g · ci = di for i = 1, 2. By construction of B∞
c
i
d
±
±
and
d
are
ends
of
A
.
As
the
action
of
G
is
transitive
on
the
apartment
system
are ends of A±
c
i
d
±
±
±
of B ± , there is a group element g0 ∈ G such that g0 · A±
c = Ad . If g0 · ci = di we are done.
±
±
±
±
±
0
0 ±
Otherwise, g0 ·c±
1 = d2 and g0 ·c2 = d1 so we define g0 = w1 g0 and see that g0 ·c1 = w1 ·d2 = d1
±
±
±
0
0

and g0 · c2 = w1 · d1 = d2 so g = g0 = w1 g0 is the desired element of G.

In this section, for G of rank 2, we extend the embedding of the twin building B ± , given in
Theorem 6.1 to the spherical building at infinity.
In the rank 2 case, the split real form of the Cartan subalgebra, hR , and the compact real form
of the Cartan subalgebra, t, have a bilinear form with signature (1, 1). So for each 0 6= r ∈ R,
tr is a hyperbola with two connected components (branches) t±
r , and t0 = ∂Lt is a pair of lines
(the asymptotes). We have Lt = ∪r≤0 tr and the real roots of g are on the hyperbolas t(αi ,αi )
for i = 1, 2. For symmetric Cartan matrices, A, all real roots have the same length, so this is
just one hyperbola. All imaginary roots of g are in L0t = {x ∈ t | (x, x) < 0}, which is the
Tits cone. We define an equivalence relation on the nonzero vectors in t0 by saying that two
nonzero points, x and y, are equivalent when x = ry for some 0 < r ∈ R. There are exactly
four equivalence classes of such points, corresponding to the four rays in t0 , which we denote by
−
∞
{x+
i , xi | i = 1, 2}. This corresponds to the “lightlike closure” B (∂Lt ) in definition 4.3. To be
±
±
−
+
−
more precise, xi denotes the ray in t0 such that x2 = −x1 , x1 = −x+
2 and
(x+
1 , α1 ) < 0,

(x+
1 , α2 ) > 0,

(x+
2 , α1 ) > 0,

(x+
2 , α2 ) < 0.

+
− −
+
We will show that {x+
1 , x2 } corresponds to the fundamental apartment A and {x1 , x2 } cor−
responds to the fundamental apartment A . Define the halos, positive and negative, of g to be
31

the union of all K conjugates of {x±
i | i = 1, 2}
± −1
Ξ±
| k ∈ K, i = 1, 2}
∞ = {kxi k
−
and let the twin halo of g be Ξ∞ = Ξ+
∞ ∪ Ξ∞ . This will be where we embed the twin spherical
building at infinity B∞ in the next theorem.

Theorem 9.2. There is a K–equivariant bijective map Ψ∞ : B∞ → Ξ∞ such that the following
diagram commutes:
B∞


g∈K

/ B∞

Ψ∞

Ξ∞



Adg

Ψ∞

/ Ξ∞

Remark 9.3. Such a spherical building at infinity exists only in the highly degenerate case of
rank 2. The analogous construction for higher rank hyperbolic Kac-Moody groups would have to
involve a new kind of structure beyond the theory of buildings, perhaps replacing Coxeter groups
with some other class of groups.
Our strategy for the proof of this result is to follow the proof of Theorem 6.1, but since chambers
consist only of points, there are some simplifications. On the other hand, we need a more precise
characterization of the Borel subgroups stabilizing chambers in Cham±
∞ . Thus we first study
the existence of a spherical BN –pair with Weyl group W∞ .
Recall that A± denotes the fundamental apartment in B ± so it is a line in that tree with two
ends. Let e±
i,∞ , i = 1, 2, denote the two ends of that line, that is, the equivalence classes of
certain rays defined as follows. As shown in Proposition 8.1, for each i = 1, 2, m ∈ Z, the
subgroup U(w1 w2 )m αi fixes the end e±
i,∞ if we define
+
e+
1,∞ = [Rray (m)],

+
e+
2,∞ = [Lray (m)],

−
e−
1,∞ = [Lray (m)],

−
e−
2,∞ = [Rray (m)].

±
Furthermore, we have shown that U−(w1 w2 )m α1 fixes e±
2,∞ and U−(w1 w2 )m α2 fixes e1,∞ . This
±
−
+
−
+
means that ω(e1,∞ ) = e2,∞ and ω(e2,∞ ) = e1,∞ . We now define the stabilizer of ei,∞ and its
complement
±
±
Bi,∞
= {g ∈ G | g · e±
i,∞ = ei,∞ }

and

±
±
Q±
i,∞ = {g ∈ G | g · ei,∞ 6= ei,∞ } .

±
Then for any m ∈ Z, we have established that U(w1 w2 )m α1 and U−(w1 w2 )m α2 are in B1,∞
, and
±
that U(w1 w2 )m α2 and U−(w1 w2 )m α1 are in B2,∞ .

Recall the non-standard partition of the real roots of g, Φre = Φ1 ∪ Φ2 where
Φ1 = W even {α1 , −α2 }

and

Φ2 = W even {−α1 , α2 } .

