A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group
Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra 𝖌 = 𝖌(A) and (adjoint) Kac-Moody group 𝐺 = 𝐺(A) = ⟨exp(ad(teᵢ)),exp(ad(tfᵢ)) | t ∈ ℂ⟩ where ei and fi are the simple root vectors. Let (B⁺, B⁻, N) be the twin BN-pair naturally associated to 𝐺 and let (𝓑⁺, 𝓑⁻) be th...
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| author_facet | Carbone, Lisa Feingold, Alex J. Freyn, Walter |
| citation_txt | A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group. Lisa Carbone, Alex J. Feingold and Walter Freyn. SIGMA 16 (2020), 045, 47 pages |
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| description | Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra 𝖌 = 𝖌(A) and (adjoint) Kac-Moody group 𝐺 = 𝐺(A) = ⟨exp(ad(teᵢ)),exp(ad(tfᵢ)) | t ∈ ℂ⟩ where ei and fi are the simple root vectors. Let (B⁺, B⁻, N) be the twin BN-pair naturally associated to 𝐺 and let (𝓑⁺, 𝓑⁻) be the corresponding twin building with Weyl group W and natural 𝐺-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building (𝓑⁺, 𝓑⁻) of 𝐺 and the Kac-Moody algebra 𝖌 = 𝖌(A) in a new geometrical way. The Cartan-Chevalley involution, ω, of 𝖌 has a fixed point real subalgebra, 𝔨, the 'compact' (unitary) real form of 𝖌, and 𝔨 contains the compact Cartan t = 𝔨 ∩ h. We show that a real bilinear form (⋅,⋅) is Lorentzian with signatures (1,∞) on 𝔨, and (1, n−1) on t. We define {k ∈ 𝔨 | (k,k) ≤0 } to be the lightcone of 𝔨, and similarly for t. Let K be the compact (unitary) real form of 𝐺, that is, the fixed point subgroup of the lifting of ω to 𝐺. We construct a K-equivariant embedding of the twin building of 𝐺 into the lightcone of the compact real form 𝔨 of 𝖌. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an SU(2)-orbit of chambers stabilized by U(1), which is thus parametrized by a Riemann sphere SU(2)/U(1) ≅ S². For n = 2, the twin building is a twin tree. In this case, we construct our embedding explicitly, and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 045, 47 pages
A Lightcone Embedding of the Twin Building
of a Hyperbolic Kac–Moody Group
Lisa CARBONE †, Alex J. FEINGOLD ‡ and Walter FREYN §
† Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854, USA
E-mail: lisa.carbone@rutgers.edu
URL: https://sites.math.rutgers.edu/~carbonel/
‡ Department of Mathematical Sciences, The State University of New York,
Binghamton, New York 13902-6000, USA
E-mail: alex@math.binghamton.edu
URL: http://people.math.binghamton.edu/alex/
§ Fachbereich Mathematik, Technical University of Darmstadt, Darmstadt, Germany
E-mail: walter.freyn@googlemail.com
Received July 23, 2019, in final form May 11, 2020; Published online May 29, 2020
https://doi.org/10.3842/SIGMA.2020.045
Abstract. Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac–
Moody algebra g = g(A) and (adjoint) Kac–Moody group G = G(A) = 〈exp(ad(tei)),
exp(ad(tfi)) | t ∈ C〉 where ei and fi are the simple root vectors. Let
(
B+, B−, N
)
be the
twin BN -pair naturally associated to G and let
(
B+,B−
)
be the corresponding twin building
with Weyl group W and natural G-action, which respects the usual W -valued distance and
codistance functions. This work connects the twin building
(
B+,B−
)
of G and the Kac–
Moody algebra g = g(A) in a new geometrical way. The Cartan–Chevalley involution, ω,
of g has fixed point real subalgebra, k, the ‘compact’ (unitary) real form of g, and k contains
the compact Cartan t = k ∩ h. We show that a real bilinear form (·, ·) is Lorentzian with
signatures (1,∞) on k, and (1, n− 1) on t. We define {k ∈ k | (k, k) ≤ 0} to be the lightcone
of k, and similarly for t. Let K be the compact (unitary) real form of G, that is, the fixed
point subgroup of the lifting of ω to G. We construct a K-equivariant embedding of the
twin building of G into the lightcone of the compact real form k of g. Our embedding gives
a geometric model of part of the twin building, where each half consists of infinitely many
copies of a W -tessellated hyperbolic space glued together along hyperplanes of the faces.
Locally, at each such face, we find an SU(2)-orbit of chambers stabilized by U(1) which is
thus parametrized by a Riemann sphere SU(2)/U(1) ∼= S2. For n = 2 the twin building is
a twin tree. In this case, we construct our embedding explicitly and we describe the action of
the real root groups on the fundamental twin apartment. We also construct a spherical twin
building at infinity, and construct an embedding of it into the set of rays on the boundary
of the lightcone.
Key words: Kac–Moody Lie algebra; Kac–Moody group; twin Tits building
2020 Mathematics Subject Classification: 20G44; 20E42; 20F05; 51E24
1 Introduction
Buildings were introduced by J. Tits in the 1950’s to provide a ‘geometric’ interpretation of
simple algebraic groups and finite groups of Lie type. Following ideas originally developed by
F. Klein in his Erlangen program, Tits aimed at understanding a large class of groups, including
simple algebraic groups and finite groups of Lie type, as the automorphism groups of carefully
constructed geometric objects he called ‘buildings’. It turns out that the buildings of Tits
mailto:lisa.carbone@rutgers.edu
https://sites.math.rutgers.edu/~carbonel/
mailto:alex@math.binghamton.edu
http://people.math.binghamton.edu/alex/
mailto:walter.freyn@googlemail.com
https://doi.org/10.3842/SIGMA.2020.045
2 L. Carbone, A.J. Feingold and W. Freyn
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 [9].
In general, the building B of a group G with a BN -pair is an abstract simplicial complex
constructed from group theoretical data. The 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 incidence 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 nat-
urally 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 group.
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 [17,
18, 36, 40, 45] but certainly of interest to physicists [11, 12, 13, 14, 34, 59].
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 [26]. 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 Kac–Moody groups G over C, the associated Tits building is not
only an abstract simplicial complex admitting an action ofG, but that it admits a natural embed-
ding 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 [47]. 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 S1, can
be embedded into the unit circle in the tangent space of H2 = SL2(R)/SO(2) which is R2.
Kramer gave a topological construction of the complex twin building of type A
(1)
n and an
equivariant embedding of this building into the associated affine Kac–Moody algebra g [43].
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 [19, 20, 22, 23, 25]. A ‘twin
city’ is the natural completion of a twin building, so affine twin cities correspond to completions
of affine Kac–Moody groups 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.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 3
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 [24, 32]), 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 [19, 20, 21, 22] 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 [11, 34, 59].
The ideas in this paper may be extended to the wider class of Lorentzian Kac–Moody algebras,
but it remains to be seen how much the results are affected by the differences in the geometry
of the lightcone and the Tits cone. Some work in this direction was started by A. Tichai [55].
1.1 Summary of results
The following paragraph summarizes the main results of this paper. For G of hyperbolic type,
we construct a K-equivariant embedding of the twin building of G into the lightcone of the
compact real form k. In t, W acts on rays in the interior of the lightcone (and on certain rays
on the nullcone), tessellating a copy of hyperbolic space Hn−1 in each half of the lightcone (and
some limit points in its boundary), giving a twin apartment. The W -images of the fundamental
domain form the chambers of that apartment, so each face of any chamber is in a hyperplane
fixed by a W -conjugate of a simple reflection wi. The associated conjugate of the compact
subgroup SU(2)i fixes that hyperplane pointwise, but rotates the rest of t into a K-conjugate
of t, another apartment sharing that fixed chamber wall. This gives our geometric model of part
of the twin building where each half is infinitely many copies of a tessellated hyperbolic space
glued together along hyperplanes of the faces. Locally, at each such face, a family of chambers
meet, the orbit of an SU(2) with a U(1) stabilizer, so the family is indexed by a Riemann sphere,
SU(2)/U(1) ∼= S2.
In rank 2 the Weyl group is the infinite dihedral group, D∞, each apartment is a line, and each
building B± is a tree. We model each apartment as a copy of the real line tessellated into unit
intervals (chambers) C(n) =
[
n− 1
2 , n+ 1
2
]
for n ∈ Z, so the vertices are Z + 1
2 . At each vertex
a family of intervals (chambers) is attached, each in a line (apartment) which is tessellated, and
in each of those lines chambers are attached, on ad infinitum. We describe the action of the
real root groups on the fundamental twin apartment. The family of chambers attached at any
vertex is the projective space P1(F), where F is the field over which the group is defined, so for
F = C, we have P1(C) = Ĉ is the Riemann sphere. Thus, in rank 2 each building B± is a Ĉ-tree.
In that case, we also find a spherical twin building at infinity, and construct an embedding of it
into the set of rays on the boundary of the lightcone.
In rank 3 the Weyl group is a hyperbolic triangle group, each apartment is a copy of the
Poincaré disk, P, tessellated into hyperbolic triangles by W . The boundary of each triangle
is a segment in a hyperbolic geodesic. Along each geodesic segment we have a Ĉ-family of
attached triangles, each in a copy of P, which is tessellated and has attached disks along each
geodesic, on ad infinitum. The geometrical embedding we found is only K-equivariant, so from
a fundamental apartment we get only to those apartments all of whose chambers are in some
K-conjugate, ktk−1. It means that in such apartments, pairs of triangles attached to a geodesic
4 L. Carbone, A.J. Feingold and W. Freyn
edge are ‘balanced’ opposite each other, corresponding to antipodal points in Ĉ. So under the
action of K, the fundamental P can be rigidly rotated along any geodesic into another one,
having only that geodesic line in common. But the full Kac–Moody group G can leave a half-
apartment fixed and rotate the other half up, creating a ‘hinge’. The complete apartment system
would then be ‘hinged’ copies of P made up of pieces glued along geodesics.
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 = (diaij) 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.
– Strictly hyperbolic type: A is hyperbolic type, but every proper, indecomposable submatrix
is of finite type.
The terminology “hyperbolic” goes back to the original (independent) papers of Kac and
Moody, and comes from the geometry of the root systems for the corresponding Kac–Moody Lie
algebras. The Weyl group orbits of roots lie on hyperbolas in the rank 2 hyperbolic case and
on hyperboloids in higher rank cases. We have included in Section 7 two figures illustrating the
rank 2 root systems which show this hyperbolic geometry clearly. The definitions of roots, root
systems and Weyl groups are below.
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 [3]). 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∗ × h→ F
given by 〈α, h〉 = 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 [27, 35]:
– [hi, hj ] = 0,
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 5
– [h, ei] = 〈αi, h〉ei, h ∈ h and [h, fi] = −〈αi, h〉fi, h ∈ h,
– [ei, fj ] = δijhi,
– (adei)
1−aij (ej) = 0, i 6= j and (adfi)
1−aij (fj) = 0, i 6= j,
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 [35,
Section II], and which extends the form on h given by 2(hi, hj)/(hj , hj) = αi(hj) = aij . The
nondegeneracy of the pairing 〈·, ·〉 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 = (diaij) = (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
Z-module. Every α ∈ Φ can be written uniquely as α =
∑̀
i=1
kiαi 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. (2.1)
The set S = {wi | 1 ≤ i ≤ `} generates 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 ] = adei1adei2 · · · adein−1
ein is in the root space gα for α =
n∑
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 [35, Theorem 1.2]
g = h⊕
⊕
α∈Φ
gα = h⊕ g+ ⊕ g−,
where
g+ =
⊕
α∈Φ+
gα, g− =
⊕
α∈Φ−
gα.
The standard positive Borel subalgebra b ≡ b+ is defined by b+ = h⊕ g+ and the standard
negative Borel subalgebra by b− = h⊕ g−.
6 L. Carbone, A.J. Feingold and W. Freyn
2.2 Kac–Moody groups
There are various ways to define abstract Kac–Moody groups (see for example [37, 46, 52,
58]). 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
PSL(2,C), and that is what we get by using the definition of the adjoint Kac–Moody group
given in equation (2.2) for the Cartan matrix A = [2] of type A1.
By the definition of the abstract Kac–Moody group, there is a surjective group homomor-
phism: Ad : G −→ Gad from an abstract Kac–Moody group onto the adjoint Kac–Moody group
whose kernel is exactly the center of G (see [52, Proposition 9.6.2]). As we will see in Section 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 [6, 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 [37, 44, 46, 48].
Let g be a symmetrizable Kac–Moody algebra over C, L be a complex vector space and let
φ : g→ End(L) be any integrable representation, so that all φ(ei) and φ(fi) are locally nilpotent
on L and the linear operators
χφαi(t) = exp(φ(tei)) and χφ−αi(t) = exp(φ(tfi)), for t ∈ C,
are well-defined in GL(L). In fact, for any x ∈ gα, α ∈ Φre the operator φ(x) is locally nilpotent
on L, so χφx = exp(φ(x)) is well-defined and these give the real root groups Uφα . In particular,
the adjoint representation ad: g → End(g), is integrable, and for all x ∈ gα, α ∈ Φre we have
well-defined operators χad
x = exp(adx) ∈ GL(g) giving the real root groups Uad
α .
Definition 2.1 (minimal Kac–Moody groups). Let the minimal Kac–Moody group associated
to an integrable representation φ be the group generated by these operators,
Gφ = Gφ(C) =
〈
χφαi(t), χ
φ
−αi(t) | i ∈ I, t ∈ C
〉
≤ GL(L).
In particular, this defines the minimal adjoint Kac–Moody group
Gad = Gad(C) =
〈
χad
αi (t), χ
ad
−αi(t) | i ∈ I, t ∈ C
〉
≤ GL(g). (2.2)
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 φ(h) ∈ End(L) for h ∈ h are semisimple so they can also be exponentiated to
give a commutative group of operators T φC = T φC (G) =
{
χφh(t) = exp(φ(th)) |h ∈ h, t ∈ C
}
≤ Gφ
which is called the standard maximal torus of Gφ. We also define the standard Borel subgroups
(Bφ)± = T φC
〈
Uφα |α ∈ (Φre)±
〉
and the normalizer of T φC denoted by Nφ
C .
It is well known that the operators
w̃ad
i = exp(adei) exp(ad−fi) exp(adei) = exp(ad−fi) exp(adei) exp(ad−fi), 1 ≤ i ≤ `,
in Gad, generate a subgroup W̃ ad in Gad such that the restriction of w̃ad
i to the standard Cartan
subalgebra h equals the simple Weyl group reflection wi and w̃ad
i (ei) = −fi. It means that W
is a homomorphic image of W̃ ad. Note that W̃ ad is a subgroup of Nad
C and that Nad
C /T ad
C
∼= W .
In Theorem A.1 we prove a formula in any integrable representation φ for
w̃φi = exp(φ(ei)) exp(φ(−fi) exp(φ(ei)) = exp(φ(π(ei − fi)/2)).
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 7
2.3 Twin BN -pair and twin Tits building of a minimal Kac–Moody group
Our references for this section are [1, Sections 6.2 and 6.3] and [53, Chapters 5 and 11].
Definition 2.2 (BN-pair). A group G is said to have a BN -pair if G has subgroups B and N
such that
T1: G = 〈B,N〉, T = B ∩N �N , W = N/T is generated by a set S.
T2: For s ∈ S and w ∈W we have sBw ⊆ BswB ∪BwB .
T3: For s ∈ S we have sBs−1 6⊆ B .
The group W is called the Weyl group of the BN -pair, and (G,B,N, S) is called a Tits system.
Furthermore, (W,S) is a Coxeter system and there is a length function ` : W → N.
Definition 2.3 (twin-BN-pair). A group G is said to have a twin BN -pair with Weyl group W
if G has subgroups B+, B− and N such that
TW1:
(
G,B±, N, S
)
is a Tits system.
TW2: If `(sw) < `(w) for s ∈ S and w ∈W , then B±sB±wB∓ = B±swB∓.
TW3: B+s ∩B− = ∅.
In this case, (G,B+, B−, N, S) is called a twin Tits system.
For a hyperbolic adjoint Kac–Moody group G = Gad
C (A) we have standard Borel subgroups,
B± = (Bad)±, the standard maximal torus, T = T ad
C , and its normalizer in G, N = Nad
C . Thus
the group T = N ∩ B± is a normal subgroup of N . The group W = NG(T )/T generated
by S =
{
w̃ad
i | 1 ≤ i ≤ `
}
, is isomorphic to our earlier definition in Section 2.1 of the Weyl
group W generated by simple reflections as a group of orthogonal transformations of h∗ given
by formula (2.1). Thus, we have a twin BN -pair or twin Tits system, for the hyperbolic adjoint
Kac–Moody group G.
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:
G =
∐
w∈W
B±wB∓ =
∐
w∈W
B±wB±.
In these double cosets, w ∈ W = N/T is a coset nT for n ∈ N , but for any two representatives
of the same coset, w = nT = n′T , we have
B±nB± = B±nTB± = B±n′TB± = B±n′B±,
so we can label a double coset by w ∈W .
We can use the Bruhat decomposition to define a W -valued distance function on G/B±,
δ± : G/B± ×G/B± →W
by δ±
(
g1B
±, g2B
±) = w when g−1
1 g2 ∈ B±wB±. Similarly, we can use the Birkhoff decomposi-
tion to define a W -valued codistance function
δ∗ : G/B± ×G/B∓ →W
by δ∗
(
g1B
±, g2B
∓) = w when g−1
1 g2 ∈ B±wB∓.