Definition 9.4. For i = 1, 2, we set Ui = hUα | α ∈ Φi i.
We have just shown the following result.
±
Proposition 9.5. For i = 1, 2 we have Ui ≤ Bi,∞
.
32

±
±
Lemma 9.6. We have the decomposition Q±
i,∞ = Bi,∞ (−1)Bi,∞ or equivalently a Bruhat decomposition
±
±
±
G = Bi,∞
t Bi,∞
(−1)Bi,∞
.

(14)

±
±
Proof. The action of G on Bi,∞
is doubly transitive by lemma 9.1, and by definition Bi,∞
is the
±
stabilizer of ei,∞ . We continue the proof for the case i = 1, since the other case is similar.
±
±
±
±
We will show that for any pair of chambers c±
1 , c2 ∈ Cham∞ there is a g ∈ Q1,∞ mapping c1
±
±
±
±
onto the end e±
1,∞ of the fundamental apartment A and c2 onto the other end e2,∞ of A .
±
±
±
To do this, we chose a (uniquely defined) apartment A±
c ∈ B , whose ends are c1 and c2 . We
±
±
±
chose first b1 ∈ B1,∞ such that b1 c1 = e2,∞ . Then we apply the Weyl group element −1 ∈ W∞ ,
±
±
±
so (−1)b1 c±
1 = e1,∞ . Transitivity of B1,∞ on ends other than e1,∞ assures the existence of an
±
±
±
±
±
element b2 ∈ B1,∞ such that b2 (−1)b1 c2 = e2,∞ . As b2 ∈ B1,∞ , we have e±
1,∞ = b2 e1,∞ =
±
±
±
±
b2 (−1)b1 c±
1 . We now see that g = b2 (−1)b1 ∈ Q1,∞ because b2 (−1)b1 e1,∞ = b2 e2,∞ 6= e1,∞ since
the action of group elements is bijective. See Figure 3.


(-1)b1c2

(-1)b1Ac

e2,

A

e1,

c1

Ac

b1Ac

c2

b1c2

± with superscript ± suppressed.
Figure 3: Apartments and Ends in B∞

Theorem 9.7. For each i = 1, 2, a rank 2 hyperbolic Kac–Moody group G = GC (A) has a
±
spherical BN –pair at infinity with B = Bi,∞
, N = NG (TC ) and Weyl group W∞ = Z/2Z.
±
Proof. Note that H = N ∩ Bi,∞
= W even TC giving

(15)

W∞ = N/H = Z/2Z .

The theorem follows from Lemma 9.6. (See Theorem 11.96 and Corollary 11.98 in [AB08].) 
33

Recall our description of the simplicial complex B ± in Section 2.3, equation (3) in the general
case. Applying that description to the spherical BN -pair at infinity in Theorem 9.7, we have
±
±
B∞
= (G/Bi,∞
× ∆±
∞ )/ ∼

(16)

where ∆±
∞ is the fundamental chamber in the fundamental apartment and ∼ is the equivalence
relation which would only be non-trivial on simplices in the boundary of ∆±
∞ , so that relation
±
±
±
is trivial. The fundamental apartment of B∞ is the two-element set A∞ = {e±
1,∞ , e2,∞ }, corre±
±
sponding to the ends of the fundamental apartment A in B . Either of these two points could
± is the collection of twobe chosen as the fundamental chamber. The apartment system of B∞
element sets consisting of the ends of apartments in the apartment system A± = {gA± | g ∈ G}
± ±
of B ± . The action of −1 ∈ W∞ on each apartment A±
c = {c1 , c2 } interchanges those two
±
±
±
chambers, (−1) · c1 = c2 . So the set of all chambers of B∞ is the G orbit of any one chamber,
±
±
±
±
{ge±
1,∞ | g ∈ G}. Since G = Bi,∞ t Bi,∞ (−1)Bi,∞ , for each i = 1, 2, the chambers of B∞ are
±
±
±
±
±
±
±
e±
B1,∞
1,∞ ∪ B1,∞ (−1)B1,∞ e1,∞ = {e1,∞ } ∪ B1,∞ e2,∞ .

Another interpretation of this result is that the set of ends of apartments in B ± consists of one
end of fundamental apartment A± (the class of any right ray in A+ , any left ray in A− ), and the
images of the opposite end of A± under the stabilizer of the first end. To get all apartments in
± we apply the same reasoning to pairs, starting with the fundamental
the apartment system of B∞
±
±
±
apartment A∞ = [e1,∞ , e2,∞ ]. Its orbit under G consists of two parts,
±
±
±
±
±
±
±
±
±
[e±
B1,∞
1,∞ , e2,∞ ] = [B1,∞ e1,∞ , B1,∞ e2,∞ ] = [e1,∞ , B1,∞ e2,∞ ]

and
±
±
±
±
±
±
±
±
±
±
±
±
(−1)[e±
B1,∞
1,∞ , B1,∞ e2,∞ ] = B1,∞ [e2,∞ , B2,∞ e1,∞ ] = [B1,∞ e2,∞ , B1,∞ B2,∞ e1,∞ ].