8 L. Carbone, A.J. Feingold and W. Freyn
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 = 〈wj | j ∈ J〉 of W and the corresponding subgroups of G,
P±J =
∐
w∈WJ
B±wB±.
Note that P±I = G is not parabolic since it is not proper, P±∅ = B±, and we write P±i = P±{i}. For
J ( I we call P±J 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 ′± such that P± ( P ′±. For each i ∈ I, P±[i] = P±I\{i} is a maximal standard
parabolic subgroup, so there are 2` conjugacy classes of maximal parabolic subgroups, ` positive
and ` negative.
Definition 2.4 (Tits building). A Tits building of type (W,S) consists of a simplicial complex B
together with a collection A of subcomplexes, each of which is called an apartment, such that
1) each apartment is a Coxeter complex for the Coxeter system (W,S),
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 A′ there is an isomorphism ϕ : A → A′, fixing the intersection
A ∩A′.
A Coxeter complex for (W,S) is a simplicial complex on which there is a simply transitive
action of a Coxeter group W on the simplices of maximal dimension ` − 1 (chambers). The
simplices of dimension ` − 2 are called panels, and are each labeled by a generator, s ∈ S. We
say that two chambers C1 and C2 are s-adjacent when sC1 = C2, which means their intersection
is an s-panel. Each element w ∈W is a product of generators from S, so we define the length |w|
to be the minimal number of generators in an expression for w. Suppose Ci for 0 ≤ i ≤ d is
a sequence of chambers such that Ci−1 and Ci are si-adjacent for 1 ≤ i ≤ d, so that Ci = riC0 for
ri = s1 · · · si. Then we define the W -valued distance function δ(C0, Cd) = rd = s1 · · · sd, so for
any w1, w2 ∈ W and any chamber C, we have δ(w1C,w2C) = w−1
1 w2. Since any two chambers
in the building B are in a common apartment, we have defined the W -valued distance function
δ : C × C → W where C = C(B) is the set of all chambers of B. One may choose a pair (A,C)
consisting of an apartment A and a chamber C in A, which we call fundamental, so that the
chambers of A are uniquely labeled by the elements of W .
From [1, Proposition 4.84], we have the following properties of δ. For any chambers C,C ′, D ∈
C(B) we have
1. δ(C,D) = 1 iff C = D.
2. δ(D,C) = δ(C,D)−1.
3. If δ(C ′, C) = s ∈ S and δ(C,D) = w ∈ W , then δ(C ′, D) = sw or δ(C ′, D) = w. If, in
addition, |sw| = |w|+ 1 then δ(C ′, D) = sw.
4. If δ(C,D) = w then for any s ∈ S, there exists a chamber C ′ such that δ(C ′, C) = s and
δ(C ′, D) = sw. If |sw| = |w| − 1 then there exists a unique such C ′.
From [1, Definition 5.133], we have the following definition of a twin building.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 9
Definition 2.5 (twin building). A twin building of type (W,S) is a triple (B+,B−, δ∗) consisting
of two buildings (B+, δ+) and (B−, δ−), each of type (W,S) and each with its own W -valued
distance function, along with a codistance function
δ∗ :
(
C+ × C−
)
∪
(
C− × C+
)
→W
where C± is the set of chambers of B±, satisfying the following conditions for each ε ∈ {+,−},
any C ∈ Cε, and any D ∈ C−ε, where w = δ∗(C,D):
(Tw1) δ∗(C,D) = δ∗(D,C)−1.
(Tw2) If C ′ ∈ Cε satisfies δε(C ′, C) = s ∈ S and |sw| < |w| then δ∗(C ′, D) = sw.
(Tw3) For any s ∈ S there exists a chamber C ′ ∈ Cε with δε(C ′, C) = s and δ∗(C ′, D) = sw.
Let us state Lemma 5.139 from [1] for later reference:
Lemma 2.6. With the notation above:
1. δ∗(C ′, D) ∈ {w, sw} for all C ′ ∈ Cε with δε(C ′, C) = s.
2. If l(sw) > l(w), then there exists precisely one chamber C ′ ∈ Cε satisfying δε(C ′, C) = s
and δ∗(C ′, D) = sw.
This leads to the following definition of the opposition relation between C+ and C−.
Definition 2.7 (opposition relation). For C ∈ Cε and D ∈ C−ε we say C and D are opposite,
denoted by C op D, when δ∗(C,D) = 1.
Finally we define a twin apartment in a twin building as in [1, Definition 5.171].
Definition 2.8 (twin apartment). A twin apartment in a twin building
(
B+,B−, δ∗
)
is an
ordered pair
(
A+, A−
)
where A± ∈ A± is an apartment in B± such that every chamber in
A+ ∪A− is opposite to precisely one chamber in A+ ∪A−.
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 simplices in the building can be equivalently indexed by the
coset spaces G/P±J , where
{
P±J | 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±[i1], . . . , P
±
[ir+1] span an r-simplex if and only if the
intersection P±[i1] ∩ · · · ∩P
±
[ir+1] = P±[i1,...,ir+1] is a parabolic subgroup, so B is a simplicial complex
of dimension dim(B) = ` − 1. For non-finite type Kac–Moody groups such as the hyperbolic
type we consider, B+ and B− are not conjugate in G, so the intersection of a positive parabolic
subgroup with a negative parabolic subgroup never contains a Borel subgroup. Hence, from the
twin BN -pair
(
B+, B−, N
)
we get two buildings
(
B+, δ+
)
and (B−, δ−), each equipped with
a W -valued distance function, δ±, each of type (W,S), and a codistance function δ∗, yielding
a twin building B =
(
B+,B−, δ∗
)
. 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 = P±J , and has closure
∆± =
⋃
J(I
P±J .
10 L. Carbone, A.J. Feingold and W. Freyn
We also choose a fundamental apartment A±fund in B± to be the one whose chambers are
(
NB±
)
/B± =
{
wB± |w ∈W
}
.
Note that δ±
(
w1B
±, w2B
±) = w−1
1 w2 and δ∗
(
w1B
±, w2B
∓) = w−1
1 w2, so that
(
A+
fund, A
−
fund
)
is
a twin apartment.
Using the property that the simplices in each building are in bijection with the union of the
coset spaces
⋃
J(I G/P
±
J , we describe the buildings B+ and B− associated to a twin BN -pair,(
B+, B−, N
)
for a Kac–Moody group G as follows:
B± :=
(
G/B± ×∆±
)
/∼. (2.3)
The equivalence relation ∼ is defined by
(
fB±,∆±J
)
∼
(
gB±,∆±J ′
)
in
(
G/B±,∆±
)
if and
only if ∆±J = ∆±J ′ (so J = J ′) and f−1gP±J ⊂ P±J .
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 [15, 33])
φ
(
gB±,∆±J
)
=
(
φ(g)B∓,∆∓J
)
. (2.4)
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 :=
{
x =
r+1∑
i=1
aiei ∈ Rr+1
∣∣ 0 ≤ ai ≤ 1,
r+1∑
i=1
ai = 1
}
which inherits the topology from Rr+1. 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 [1].
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 tessellated by the action of the hyperbolic Weyl group W . In the case when
the Cartan matrix is strictly hyperbolic, that tessellation is by compact simplices, but otherwise
these simplices have ideal vertices stabilized by affine type subgroups of 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. Com-
posing 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 [44, p. 243].
Note that ω0 and ω both centralize the Weyl group, so they each induce twin building
involutions via formula (2.4).
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
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 11
generated by the subalgebras gj and ω(gj) = gj . Then, using the notation i =
√
−1 ∈ C, the
real Lie algebra of fixed points of ω on gj , kj = suj2 has basis
xj =
1
2
(ej − fj), yj =
i
2
(ej + fj), zj =
i
2
(hj)
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 [4, Proposition 1]). 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
any Lie algebra automorphism, which we will apply to ω as well as to w̃ad
i ∈ W̃ ad.
Lemma 2.9. For any α ∈ Φre, eα ∈ gα and any φ ∈ Aut(g), we have φ ◦ exp(adeα) ◦ φ−1 =
exp(adφ(eα)) so that, in particular, ωUad
α ω−1 = Uad
−α and w̃ad
i U
ad
αi
(
w̃ad
i
)−1
= Uad
−αi for 1 ≤ i ≤ `.
Proof. For any x ∈ g, since φ is a Lie algebra automorphism, we have
φ(exp(adeα)x) = φ
∑
k≥0
1
k!
(adeα)k(x)
=
∑
k≥0
1
k!
(adφ(eα))
k(φ(x)) = exp(adφ(eα))(φ(x)).
The formula with φ = ω gives ωUad
α ω−1 = Uad
−α since ω(gα) = g−α, and with φ = w̃ad
i gives
w̃ad
i U
ad
αi
(
w̃ad
i
)−1
= Uad
−αi since w̃ad
i (ei) = −fi. �
We set K = FixG(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, . . . , ` [6, 37, 56].
For each v = a+ bi ∈ C and 1 ≤ j ≤ ` we write a generator of T ad
C as
exp(advhj ) = exp(adahj ) exp(adbihj ) = exp(adahj ) exp(adb2zj ).
This gives the decomposition T ad
C = T ad
R T where
T ad
R = 〈exp(adahj ) | a ∈ R, 1 ≤ j ≤ `〉 (2.5)
is the split real torus and T = 〈exp(adbzj ) | b ∈ R, 1 ≤ j ≤ `〉 is the compact real torus.
It is clear from the two expressions for w̃ad
i that W̃ ad ≤ K, but it is not so obvious that these
operators can be expressed as a single exponential w̃ad
i = exp(adπ xi). This is a special case
of the formula w̃φi = exp(φ(π xi)) for any integrable representation φ : g → End(V ), where the
inner π is in R, proven in Theorem A.1 and first found in certain important cases by [12]. See
also [29, 41].
Proposition 2.10. All Cartan subalgebras of k are conjugate under the action of K.
This result follows from [6, Proposition 8.1(iii)],. See also [39, Corollary 5.33] and [38, Propo-
sition 3.5].
12 L. Carbone, A.J. Feingold and W. Freyn
3 Tits cone and lightcone of hyperbolic Kac–Moody algebras
of compact type
Let g = gC(A) be a hyperbolic 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 TL+ and the other called backward and
denoted TL−. We have TL− = −TL+.
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 W
on each of the timelike components is defined by
C± = {x ∈ TL± |αi(x) ≥ 0, 1 ≤ i ≤ `}.
The union
X± =
⋃
w∈W
w
(
C±
)
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± ⊇ TL±, 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 −1
0 −1 2
and
2 −2 −2
−2 2 −2
−2 −2 2
corresponding to the hyperbolic Kac–Moody algebra F [18] 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. Another such an example is E10
because it contains the affine Kac–Moody algebra E9 = E
(1)
8 . For A strictly hyperbolic, that is,
whose principal minors are of finite type, we have X± = TL±.
We have the following description of the closure of the Tits cone (see [35, equation (5.10.2)])
X = X+ ∪X− = {h ∈ hR | (h, h) ≤ 0}.
We introduce the notations
LhR = {h ∈ hR | (h, h) ≤ 0}, L0
hR
= {h ∈ hR | (h, h) < 0} and
∂LhR = {h ∈ hR | (h, h) = 0}.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 13
Proposition 3.1. Let g be a hyperbolic Kac–Moody algebra over C, 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 = {ix |x ∈ hR}, where i2 = −1. The
signature of t ∼= R1,`−1 is (1, `− 1).
Proof. The definition of a hyperbolic Cartan matrix gives that the bilinear form on the split
real form of the Cartan subalgebra (the real span of the generators hi) is Lorentzian, so the
statements in the proposition are clear. �
We use a sign convention on k adopted from the theory of finite-dimensional Riemannian
symmetric spaces [32] and set
(·, ·)k = −(·, ·)|k. (3.1)
Note that in the affine case (·, ·)k, this sign convention naturally occurs in the loop group re-
alizations [19, 51]. 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).
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 = i
2hi, i = 1, . . . , `, and the elements obtained from basis
multibrackets above
1
2
([ei1 , ei2 , . . . , ein ] + [fi1 , fi2 , . . . , fin ]) and
i
2
([ei1 , ei2 , . . . , ein ]− [fi1 , fi2 , . . . , fin ]) for n even,
1
2
([ei1 , ei2 , . . . , ein ]− [fi1 , fi2 , . . . , fin ]) and
i
2
([ei1 , ei2 , . . . , ein ] + [fi1 , fi2 , . . . , fin ]) for n odd.
In particular, for n = 1 these include the elements xi = 1
2(ei − fi) and yi = i
2(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 [35, 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 [35,
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
{
f jα = ej−α | 1 ≤ j ≤ dim(gα)
}
, respectively
such that
(
f jα, e
m
β
)
= δj,mδα,β.
As a consequence, for positive roots α, the basis elements of k, xjα = 1
2
(
ejα − f jα
)
and yjα =
i
2
(
ejα + f jα
)
satisfy
(
xjα, x
m
β
)
= −1
2
δj,mδα,β =
(
yjα, y
m
β
)
and
(
xjα, y
m
β
)
= 0.
14 L. Carbone, A.J. Feingold and W. Freyn
Hence it follows from (3.1) that
(
xjα, x
m
β
)
k
=
1
2
δj,mδα,β =
(
yjα, y
m
β
)
k
and
(
xjα, y
m
β
)
k
= 0.
The Cartan subalgebra has signature (`− 1, 1) and for each positive root α the subspaces
〈
xjα | 1 ≤ j ≤ dim(gα)
〉
and
〈
yjα | 1 ≤ j ≤ dim(gα)
〉
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). �
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, f
j
α
}
forms a hyperbolic plane, and for different positive α and distinct j these planes are orthogonal.
We recall that the ad-invariant bilinear form on hR extends C-bilinearly to the complexifica-
tion hC. In particular, for x ∈ hR we have
(ix, ix) = −(x, x) = (x, x)k
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),
L0
k = {x ∈ k | (x, x)k < 0}, L0
t = L0
k ∩ t = ϕ
(
L0
hR
)
,
∂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
(
TL±
)
= TL±t , ϕ
(
C±
)
= C±t , ϕ
(
X±
)
= X±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 =
⋃
k∈K
ktk−1 =
{
kxk−1 ∈ k | k ∈ K,x ∈ t
}
.
By Proposition 2.10 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
⋃
α∈Φ+
{
xjα | 1 ≤ j ≤ dim(gα)
}
∪
{
yjα | 1 ≤ j ≤ dim(gα)
}
.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 15
Proof. For α ∈ Φ+ and 1 ≤ j ≤ dim(gα) we have
[
xjα, z
]
= −
[
ih,
1
2
(ejα − f jα)
]
= − i
2
[
h,
(
ejα − f jα
)]
= − i
2
α(h)
(
ejα + f jα
)
= −α(h)yjα,
[
yjα, z
]
= −
[
ih,
i
2
(
ejα + f jα
)]
=
1
2
[
h,
(
ejα + f jα
)]
=
1
2
α(h)
(
ejα − f jα
)
= 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.
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′ ∈ t we obtain
(w, z′)k = ([y, z], z′)k = (y, [z, z′])k = (y, 0)k = 0. �
Definition 4.3 (surface notation). 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 = − 1
r2
whose connected components we call X+
r and X−r . In particular we will use tr ⊂
L0
t ⊂ t, kr ⊂ L0
k ⊂ k and Hr ⊂ H and note that k0 = ∂Lk and t0 = ∂Lt. For ` > 2 we find
t+−1
∼= t−−1
∼= H`−1, (`− 1)-dimensional hyperbolic space.
Definition 4.4 (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
V̂ = V ∪B∞(∂LV ).
For each r < 0, define B∞(V ±r ), the boundary at infinity of V ±r , 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 [16]. Note that for each r < 0, B∞(V ±r ) can be identified with B∞(∂LV )±.
Using this observation we define the lightlike closure of V ±r to be
V̂ ±r = V ±r ∪B∞(∂LV )±. (4.1)
Since the bilinear form (·, ·)k is ad-invariant, the adjoint action of K on k preserves the
surfaces Hr ⊂ H defined in Definition 4.3. 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 t̂ and on t̂±r .
16 L. Carbone, A.J. Feingold and W. Freyn
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
as well as on the lightlike closure Ĥr.
Since all Cartan subalgebras are conjugate, we can define the following surjective map
ψ : K × tr −→ Hr, ψ(k, x) = kxk−1.
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−1k−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 t′ = ktk−1 of k, Lt′ coincides with the closure of the Tits cone
of t′, X t′ . Hence X t′ is the closure of the cone {su ∈ t′ | s > 0, u ∈ t′r}, for any fixed r < 0,
which includes the boundary ∂Lt′ . 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 Xt′ = L0
t′ and there is a bijection between each sur-
face (t′r)
± for r < 0 and the set of rays in L0
t′ . In this case we do not need to consider the
lightlike closure Ĥr in order to embed the building in it.
2. When A contains an affine Cartan submatrix the fundamental chambers C± contain rays
that accumulate at rays on the lightcone ∂Lt′ . So the corresponding points on the sur-
face (t′−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 (t′−1)± ∼= H2 are isometric to
the Poincaré disk, and the tessellation 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
use Ĥr.