NOTE: These statements are still under construction.
Recall the even subgroup of the Weyl group W has been defined to be
W even = {tm = (w1 w2 )m | m ∈ Z} ∼
=Z
so that W = W even ∪ W odd where W odd = w1 W even . The action of W even on an apartment
in B ± , {C(n) | n ∈ Z}, with chambers C(n), (a line in the tree) is by translation of chambers,
±
tm C(n) = C(n + 2m), leaving the two ends fixed. In particular we have W even ⊂ Bi,∞
for
i = 1, 2.
We define
±
BW
=

\

wB ± w−1 .

w∈W

As B ± consists of all elements in G fixing the fundamental chamber C ± ⊂ A± , its conjugate
±
wB ± w−1 fixes the chamber wC ± ∈ A± . Hence BW
can be characterized as follows:
±
BW
= {g ∈ G | g|A± = Id}.
±
±
In particular we have BW
⊂ Bi,∞
for i = 1, 2.
±
Recall the definition of U ± from section 4.3. As we defined BW
, we similarly define the group
\
±
UW
=
wU ± w−1
w∈W
34

and since B ± = TC U ± , we have the relation
±
±
BW
= TC UW
.

(17)

Remark 9.8. Each element w ∈ W has a length, the minimum number of simple reflections
in an expression for w. But that number is also the number of positive real roots sent by w to
Φre− . Since wU
e αw
e−1 = Uwα , as longer elements are applied, more positive root generators in
+
U are eliminated from the intersection (they are not in U + ), and in the complete intersection
all positive root generators are eliminated. But that means that what is left must correspond
to the imaginary root spaces, which can only be the result of complicated commutators of real
root generators. Hidden inside this intersection may be the structures giving the imaginary root
multiplicities.
Lemma 9.9. Each group Ui , i = 1, 2, is abelian and is normalized by W even , so Ui W even =
W even Ui . Furthermore, Ui W odd = W odd Uî .
Proof. Under our assumptions about the Cartan matrix of the rank 2 hyperbolic Kac-Moody
algebra g, it can be seen that for any α, β ∈ Φi we have α + β ∈
/ Φ ∪ {0} (see [Mor88, CMS15]).
Then we have the Lie algebra commutator [xα , xβ ] = 0 for any real root vectors xα ∈ gα and
xβ ∈ gβ , so [adxα , adxβ ] = 0. Since Uα = {exp(adsxα ) | s ∈ C} and Uβ = {exp(adtxβ ) | t ∈ C},
each element of Uα commutes with each element of Uβ .
To see that W even normalizes each group Ui , note that for any w ∈ W and for any α ∈ Φi , we
have w exp(adsxα ) w−1 = exp(adsw(xα ) ) ∈ Uw(α) ≤ Ui since W even Φi = Φi . On the other hand,
W odd Φi = Φî , so we get the final statement.

±
±
, so
and UW
Lemma 9.10. W normalizes BW
±
±
W
= BW
W BW

and

±
±
W.
= UW
W UW

Proof. Obvious.



±
±
Corollary 9.11. We have W even normalizes BW
and UW
, so
±
±
±
±
= BW
W even = B1,∞
∩ B2,∞
W even BW

and

±
±
= UW
W even .
W even UW

±
Proof. By definition BW
is the pointwise stabilizer of the standard apartment A± , while W even
±
contains the translations of the standard apartment. Hence the group BW
W even contains all
elements in G which preserve both ends of the standard apartments A± in B ± . Hence we have
±
±
±
±
W even BW
= BW
W even = B1,∞
∩ B2,∞
.


Proposition 9.12. We have U2 ∼
= {f : Z → C | supp(f ) < ∞} under pointwise addition.
Proof. We assume a choice of real root vectors {eα ∈ gα | α ∈ Φre } has been made as follows. We
begin by letting eαi = ei be the simple root generators of g, and then set e−αi = w
ei (ei ) = −fi .
Application of additional generators w
ej fills out the rest of the real root vectors, and in this way
we have arranged it so that we have w
ei (eα ) = ewi α for i = 1, 2 for all α ∈ Φre . Each element of
35

u ∈ U2 is a finite product of elements exp(adzα eα ) for α ∈ Φ2 , zα ∈ C, but we have a labeling
by the integers of Φ2 , so u can be written as
Y
u=
exp(adzn eΦ2 (n) )
n∈Z

where only a finite number of the zn are non-zero, and the order of the factors can be decreasing
since U2 is abelian. We associate u with the function fu : Z → C such that fu (n) = zn , so
supp(fu ) < ∞. Conversely, each function f : Z → C with supp(f ) < ∞ defines an element
Y
uf =
exp(adf (n)eΦ (n) ) ∈ U2 .
2

n∈Z

The pointwise addition of two such functions corresponds to the product in U2 , that is, for any
u, v ∈ U2 , fuv = fu + fv .