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 C±r for
the action of W on tr gives the surjective maps
ψ± : K/T × C±r −→ H±r , ψ±(kT, u) = kuk−1.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 17
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 each 1 ≤ i ≤ `,
SU(2)i + t = SU(2)i ⊕ Lt,i since αi(zi) = i. For any s, t, u ∈ R, we have exp(adsxi+tyi+uzi) ∈
SU(2)i ≤ K and for z ∈ t we have [sxi + tyi + uzi, z] = −iαi(z)(txi − syi) = 0 when αi(z) = 0
which means exp(adsxi+tyi+uzi)(z) = z when αi(z) = 0. While (adsxi+tyi+uzi)z = (adsxi+tyi)z
for any z ∈ t, we see that
(adsxi+tyi+uzi)
2z = −iαi(z)(adsxi+tyi+uzi)(txi − syi) = −iαi(z)
(
usxi + utyi −
(
s2 + t2
)
zi
)
differs from (adsxi+tyi)
2z, and higher powers have increasingly complicated expressions in-
volving u, so that exp(adsxi+tyi+uzi)z certainly depends on u. Nevertheless, the factor αi(z)
tells us that Lt,i is the fixed point set in t of exp(adsxi+tyi+uzi), so the Cartan subalgebra
exp(adsxi+tyi+uzi)t is spanned by Lt,i and the vector exp(adsxi+tyi+uzi)zi. It appears that this
is a three-parameter family of Cartan subalgebras, each containing Lt,i, but, in fact, we can see
that the entire family is obtained from the two-parameter family with u = 0.
While these operators are defined on the entire Kac–Moody algebra, g, we are really only
interested in the orbit of zi inside su(2)i under the action of SU(2)i, so we can do this calculation
with 2 × 2 matrices. An arbitrary matrix in SU(2) is A =
[
α β
−β̄ ᾱ
]
where α, β ∈ C and
det(A) = 1. With z = i
2
[
1 0
0 −1
]
we have
AzA−1 =
i
2
[
(αᾱ− ββ̄) −2αβ
−2ᾱβ̄ −(αᾱ− ββ̄)
]
.
So the stabilizer of z consists of those A such that αβ = 0 and αᾱ− ββ̄ = 1, which implies that
β = 0 and the stabilizer of z is the diagonal U(1) =
{[
α 0
0 ᾱ
]
|αᾱ = 1
}
. The orbit SU(2) · z is
in bijection with SU(2)/U(1) which is well-known to be the 2-sphere. We will show below that
the two-parameter family exp(adsxi+tyi)zi gives a 2-sphere, so it is enough to find all the Cartan
subalgebras containing Lt,i.
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). With v =
(
s
2 + ti
2
)
∈ C we have sxi + tyi = vei − vfi, so
exp(adsxi+tyi) = exp(advei−vfi) ∈ SU(2)i
gives another parameterization of {ti(s, t) | s, t ∈ R} = {exp(advei−vfi)t | v ∈ C}. For fixed i,
distinct choices of (s, t) ∈ R can give the same Cartan subalgebra ti(s, t), so we would like
to know exactly how to parameterize the distinct Cartan subalgebras in this set. The first
part of the following theorem shows that for each 1 ≤ i ≤ ` the family of distinct Cartan
subalgebras obtained this way can be parameterized by a 2-sphere with antipodes identified, the
real projective space P2(R). A more careful interpretation of this calculation will later give us
information related to the set of chambers in the building which share a common panel.
18 L. Carbone, A.J. Feingold and W. Freyn
Theorem 4.5. 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)
r (txi − syi),
2) exp(adsxi+tyi)xi = xi − t sin(r)
r zi − t
r2
(cos(r)− 1)(txi − syi),
3) exp(adsxi+tyi)yi = yi + s sin(r)
r zi − s
r2
(cos(r)− 1)(txi − syi).
Proof. We prove only the first relation since the others are analogous. We have
(adsxi+tyi)
1z = [sxi + tyi, z] = −iαi(z)(txi − syi),
(adsxi+tyi)
2z = [sxi + tyi,−iαi(z)(txi − syi)] = iαi(z)
(
s2 + t2
)
zi,
(adsxi+tyi)
3z =
[
sxi + tyi, iαi(z)
(
s2 + t2
)
zi
]
= iαi(z)
(
s2 + t2
)
(txi − syi),
(adsxi+tyi)
4z =
[
sxi + tyi, iαi(z)
(
s2 + t2
)
(txi − syi)
]
= −iαi(z)
(
s2 + t2
)2
zi.
In the third step we have used that αi(zi) = i.
Using r2 = s2 + t2 6= 0, it is clear that for n ≥ 1 we have
(adsxi+tyi)
2nz = −iαi(z)(−1)n
(
r2
)n
zi
and for n ≥ 0 we have
(adsxi+tyi)
2n+1z = −iαi(z)(−1)n
(
r2
)n
(txi − syi)
so we get
exp(adsxi+tyi)z = z − iαi(z)
∞∑
n=1
(−1)nr2n
(2n)!
zi − iαi(z)
∞∑
n=0
(−1)nr2n
(2n+ 1)!
(txi − syi)
= z − iαi(z)(cos(r)− 1)zi − iαi(z)
sin(r)
r
(txi − syi). �
Corollary 4.6. For each 1 ≤ i ≤ `, the family of distinct Cartan subalgebras in {ti(s, t) | s, t ∈
R}, including t, is parametrized by the unit hemisphere
{
(sin(r) sin(ψ),− sin(r) cos(ψ), cos(r)) ∈ R3 | 0 ≤ r < π, 0 ≤ ψ < π
}
,
where r =
√
s2 + t2 ≥ 0 and ψ is defined when r > 0 by sin(ψ) = t
r and cos(ψ) = s
r . Also,
(s′, t′) = r−π
r (s, t) corresponds to the antipodal point determined by (π − r, ψ + π) and for any
z ∈ t, we have
exp(ads′xi+t′yi)z = exp(adsxi+tyi)wi(z).
Proof. The Cartan subalgebra ti(s, t) is spanned by Lt,i (for any choice of (s, t)) and the vector
exp(adsxi+tyi)zi = sin(r) sin(ψ)xi − sin(r) cos(ψ)yi + cos(r)zi ∈ sui2
from Theorem 4.5(1) and the fact that αi(zi) = i. With respect to basis {xi, yi, zi} of sui2, the
coordinates of this vector are
(sin(r) sin(ψ),− sin(r) cos(ψ), cos(r))
so the vector is on a unit sphere. Each point on one hemisphere is uniquely determined by
the choices 0 ≤ r < π and 0 ≤ ψ < π, and no two vectors of the above form are co-linear, so
all of the corresponding subspaces ti(s, t) are distinct. The point on the sphere determined by
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 19
(r, ψ) corresponding to (s, t) has antipodal point determined by (π − r, ψ + π) corresponding
to (s′, t′) = r−π
r (s, t). The corresponding subspaces are the same since exp(ads′xi+t′yi)zi =
− exp(adsxi+tyi)zi.
We have (s′, t′) determined by sin(ψ+π) = t′
π−r and cos(ψ+π) = s′
π−r so −tr = − sin(ψ) = t′
π−r
and −s
r = − cos(ψ) = s′
π−r , which gives t′ = r−π
r t and s′ = r−π
r s. For αi(z) = 0, that is, for
z ∈ Lt,i, we have wi(z) = z and so exp(ads′xi+t′yi)z = z = exp(adsxi+tyi)wi(z). For z = zi we
have wi(zi) = −zi, and we have shown above that exp(ads′xi+t′yi)zi = − exp(adsxi+tyi)zi. Since
these operators are linear, that proves the claimed formula for any z ∈ t. �
Remark 4.7. We may find it useful later to associate w ∈ Ĉ with a point (x, y, z) on the
unit sphere, and to record the relationship between antipodal points on the sphere and their
associated complex numbers. The standard formula for a stereographic projection from the
north pole (0, 0, 1) to a point in the complex plane z = 0 is
(x, y, z)→ x
1− z + i
y
1− z = w
so for the antipodal point
(−x,−y,−z)→ −x
1 + z
+ i
−y
1 + z
= wa.
Since x2 + y2 + z2 = 1, we then have wa = −w̄−1.
We wish to realize the twin apartments for a hyperbolic Kac–Moody group as a geometric
object inside the lightcone of the compact real form of the Kac–Moody Lie algebra. It is not
hard to see the Coxeter complex structure in each side of the Tits cone of a standard compact
real Cartan subalgebra. The tessellation of the forward and backward Tits cones by the action
of the hyperbolic Weyl group provides a pair of opposite fundamental apartments, (Σ+,Σ−),
with tessellations Σ± =
⋃
w∈W
w
(
C±
)
where C± is a fundamental chamber in Σ±. A panel is
a nonempty intersection of maximal dimension of two wi-adjacent chambers, that is, C±1 = wC±
and C±2 = wwiC±, so that wwiw
−1C±1 = C±2 . It means that the panel is contained in the wall
of all fixed points of the Weyl group reflection wwiw
−1 which is a W -conjugate of the simple
reflection wi. We may simplify the discussion by considering just the panels of a fundamental
chamber, C±, contained in the wall of fixed points of the simple reflection wi. The hyperplane Lt,i
intersects both sides of the Tits cone and determines a wall containing a panel of C± in each side
of the twin apartment. The operators exp(adsxi+tyi) ∈ SUi
2 for any s, t ∈ R fix Lt,i pointwise,
but take zi to a vector in sui2 not in Lt,i, giving a family of compact Cartan subalgebras,
exp(adsxi+tyi)t = ti(s, t), each of which contains a Tits cone, and all of which share the common
subspace Lt,i. Consider the distinct chambers in this family of apartments which share a common
panel with the fundamental chamber in the fundamental apartment (in either side of the twin
fundamental apartment). For now, let us only think about Σ+, and its images under this family
of operators. The fundamental chamber, C+, has a unique panel fixed by wi, and that panel is
contained in Lt,i. So the family of distinct chambers exp(adsxi+tyi)C+ obtained as s and t vary,
all contain that wi-fixed panel. The same can be said of the chamber wiC+ in Σ+, and we may
wish to consider its orbit under this family of operators. But we know that with (s, t) = (π, 0),
which corresponds to (r, ψ) = (π, 0), that operator restricted to t equals wi, so we should be
able to understand that orbit starting from either chamber.
The point of the first part of Corollary 4.6 is that for (s, t) corresponding to (r, ψ) giving dis-
tinct points on the hemisphere 0 ≤ r < π, 0 ≤ ψ < π, the real compact Cartan subalgebras ti(s, t)
are all distinct, so the chambers exp(adsxi+tyi)C+ must all be distinct. But the second part of
the corollary says that for (s′, t′) = r−π
r (s, t) corresponding to the antipodal point determined
20 L. Carbone, A.J. Feingold and W. Freyn
by (π− r, ψ+π), and for any z ∈ t, we have exp(ads′xi+t′yi)z = exp(adsxi+tyi)wi(z). This means
that exp(ads′xi+t′yi)C+ = exp(adsxi+tyi)wi(C+) is the chamber adjacent to exp(adsxi+tyi)C+ in
the apartment exp(adsxi+tyi)Σ
+, sharing the wi-fixed panel. Thus, the complete set of all dis-
tinct chambers exp(adsxi+tyi)C+ is obtained when (s, t) varies so that the corresponding (r, ψ)
gives all points on the unit sphere. Antipodal points give distinct chambers in the same apart-
ment, and the set of all chambers sharing the common panel is parametrized by a real 2-sphere,
exactly corresponding to the abstract building picture, where the answer is the Riemann sphere.
From the remark above, if w ∈ Ĉ is the label of a chamber sharing the common panel, the label
wa = −w̄−1 ∈ Ĉ denotes the wi reflection of the first chamber.
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 [15] 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) = TA, the decomposition B± = TC(G)U± and the Iwasawa decomposition,
we have a bijection between coset spaces
G/B± ↔ K/T. (4.2)
Recalling the description of the building from equation (2.3) in Section 2.3
B± =
(
G/B± ×∆±
)
/∼, (4.3)
we obtain an equivalent new description by equation (4.2):
B± =
(
K/T ×∆±
)
/∼. (4.4)
In the description of the twin building B = B+∪B− given by equation (4.3), the natural action
of G via left multiplication is apparent, while in the description of B as in equation (4.4) the G-
symmetry is broken to K-symmetry. Note that in equation (4.3) 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 B+
g = gB+g−1, but that
subgroup acts transitively on
{
fB− ∈ G/B− | g−1f ∈ B+B−
}
which certainly contains gB−.
But the description of B± in equation (4.4) 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 =
(
fB±,∆±∅
)
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-panel of c is
(
fB±,∆±i
)
and the i-residue
of c is defined to be Ri(c) =
{
d =
(
gB±,∆±∅
)
∈ Cham
(
B±
)
|
(
fB±,∆±i
)
∼
(
gB±,∆±i
)}
. Then
we have Ri(c) ∼= P1(C). Identifying P1(C) with the Riemann sphere Ĉ = C ∪ ∞, the action
of the subgroup SU(2)i on Ri(c) can be identified with the action of SU(2) on Ĉ by Möbius
transformations. Additional details may be found in [1, Chapter 6],.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 21
5 Simplicial complex, distance and codistance on Ĥr
In this section we define for each r < 0 a simplicial complex on the set Ĥr, where we are using the
notations in Definitions 4.3 and 4.4. We begin in the standard Cartan subalgebra with tr ⊂ t.
Recall, that tr = t+r ∪ t−r is (up to rescaling) isometric to a pair of hyperbolic spaces (for ` > 2)
which are both tessellated by the action of the Weyl group W = 〈S〉. 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 tsinr be
the set of all singular elements in tr and let treg
r be the set of all regular elements, and similarly
we have sets
(
tsinr
)±
and
(
treg
r
)±
. Then we have the decomposition
tr = tsinr ∪ treg
r .
We denote by Comp±r the set of connected components of treg
r . In each sheet one connected
component, C±r = C± ∩
(
treg
r
)±
, corresponds to 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±r as follows: Let 1 ∈ W denote the identity element. In each
sheet t±r we index the fundamental chamber C±r with 1 such that C−r = −C+
r . Then we index the
connected component wC±r ∈ Comp±r by w yielding
Comp±r =
{
wC±r |w ∈W
}
⊂
(
treg
r
)±
.
Let wC±r denote the closure of the component wC±r and let U± denote the union
U± =
⋃
w∈W
wC±r
so U± covers t±r . It may happen that the closure C±r includes points at infinity, that is, ideal
points not on the surface tr, but which correspond to rays in the null cone as discussed in
Definition 4.4. In that case, some vertices (0-simplices) will be in t̂±r .
For any subset J ( S recall that WJ = 〈s | s ∈ J〉, and define the intersection
Simp±J =
⋂
w∈WJ
wC±r ⊂ t̂±r .
We identify Simp±J with a simplex of dimension `− 1− |J |.
For example, when J = ∅, Simp±J = C±r is an `− 1 simplex, and when J = S\{i}, Simp±J is
a 0-simplex.
Theorem 5.1 (simplicial structure on Ĥr). For each r < 0, with the notations above, we have
a Coxeter complex on
(
k̂tk−1
)±
r
for each k ∈ K and their union over all k ∈ K forms a simplicial
complex in Ĥ±r .
Proof. It is straightforward to check that S± =
{
Simp±J | J ( S
}
is a Coxeter complex [1] for
(W,S). 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 face of codimension 1.
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. We
have now defined a Coxeter complex on
(
k̂tk−1
)±
r
for each k ∈ K.
The union of all these Coxeter complexes forms a simplicial complex in Ĥ±r as follows. We
must only check that the Weyl group tessellations on different Cartan subalgebras fit together
22 L. Carbone, A.J. Feingold and W. Freyn
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 complexes in
different Cartan subalgebras fit together, leading to a simplicial complex in Ĥ±r .
Each simplex of dimension `−2 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 dimension
`− n lie in the intersection of n− 1 such hyperplanes with the surface. The stabilizer in K of t
is the Weyl group W since W = N/T is a quotient of the normalizer of T . Hence simplicies of
lower dimension are fixed by nontrivial subgroups of W . �
Before we can state and prove the main embedding theorem in the next section, we must
define the distance and codistance functions on Hr. For any two chambers k1C±r k−1
1 and k2C±r k−1
2
in Ĥ±r , define the W -valued distance function
δ±
(
k1C±r k−1
1 , k2C±r k−1
2
)
= w ∈W when k−1
1 k2 ∈ B±wB±
and the codistance function
δ∗
(
k1C±r k−1
1 , k2C∓r k−1
2
)
= w ∈W when k−1
1 k2 ∈ B±wB∓.
Note that
(
Ĥ±r , δ
±) is a Tits building with an apartment system determined by δ±, and(
Ĥ+
r , Ĥ
−
r , δ
∗) is a twin building. We choose C±r to be a fundamental chamber in t±r and then
{wC±r |w ∈W} is a fundamental apartment in Ĥ±r .
6 The main embedding theorem
The main result of this paper, given in this section, is an embedding of the twin building
B = (B+,B−, δ∗) of the Kac–Moody group G with twin BN -pair (B+, B−, N) into the compact
real form k. This is a bijective simplicial map from B± onto the simplicial complex defined in Ĥ±r
for each r < 0 defined in Section 5, such that the W -valued distance and codistance functions
are respected, and the map is K-equivariant, but not G-equivariant.