From now on we use the functional notation fu for u ∈ U2 , and introduce some new notations
for supp(fu ) in order to express its action on A± . Define M ax(fu ) = M ax{n ∈ Z | fu (n) 6= 0}
and M in(fu ) = M in{n ∈ Z | fu (n) 6= 0} so that
supp(fu ) = {M ax(fu ) = m1 > m2 > · · · > mr = M in(fu )}.
We call these indices the hinges of u and for each hinge mj , we call fu (mj ) the direction of that
hinge, so that this information determines the apartment uA± with end e±
2,∞ as a deformation
±
of A . For fu for u ∈ U2 , note that
±
±
L±
ray (m1 ) ⊂ Lray (m2 ) ⊂ · · · ⊂ Lray (mr )
±
and U±Φ2 (mj ) fixes L±
ray (mj ), so u fixes Lray (m1 ) and moves the segments of chambers (m1 , m2 ],
± (m + 1). So
(m2 , m3 ], · · · , (mr−1 , mr ], (mr , e1 ), whose union is the ray Rray
1
±
uA± = L±
ray (m1 ) ∪ u(m1 , m2 ] ∪ u(m2 , m3 ] ∪ · · · ∪ u(mr−1 , mr ] ∪ uRray (m1 + 1)

and at each hinge mj we have a direction fu (mj ) = zj ∈ C. Thus, uA± is the apartment
with ends e2 and ue1 . We have used the abbreviations (mj , mj+1 ] for a sequence of connected
chambers with labels corresponding to that interval of labels, and ei for e±
i,∞ . See Figure 4 for
an illustration.

ue1,
uA
...

z2
z3
e2,

zr
zr-1

z1
m1

m2

m3

A

36

mr-1 mr

e1,

Figure 4: Apartments A = A± and uA for u ∈ U2
with supp(fu ) = {m1 > m2 > · · · > mr } and fu (mj ) = zj .
We could have associated a function f : Z → C with finite support with an element of U1 or U2 ,
but that would have caused some confusion about the choice. We know that wi , i = 1, 2, each
switch the ends of A± , their action on the chambers is by
w1 C ± (n) = C ± (1 − n)

and w2 C ± (n) = C ± (−1 − n),

and for any n ∈ Z we have
w
e1 exp(adf (n)eΦ

2 (n)

)w
e1−1 = exp(adf (n)we1 eΦ

2 (n)

) = exp(adf (n)ew

1 Φ2 (n)

) = exp(adf (n)eΦ

1 (1−n)

)

so
w
e1 uf w
e1−1 =

Y

exp(adf (n)eΦ

1 (1−n)

) ∈ U1

n∈Z

e2 :
allows us to write elements of U1 as w
e1 uf w
e1−1 . There is a similar formula using w
Y
w
e2 uf w
e2−1 =
exp(adf (n)eΦ (−1−n) ) ∈ U1 .
1

n∈Z

We wish to use these descriptions of U1 and U2 , and their actions on the tree, to understand
what are all possible ends of apartments in B ± . The problem is that G = hU1 , U2 i so that
in principle, we have to consider all finite sequences of products U1 U2 U1 U2 · · · U1 ei to get all
possible ends. These divide into those coming from U1 e2 and those coming from U2 e1 . The
situation is symmetric so it would be enough to understand whether U1 U2 e1 reaches more ends
than U2 e1 . It seems unlikely that U1 U2 e1 = U2 e1 , that is, the ends that can be reached by the
action of U2 on e1 are all the ends of the building other than e2 . The ends {ue1 | u ∈ U2 } must
all be distinct, since otherwise there would be a loop in the tree, and none of them can be e2
since ue2 = e2 = ue1 would give e1 = e2 . Similarly, the ends {ue2 | u ∈ U1 } must all be distinct,
and none of them can be e1 . Thus it appears that the set of all ends of the building B ± , denoted
by Ends(B ± ), includes at least
±
{e±
2,∞ } ∪ {u2 e1,∞ | u2 ∈ U2 }

±
and {e±
1,∞ } ∪ {u1 e2,∞ | u1 ∈ U1 }.

±
±
Conjecture 9.13. Let A±
g = g · A for some g ∈ Bi,∞ . Then there exists some u ∈ Ui such
±
that u · A±
g =A .
+
+
Attempt to prove conjecture: We will do the case A+
g = g · A for some g ∈ B2,∞ since the other
cases are similar. We write more briefly A for A+ and we write its ends as ei = e+
i,∞ for i = 1, 2.
We wish to show there exists some u ∈ U2 such that uAg = A. This is trivially true with u = 1
when g = 1. Assume g 6= 1 and that the result has been established for all shorter elements
+
g = gi2 · · · gin ∈ B2,∞
. Using the correspondence between lines in the tree B + and pairs of ends
+ , we write A = [e , e ] and A = gA = [ge , ge ] = [e , ge ] because g ∈ B + fixes the end
in B∞
2 1
g
2
1
2
1
2,∞
e2 .
+
+
If gi1 ∈ U2 ⊂ B2,∞
then gi−1
g = gi2 · · · gin ∈ B2,∞
so by induction there is an element u2 ∈ U2
1
−1
such that u2 gi2 · · · gin A = A. Then u = u2 gi1 ∈ U2 satisfies uAg = A.
+
+
Now assume gi1 ∈ U1 ⊂ B1,∞
then gi−1
g = gi2 · · · gin ∈ U1 B2,∞
. The problem is that we have
1
too little information about gi2 · · · gin so we cannot apply the inductive hypothesis to it.

37

If we could prove the conjecture above, we would have the following two results:
±
Conjecture 9.14. For i = 1, 2 we have the decomposition Bi,∞
=
±
±
±
±
±
±
BW
Ui W even = BW
W even Ui = Ui W even BW
= Ui BW
W even = W even Ui BW
= W even BW
Ui .

Proof.

±
(1) We have to establish the stated decomposition of Bi,∞
into the product of the
±
±
even
subgroups BW = TC UW , Ui and W
.
±
Hence there is some bw ∈ BW
W even such that un . . . u1 g = bw. Thus we find the
−1
product expression g = u−1
1 . . . un bw giving the decomposition
±
±
Bi,∞
= Ui BW
W even .

(2) Using lemmas 9.9 and 9.10 yields the product decompositions
±
±
±
±
.
= W even Ui BW
W even = Ui W even BW
Bi,∞
= Ui B W
±
±
±
.
(3) To get the remaining three decompositions for Bi,∞
we have to show that BW
Ui = Ui BW
±
To do this use the notation of part 1. Assume g ∈ Bi,∞ has a decomposition g = ubw
±
and w ∈ W even . We start with the apartment A±
= g −1 ·A± .
such that u ∈ Ui , b ∈ BW
g −1
Clearly g · A±
= A± . The argument in step 1 shows that there is some u ∈ Ui such
g −1
±
such that
= A± . Hence there is some element wb ∈ W even BW
that ug −1 · A± = u · A±
g −1
−1

±
W even Ui . The use of lemmas 9.9 and 9.10 establishes
ug −1 = wb so g = b w−1 u ∈ BW
the other decompositions.


±
±
Ui .
For 1 ≤ i ≤ 2 we define Ui,∞
= UW

Conjecture 9.15. The hyperbolic Kac–Moody group G admits a spherical Iwasawa decomposition
±
.
G = KTRad Ui,∞
± . Let
Proof. We know that K acts transitively on chambers of B ± so it acts transitively on B∞
±
±
±
g ∈ G. This gives chambers ci = g · ei,∞ , i = 1, 2. Then by transitivity of K on B∞ we find
±
±
±
±
elements ki ∈ K such that ki ·c±
i = ei,∞ , so that (ki g)·ei,∞ = ei,∞ . Thus ki g ∈ Bi,∞ , establishing
±
G = K Bi,∞

(18)

±
±
±
±
From Bi,∞
= W even TC Ui,∞
= W even T TRad Ui,∞
we get K ∩ Bi,∞
T W even and in consequence a
decomposition

(19)

±
G = KTRad Ui,∞

±
±
such that K ∩ TRad = TRad ∩ Ui,∞
= K ∩ Ui,∞
=1


Remark 9.16. A spherical Iwasawa decomposition was established in Proposition 11.99 in [AB08]
for Euclidean buildings and follows from a spherical Bruhat decomposition. For our Kac–Moody
group G over C, we proved a spherical Bruhat decomposition (lemma 9.6). In rank 2, we may
use the fact that the hyperbolic and affine spherical buildings coincide. Hence we may apply
Proposition 11.99 in [AB08] to deduce the theorem.
38

±
±
Let Bi,K
= Bi,∞
∩ K. In our next step towards the proof of Theorem 9.2, we show that
+
−
+
−
B1,K = B1,K = B2,K = B2,K
= BK is independent of the choice of sign ± and of the choice of
±
i, and establish a K-equivariant bijection between G/Bi,∞
and K/BK .
±
±
Lemma 9.17. We have Bi,K
= T W even , so BK = Bi,K
is independent of the choice of sign ±
and of i = 1, 2.
±
±
Proof. By lemma ?? we have Bi,∞
= BW
Ui W even . We calculate the intersection of each com±
±
ponent with K. Recall that B ± ∩ K = T . As T ⊂ BW
⊂ B ± , we see that BW
∩ K = T.

For i = 1, 2, set (Ui )K = Ui ∩ K = {u ∈ Ui | ω(u) = u} where ω denotes the Cartan involution.
Since ω(gα ) = g−α we get ω ◦ (Uα ) ◦ (ω)−1 = U−α , and if 1 6= u = exp(adeα ) ∈ (Ui )K for a basis
vector eα ∈ gα then u = exp(ade−α ) for some basis vector e−α ∈ g−α . Then for any element
h ∈ h such that α(h) 6= 0, we get u(h) = h + [eα , h] = h − [e−α , h] so α(h)eα = α(h)e−α which
is impossible, so u = 1 and (Ui )K = {1}. We have W even ⊂ W ⊂ K.

± are in bijection with the elements of K/B , hence from
Corollary 9.18. The chambers of B∞
K
±
equation (16) we get B∞ = K/BK × {e±
}
for
either
choice
of
i
=
1,
2.
i,∞
± . Hence chambers
Proof. The group K acts transitively on chambers which are the points of of B∞
correspond to the quotient of K modulo the subgroup BK fixing the chamber e±
i,∞ for either
choice of i = 1, 2.

±
−
±
Lemma 9.19. For Ξ∞ = Ξ+
∞ ∪Ξ∞ the twin halo of g, each element the set {xi | 1 = 1, 2} ⊂ Ξ∞
is fixed by BK , which is the pointwise stabilizer of that set.