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 with twin BN -pair (B+, B−, N), W -
valued distance functions, δ±, W -valued codistance function, δ∗, and let r < 0. Let K be the
compact real form of G and let k be its Lie algebra. Let B = (B+,B−, δ∗) be the geometric
realization (Section 2.3) of the twin building of G over C, with Tits buildings, (B±, δ±) and
codistance function δ∗ in B. We use the same notations, δ± and δ∗, for distance functions
between chambers in H±r and codistance function in Hr.
1. There is a K-equivariant simplicial map Ψr : B ↪→ Ĥr ⊂ k̂, that is, the following diagram
commutes:
B K //
Ψr
��
B
Ψr
��
Ĥr
AdK // Ĥr
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± ↪→ H±r are bijec-
tive, otherwise, Ψ±r : B± ↪→ Ĥ±r are injective. For any chambers C, D in B±, we have
δ±(C,D) = δ±
(
Ψ±r (C),Ψ±r (D)
)
, so Ψ±r respects the distance functions in B± and H±r .
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 23
3. There is a W -equivariant embedding of the fundamental apartment of B± into t̂±r ⊂ t̂r.
4. The map Ψr is a twin building isomorphism. In particular, for any chamber C in B±
and any chamber D in B∓, we have δ∗(C,D) = δ∗
(
Ψ±r (C),Ψ∓r (D)
)
, so Ψr respects the
codistance functions in B and Ĥr.
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 7).
Otherwise it contains points on the boundary of the lightcone. Examples of this type are the
algebras F , I and E10.
Proof. To construct the embedding of the twin building we use the description given in equa-
tion (4.4)
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 ` panels 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
(
fT,∆±i
)
and
(
gT,∆±i
)
describe the same simplex if any only if fKi = gKi where Ki = K ∩ Pi =
K∩(BtBwiB); similarly for any subset J ( I two J-cells
(
fT,∆±J
)
and
(
fT,∆±J
)
are equivalent
if and only fKJ = gKJ for KJ = K ∩ PJ .
For fixed r < 0 we have chosen fundamental chambers C±r such that
C−r = −C+
r . (6.1)
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 t̂r 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
C±J = C±r ∩
⋂
j∈J
Lt,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
n-dimensional 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
V
: · · · : Vn
V
]
.
This yields a simplicial map Ψr :
(
1T,∆±
)
−→ C±r .
24 L. Carbone, A.J. Feingold and W. Freyn
Part 2: Let x± ∈ ∆±. We extend the map defined in Part 1 to a map
Ψr : B ↪→ Ĥr
by defining
Ψr
(
kT, x±
)
= Adk
[
Ψr
(
1T, x±
)]
.
We have to check that Ψr is well-defined. Assume we have two equivalent elements
(
k1T, x
±
1
)
∼
(
k2T, x
±
2
)
, hence x±1 = x±2 and assuming x±1 ∈ ∆±J for some subset J ( I, we have k1KJ =
k2KJ . Then there is some l ∈ KJ such that k1 = k2l and we have
Ψr
(
k1T, x
±
1
)
= Adk1
[
Ψr
(
1T, x±1
)]
= Adk2l
[
Ψr
(
1T, x±1
)]
.
Since x±1 ∈ ∆±J we have
Ψr
(
1T, x±1
)
∈
⋂
j∈J
Lt,j .
Hence AdlΨr
(
1T, x±1
)
= Ψr
(
1T, x±1
)
. Thus, from Adk2l = Adk2Adl we get
Adk2l
[
Ψr
(
1T, x±1
)]
= Adk2
[
Ψr
(
1T, x±2
)]
= Ψr
(
k2T, x
±
2
)
.
Note that Ψr maps the simplex (kT,∆±J ) for J ( I spanned by vertices
(
kT,∆±[i]
)
, i ∈ J ,
onto the simplex spanned by Ψr
(
kT,∆±[i]
)
, i ∈ J . If two simplices
(
k1T,∆
±
J
)
and
(
k2T,∆
±
J
)
share a common face
(
lT,∆±L
)
, for J ( L in B then by definition we have k1T ⊂ AdlKL and
k2T ⊂ AdlKL. But then Ψr
(
k1T,∆
±
L
)
= Ψr
(
lT,∆±L
)
= Ψr
(
k2T,∆
±
L
)
is the commonly shared
face of the simplices Ψr
(
k1T,∆
±
J
)
and Ψr
(
k2T,∆
±
J
)
. Hence Ψr preserves the simplicial structure
of B and is thus a simplicial complex map.
Part 3: We continue to check that Ψr satisfies the properties stated in the theorem.
Proof of (1): Recall the left K-action on the building B±:
K : B± −→ B±, k ·
(
fT,∆±
)
7→
(
kfT,∆±
)
.
We need to verify that for k1, k2 ∈ K and J ( I
Ψr
(
k1 ·
(
k2T,∆
±
J
))
= Adk1Ψr
(
k2T,∆
±
J
)
.
We have
Ψr(k1 ·
(
k2T,∆
±
J )
)
= Ψr
(
k1k2T,∆
±
J
)
= Adk1k2Ψr
(
1T,∆±J
)
= Adk1
(
Adk2Ψr
(
1T,∆±J
))
= Adk1
(
Ψr
(
k2T,∆
±
J
))
.
Proof of (2): Assume two simplices
(
k1T,∆
±
J
)
and
(
k2T,∆
±
J
)
satisfy
Ψr
(
k1T,∆
±
J
)
= Ψr
(
k2T,∆
±
J
)
.
Then we have
Adk1Ψr
(
1T,∆±J
)
= Adk2Ψr
(
1T,∆±J
)
,
which implies Adk−1
1 k2
Ψr
(
1T,∆±J
)
= Ψr
(
1T,∆±J
)
.
As Ψr
(
1T,∆±J
)
= C±J its stabilizer in K is KJ . Hence k−1
1 k2 ∈ KJ . Thus
(
k1T,∆
±
J
)
∼(
k2T,∆
±
J
)
which proves injectivity.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 25
Suppose we have an arbitrary element x± ∈ H±r . Then there is some group element k ∈ K such
that Adkx
± ∈ t±r so it is in the closure of some chamber, wC±r , uniquely labeled by an element
w ∈W . The action of the corresponding element w̃ ∈ W̃ ad ≤ K matches the action of w on t. So
we have Adw̃−1Adkx
± ∈ C±r . Therefore x± ∈ Adk−1Adw̃Ψr
(
1T,∆±
)
= Ψr
(
k−1w̃T,∆±
)
. 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).
Let C =
(
k1T,∆
±) andD =
(
k2T,∆
±) be chambers in B±. By definition, δ±(C,D) = w ∈W
when k−1
1 k2 ∈ B±wB±. We also have by definition,
Ψ±r (C) = Adk1
[
Ψ±r
(
1T,∆±
)]
= k1C±r k−1
1 , and
Ψ±r (D) = Adk2
[
Ψ±r
(
1T,∆±
)]
= k2C±r k−1
2 so
δ±
(
Ψ±r (C),Ψ±r (D)
)
= δ±
(
k1C±r k−1
1 , k2C±r k−1
2
)
= w,
since k−1
1 k2 ∈ B±wB±. This completes the proof of (2).
The proof of (3) is clear.
Proof of (4): Let C =
(
k1T,∆
±) be a chamber in B± and let D =
(
k2T,∆
∓) be a chamber
in B∓. By definition, δ∗(C,D) = w ∈W when k−1
1 k2 ∈ B±wB∓. We also have by definition,
Ψ±r (C) = Adk1
[
Ψ±r
(
1T,∆±
)]
= k1C±r k−1
1 , and
Ψ∓r (D) = Adk2
[
Ψ∓r
(
1T,∆∓
)]
= k2C∓r k−1
2 so
δ∗
(
Ψ±r (C),Ψ∓r (D)
)
= δ∗
(
k1C±r k−1
1 , k2C∓r k−1
2
)
= w,
since k−1
1 k2 ∈ B±wB∓. �
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 Kac–
Moody 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 the arithmetic properties of W , essentially understanding its
cusps as a modular group. Such an analysis was carried out for E10 in [42, Lemma 5.2].
Thus for each r < 0 we have established an embedding Ψr of the twin building B =(
B+,B−, δ∗
)
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
closure, Ĥ±r , otherwise.
Remark 6.4. Recall that B± is contractible since each apartment is contractible (see [53]).
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
Corollary 6.5. The spaces Ĥ±r are contractible.
Remark 6.6. The analogs of Theorem 6.1 for finite-dimensional compact Lie groups and affine
Kac–Moody groups are well-known (see [16, 21, 30, 47]). It can be generalized in these cases
26 L. Carbone, A.J. Feingold and W. Freyn
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 has only recently been
investigated in [26].