Proof. The action of the even Weyl group W even fixes the four rays of ∂Lt , so its induced
action on {x±
i | 1 = 1, 2} is trivial. The same is true of the Adjoint action of T = exp(t)
which fixes t pointwise, so this is true for BK , which is contained in the pointwise stabilizer
of this set. Now suppose that k ∈ K stabilizes the set pointwise. Then k sends two linearly
independent points on the two lines of ∂Lt to linearly independent points, a basis for t, so k
±
normalizes t. But NK (t) = T W , and the elements in the set T W odd exchange x±
1 and x2 , so
even
k ∈ TW
= BK .

We now have all technical ingredients to prove Theorem 9.2.
Proof of Theorem 9.2. We begin to define Ψ∞ by setting
+
Ψ∞ (e+
1,∞ ) = x1

and

−
Ψ∞ (e−
1,∞ ) = x1

and we extend this to all of B∞ by
±
−1
Ψ∞ (kBK , e±
1,∞ ) = kΨ∞ (e1,∞ )k

for any k ∈ K. It is clear that Ψ∞ is a well-defined K-equivariant bijection.



The results in this section suggest that on the Lie algebra level one could study a non-standard
Cartan decomposition
M
M
g=h⊕
gα
gα
α∈P1
39

α∈P2

based on a non-standard partition Φ = P1 ∪ P2 such that Φi ⊂ Pi for i = 1, 2, and such that
M
M
n1 =
gα
and
n2 =
gα
α∈P1

α∈P2

are subalgebras. Then bi = h ⊕ ni , i = 1, 2, would be non-standard Borel subalgebras related
±
to the Borel subalgebras Bi,∞
. That would be accomplished if each subset Pi is closed under
addition. Since Φ is a subset of the dual of the split real Cartan subalgebra, h∗R , one might
consider using a timelike line in the interior of the lightcone L0hR to partition all of Φ. If such a
line does not contain any roots, every root is on one side or the other, and Φ1 and Φ2 are on
opposite sides for any choice of the line. But if the line contains roots, one would have to decide
which ones go in which part of the partition, and it must be done in such a way that each Pi
is closed under addition. The solution would be to divide up the line into two rays from the
origin, and divide up the roots on the line according to which ray they are in.
There are two obvious partitions determined by the two lightcone lines themselves. For the line
im +
im −
determined by x+
2 the partition would be P1 = Φ1 ∪ (Φ ) and P2 = Φ2 ∪ (Φ ) , while for
+
the line determined by x1 the partition would be P1 = Φ1 ∪ (Φim )− and P2 = Φ2 ∪ (Φim )+ .
±
Associated to each of the four non-standard Borels at infinity, Bi,∞
, is a choice of one of these
four half-partitions.
Having found a non-standard Cartan decomposition as above, one gets a corresponding nonstandard decomposition of the universal enveloping algebra U(g) = U(n1 )U(h)U(n2 ) and can
construct induced Verma modules of two types, V ermai (λ) = U(ni )vλi , i = 1, 2, where vλi are
vectors such that h · vλi = λ(h)vλi for any h ∈ h, n1 · vλ2 = 0 and n2 · vλ1 = 0. The quotient of such
a Verma module by its maximal proper submodule would be an irreducible module, Irredi (λ)
generated by vλi . These would be integrable g-modules for λ = n1 λ1 + n2 λ2 in the weight lattice
of g (λ1 , λ2 are the fundamental weights of g such that λi (hj ) = δij ) but outside of the lightcone,
so that n1 and n2 have opposite signs. Examples of such integrable modules, in addition to the
adjoint representation, were mentioned by Borcherds in Section 6 of [Bor86]. There is evidence
that they occur in the decomposition of the rank 3 hyperbolic algebra, F, with respect to its
rank 2 hyperbolic subalgebras. That decomposition is currently under study [Pen16].

10. Conclusion and further directions
Our lightcone embedding of the twin building of a hyperbolic Kac–Moody group is motivated
by the conjectural existence of hyperbolic Kac–Moody symmetric spaces. There have been some
efforts recently to develop the geometry of hyperbolic Kac–Moody symmetric spaces, building
on work of the third author on the construction of affine Kac–Moody symmetric spaces [Fre09,
Fre11, Fre12b, Fre13c, Fre13b].
Recalling the well–known finite dimensional theory (see for example [Hel01, Ebe96]), the boundary of a symmetric space M of non–compact type IV corresponding to a complex simple Lie
group G, can be identified with the building over C associated to G. Via the isotropy representation at a point p ∈ M , the building can be embedded into the unit sphere of the tangent space
Tp M . In this way, points in the building get identified with directions in the tangent space of
the symmetric space. Via the duality between the compact type and non–compact types, we
40

can identify spaces of type IV with spaces of type II and thus also obtain an embedding of the
building into the tangent space of a compact symmetric space.
In the absence of hyperbolic Kac–Moody symmetric spaces, important properties of the local
geometry are captured via the embedding of the building into the Lie algebra. Our embedding
of the building into the Lie algebra gives local pictures of the tangent spaces of conjectural
hyperbolic Kac–Moody symmetric spaces of types II and IV . We note however, that since our
twin building embeds into the lightcone of the compact form of the Lie algebra, it captures only
the timelike directions in the tangent space.
Via Proposition 4.2, we have also obtained a hyperbolic analog of the notion of a polar representation by the group K on the K–conjugacy class of a Cartan subalgebra. Recall that a
group representation G : V −→ V on a vector space V is called polar if there exists a subspace
Σ ⊂ V , called a section, such that each orbit G · v for v ∈ V intersects Σ orthogonally. Finite
dimensional polar representations are orbit equivalent to isotropy representations of finite dimensonal Riemannian symmetric spaces [Dad85]. Similar observations were made in the affine
case [HPTT95, Gro00]. Our Proposition 4.2 shows that the action of K on H is a polar action
with section t.
Further questions about polar representations for hyperbolic Kac–Moody groups remain open
and the full differential geometry need to develop hyperbolic Kac–Moody symmetric spaces
remains elusive. We hope to take this up elsewhere.

References
[AB08]

Peter Abramenko and Kenneth Brown, Buildings, Graduate Texts in Mathematics, vol. 248,
Springer Verlag, New York, 2008.
[AR98]
Peter Abramenko and Mark Ronan, A characterization of twin buildings by twin apartments,
Geom. Dedicata 73 (1998), no. 1, 1–9. MR 1651854 (99m:51016)
[Ber85]
Stephen Berman, Real forms of universal Kac-Moody Lie algebras, Algebras Groups Geom. 2
(1985), no. 1, 10–25. MR 803743 (87a:17021)
[Bor86]
Richard E. Borcherds, Vertex algebras, kac-moody algebras, and the monster, Proc. Nat. Acad.
Sci. U.S.A. 83 (1986), 3068–3071.
[BVBPBMR95] Valérie Back-Valente, Nicole Bardy-Panse, Hechmi Ben Messaoud, and Guy Rousseau, Formes
presque-déployées des algèbres de Kac-Moody: classification et racines relatives, J. Algebra 171
(1995), no. 1, 43–96. MR 1314093 (96d:17022)
[Cap09]
Pierre-Emmanuel Caprace, “Abstract” homomorphisms of split Kac-Moody groups, Mem. Amer.
Math. Soc. 198 (2009), no. 924, xvi+84. MR 2499773 (2010d:20057)
[CG03]
Lisa Carbone and Howard Garland, Existence of lattices in Kac-Moody groups over finite fields,
Commun. Contemp. Math. 5 (2003), no. 5, 813–867. MR 2017720 (2004m:17031)
[Che55]
C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. (2) 7 (1955), 14–66. MR 0073602
(17,457c)
[CMS15]
Lisa Carbone, Scott H. Murray, and Sowmya Srinivasan, The geometry of rank 2 hyperbolic root
systems, preprint, 2015.
[Dad85]
Jiri Dadok, Polar coordinates induced by actions of compact lie groups, Trans. Amer. Math. Soc.
288 (1985), 125–137.
[DH09]
Thibault Damour and Christian Hillmann, Fermionic Kac-Moody billiards and supergravity, J.
High Energy Phys. (2009), no. 8, 100, 55. MR 2580182 (2011a:83112)
[DHN02]
T. Damour, M. Henneaux, and H. Nicolai, E10 and a small tension expansion of M theory, Phys.
Rev. Lett. 89 (2002), no. 22, 221601, 4. MR 1949931 (2003k:83130)
[DMGH09]
Tom De Medts, Ralf Gramlich, and Max Horn, Iwasawa decompositions of split Kac-Moody
groups, J. Lie Theory 19 (2009), no. 2, 311–337. MR 2572132 (2010j:20075)
41

[Ebe96]
[Fei80]
[FF83]
[FF16]
[Fre09]
[Fre11]
[Fre12a]
[Fre12b]
[Fre13a]
[Fre13b]
[Fre13c]
[GK81]
[Gro00]
[Hei06]
[Hel01]

[Hor09]
[HPTT95]

[Jul85]
[Kac90]
[KM95]
[KMW88]

[KP85]

[KP87]

[KPN12]
[Kra02]

Patrick B. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, Chicago, 1996.
Alex J. Feingold, A hyperbolic GCM Lie algebra and the Fibonacci numbers, Proc. Amer. Math.
Soc. 80 (1980), no. 3, 379–385. MR 580988 (81k:17009)
Alex J. Feingold and Igor B. Frenkel, A hyperbolic Kac-Moody algebra and the theory of Siegel
modular forms of genus 2., Math. Ann. 263 (1983), 87–144.
Alex Feingold and Walter Freyn, Spherical parametrizations of cartan subalgebras, in preparation,
2016.
Walter Freyn, Kac-Moody symmetric spaces and universal twin buildings, Ph.D. thesis, Universität Augsburg, 2009.
, Functional analytic methods for cities, submitted, 2011.
, Kac-Moody geometry, Global Differential Geometry, Springer, Heidelberg, 2012, pp. 55–
92.
, Kac-Moody groups, infinite dimensional differential geometry and cities, Asian journal
of mathematics 16 (2012), no. 4, 607–636.
, Orthogonal-Symmetric affine Kac-Moody-algebras, accepted for publication in Transactions of the AMS, 2013.
, Tame Fréchet structures for affine Kac-Moody groups, accepted for publication in Asian
Journal of mathematics, 2013.
, Tame submanifolds of co-Banach type, accepted for publication in Forum Mathematicum, 2013.
Ofer Gabber and Victor G. Kac, On defining relations of certain infinite-dimensional Lie algebras,
Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 185–189. MR 621889 (84b:17011)
Christian Groß, s-Representations for involutions on affine Kac-Moody algebras are polar,
manuscripta math. 103 (2000), 339–350.
Ernst Heintze, Towards symmetric spaces of affine Kac-Moody type, Int. J. Geom. Methods Mod.
Phys. 3 (2006), no. 5-6, 881–898. MR MR2264395 (2007g:17024)
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in
Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001, Corrected reprint
of the 1978 original. MR MR1834454 (2002b:53081)
Max Horn, Involutions of Kac-Moody groups, Ph.D. thesis, Technische Universitẗ Darmstadt,
2009.
Ernst Heintze, Richard S. Palais, Chuu-Lian Terng, and Gudlaugur Thorbergsson, Hyperpolar
actions on symmetric spaces, Conf. Proc. Lecture Notes Geom. Topology, IV, pp. 214–245, Int.
Press, Cambridge, MA, 1995. MR MR1358619 (96i:53052)
Bernard Julia, Kac-Moody symmetry of gravitation and supergravity theories, Lectures in Applied
Mathematics, vol. 21, American Mathematical Society, Providence, 1985.
Victor G. Kac, Infinite-dimensional Lie algebras, third ed., Cambridge University Press, Cambridge, 1990. MR MR1104219 (92k:17038)
Seok-Jin Kang and D.J. Melville, Rank 2 symmetric hyperbolic Kac-Moody algebras, Nagoya
Math. J. 140 (1995), 41–75.
V.G. Kac, R.V. Moody, and M. Wakimoto, On e10 ., Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI
Ser., Ser. C 250., 1988, pp. 109–128.
V. G. Kac and D. H. Peterson, Defining relations of certain infinite-dimensional groups,
Astérisque (1985), no. Numero Hors Serie, 165–208, The mathematical heritage of Élie Cartan
(Lyon, 1984). MR 837201 (87j:22027)
Victor G. Kac and Dale H. Peterson, On geometric invariant theory for infinite-dimensional
groups, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987,
pp. 109–142. MR 911137 (89b:22028)
Axel Kleinschmidt, Jakob Palmkvist, and Hermann Nicolai, Modular realizations of hyperbolic
weyl groups, Advances in Theoretical and Mathematical Physics 16 (2012), no. 1, 97–148.
Linus Kramer, Loop groups and twin buildings, Geometriae Dedicata 92 (2002), 145–178.
42

[Kum02]

[KW92]
[LM79]
[Mit88]
[Mor88]
[MP95]
[Pen16]
[PS86]

[RC89]

[Rém02]
[Ron89]
[Tic14]
[Tit86]
[Tit87]
[Wes01]

Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress
in Mathematics, vol. 204, Birkhäuser Boston Inc., Boston, MA, 2002. MR MR1923198
(2003k:22022)
V. G. Kac and S. P. Wang, On automorphisms of Kac-Moody algebras and groups, Adv. Math.
92 (1992), no. 2, 129–195. MR 1155464 (93f:17041)
James Lepowsky and Robert V. Moody, Hyperbolic Lie algebras and quasiregular cusps on Hilbert
modular surfaces, Math. Ann. 245 (1979), no. 1, 63–88. MR 552580 (81c:10030)
Stephen A. Mitchell, Quillen’s theorem on buildings and the loops on a symmetric space,
L’Enseignement Mathematique 34 (1988), 123–166.
Jun Morita, Root strings with three or four real roots in Kac-Moody root systems, Tohoku Math.
J. (2) 40 (1988), no. 4, 645–650. MR 972252 (89k:17044)
Robert V. Moody and Arturo Pianzola, Lie algebras with triangular decomposition, John Wiley
Sons, New York, 1995.
Diego Penta, The decomposition of the hyperbolic Kac-Moody algebra F with respect to its rank
2 Fibonacci subalgebra, Ph.D. thesis, Binghamton University, SUNY, 2016.
Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1986, Oxford Science Publications. MR MR900587
(88i:22049)
Enriqueta Rodriguez-Carrington, Kac-Moody groups: An analytic approach, ProQuest LLC, Ann
Arbor, MI, 1989, Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick.
MR 2638155
Betrand Rémy, Groupes de Kac-Moody déployés et presque déployés, Société mathematiques de
France, Paris, 2002.
Mark Ronan, Lectures on buildings, Perspectives in Mathematics, vol. 7, Academic Press Inc,
San Diego, 1989.
Alexander Tichai, Some lightcone embeddings of Lorentzian Kac–Moody algebras, Masters thesis,
TU Darmstadt, 2014.
Jacques Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg.
Sér. A 38 (1986), 367–387 (1987). MR 885545 (88j:20041)
, Uniqueness and presentation of Kac-Moody groups over fields, Journal of Algebra 105
(1987), 542–573.
Peter West, e11 and m-theory, Classical and Quantum Gravity 18 (2001), 4443–4460.

Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA
E-mail address: lisajcarbone@gmail.com
Dept. of Math. Sci., The State University of New York, Binghamton, New York 13902-6000
E-mail address: alex@math.binghamton.edu
Formerly at Fachbereich Mathematik, Technical University of Darmstadt, Darmstadt, Germany.
E-mail address: walter.freyn@googlemail.com

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