We conclude this section with a discussion of 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∓
as well as
ω0
(
t±r
)
= t∓r and − ω
(
t±r
)
= t∓r
for each r ∈ R, so that
ω0
(̂
t±r
)
= t̂∓r and − ω
(̂
t±r
)
= t̂∓r .
Proposition 6.7. The following diagram is commutative:
B ω //
Ψr
��
B
Ψr
��
k
−ω // k
Furthermore, for any chamber C in B± and D in B∓ we have δ∗(C,D) = δ∗(ω(C), ω(D)).
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
)
.
Let (kT,∆±J ) ∈ B±. Then we have
−ω
(
Ψr
(
kT,∆±J
))
= −ω
(
AdkΨr
(
1T,∆±J
))
= −Adω(k)ω
(
Ψr
(
1T,∆±J
))
= −AdkΨr
((
1T,∆±J
))
.
On the other hand
Ψr
(
ω
(
kT,∆±J
))
= Ψr
(
kT,∆∓J
)
= AdkΨr
((
1T,∆∓J
))
= −AdkΨr
((
1T,∆±J
))
,
where the last equality comes from equation (6.1). The statement about δ∗ is clear from the
definitions. �
Remark 6.8. 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.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 27
7 Special results for the twin building in rank 2
In this section assume that g = g(A) is a rank 2 hyperbolic Kac–Moody Lie algebra, so that
A =
[
2 −a
−b 2
]
, ab > 4.
Rank 2 hyperbolic Kac–Moody algebras were studied intensively by Lepowsky and Moody [45],
by Feingold [17] for the “Fibonacci hyperbolic” (a = b = 3), F ib, by Kang and Melville [40],
by Carbone, Kownacki, Murray and Srinivasan [8] and by Andersen, Carbone and Penta [2].
Twin trees and their relationship to Kac–Moody groups of rank 2 are studied in [57]. 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 provide a detailed construction of the twin building, B =
(
B+,B−, δ∗
)
,
whose simplicial structure is a pair of trees, each equipped with a W -valued distance function,
W -valued codistance function, 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 apartments in the K-orbit
of the fundamental twin apartment, which will provide more details of the embedding of the
twin 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 (3.1), 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 =
〈
w1, w2 |w2
1 = 1 = w2
2
〉 ∼= Z/2Z ∗ Z/2Z ∼= Z o {±1},
which acts on h ⊂ g as well as on hR ⊂ gR and on t ⊂ k. Denote the index 2 maximal infinite
cyclic subgroup of W by
W even =
{
(w2w1)m |m ∈ Z
} ∼= Z
so that for i = 1, 2,
W odd =
{
(w2w1)mwi |m ∈ Z
}
is the other coset, and we have the relations
wi(w2w1)mw−1
i = (w2w1)−m for m ∈ Z, i = 1, 2.
With t = w2w1 and r = wi this gives the presentation of W as the infinite dihedral group
W =
〈
t, r | r2 = 1, rtr−1 = t−1
〉
. 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):
w(n) =
{
(w2w1)m if n = 2m,
(w2w1)mw2 if n = 2m+ 1.
Then we have w1w(n) = w(−1− n) and w2w(n) = w(1− n) for n ∈ Z, and
w(n)−1 =
{
(w2w1)−m if n = 2m,
w2(w2w1)−m if n = 2m+ 1
=
{
w(−n) if n = 2m,
w(n) if n = 2m+ 1.
28 L. Carbone, A.J. Feingold and W. Freyn
It is straightforward to check that for any n, k ∈ Z, we have
w(n)w(k) =
{
w(n+ k) if n = 2m,
w(n− k) if n = 2m+ 1.
We will also find it useful to similarly label certain elements of W̃ by the integers in the same
way:
w̃(n) =
{
(w̃2w̃1)m if n = 2m,
(w̃2w̃1)mw̃2 if n = 2m+ 1.
We define the non-standard partition of the real roots of g, Φre = Φ1 ∪ Φ2 where
Φ1 = W evenα1 ∪W oddα2 = W even{α1,−α2}
and
Φ2 = W evenα2 ∪W oddα1 = W even{−α1, α2}.
See Fig. 1 as an illustration for the “Fibonacci” rank 2 hyperbolic root system, and see Fig. 2
for the rank 2 hyperbolic root system coming from the Cartan matrix A given at the beginning
of Section 7, 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. Imaginary roots of F ib are the blue dots inside the Tits
cone (gray line asymptotes). The two green lines closest to the vertical y-axis are the lines
fixed by w1 (α1(h) = 0) and fixed by w2 (α2(h) = 0), so the fundamental domains C+ and C−
(previously denoted by C±) above and below the x-axis, respectively, are the wedges bounded
by those green lines. Then
⋃
w∈W
w ·C± fills up each component of the Tits cone. In contrast, the
action of W on the real roots has two orbits, and there are elements of each orbit on both left
and right branches. The intersection of each timelike branch tr, r < 0, with the tessellation of
the Tits cone gives the tessellation of each timelike branch into intervals, so we can map a pair
of timelike branches, tr, onto a pair of real lines, A±, each tessellated by unit intervals centered
on the integers. Then the fundamental domain (chamber) in A± is the interval C±0 =
[−1
2 ,
1
2
]
centered on 0, and we label the chamber centered on n ∈ Z by C±n = w(n)C±0 =
[
n− 1
2 , n+ 1
2
]
as shown in Fig. 3. We may label the vertices v±
n+ 1
2
= C±n ∩ C±n+1 (chamber walls) by elements
in Z+ 1
2 . The action of the generators of W on the real lines A± are easily given by the formulas
w1(x) = −1− x and w2(x) = 1− x
so their action on the chambers and vertices of Σ± is given by
w1C
±
n = C±(−1−n), w2C
±
n = C±(1−n), w1v
±
n+ 1
2
= v±−n− 3
2
, w2v
±
n+ 1
2
= v±−n+ 1
2
for any n ∈ Z, so that (w2w1)mC±n = C±(n+2m) and (w2w1)mv±
n+ 1
2
= v±
n+2m+ 1
2
for any m,n ∈ Z.
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 R±ray(n) =
{
C±m |m ≥ n
}
.
The action on rays is
w1L
±
ray(n) = R±ray(−1− n) and w2L
±
ray(n) = R±ray(1− n)
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 29
so for each m,n ∈ Z we have
(w2w1)mL±ray(n) = L±ray(n+ 2m) and (w2w1)mR±ray(n) = R±ray(n+ 2m).
Note that left and right rays in A+ are consistent with the usual orientation of the real line, but
are reversed in A−, so we should think of viewing A− from above, facing down.
Let Cham± =
{
C±m |m ∈ Z
}
be the set of chambers of A±, and define a W -distance function
δ± : Cham± × Cham± → W by δ±
(
wC±, w′C±
)
= w−1w′, so δ±
(
w′C±, wC±
)
= (w′)−1w =
(w−1w′)−1 and
δ±
(
C±m, C
±
n
)
= w(m)−1w(n) =
{
w(n−m) if m is even,
w(m− n) if m is odd.
Now define a W -codistance function δ∗ : Cham± × Cham∓ → W by δ∗
(
wC±, w′C∓
)
= w−1w′,
so
δ∗
(
C±m, C
∓
n
)
= w(m)−1w(n) =
{
w(n−m) if m is even,
w(m− n) if m is odd.
Recall that a codistance function gives an opposition relation by C±m op C∓n when δ∗
(
C±m, C
∓
n
)
=
1, but this is true iff w(m) = w(n) which means m = n. Thus, each chamber C±m in A± is
opposite to precisely one chamber, C∓m in A∓, making A =
(
A+, A−
)
a twin apartment in B.
Let Vert± =
{
v±
m− 1
2
|m ∈ Z
}
be the set of vertices of A±, and define N-valued distance
functions and codistance function
d± : Vert± ×Vert± → N by d±
(
v±
m− 1
2
, v±
n− 1
2
)
= |m− n|,
d∗ : Vert± ×Vert∓ → N by d±
(
v±
m− 1
2
, v∓
n− 1
2
)
= |m− n|.
Note that these functions are consistent with the W -valued functions if we define |w(n)| = |n|
so that
∣∣w(m)−1w(n)
∣∣ = |w(±(m− n))| = |m− n|.
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 real projective space P2(R) (Corol-
lary 4.6).
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 Figs. 1 and 2 those fixed lines are the inner green lines.
30 L. Carbone, A.J. Feingold and W. Freyn
α1 α2
w2α1w1α2
w2w1α2w1w2α1
-α1-α2
-w1α2-w2α1
-w1w2α1-w2w1α2
Φ2Φ1
-6 -4 -2 2 4 6
-10
-5
5
10
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 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
(
B+, B−, N
)
be a twin BN -pair for the complex Kac–Moody group G = GC(A). The
standard parabolic subgroups P±J for J ( {1, 2} are
P±∅ = B±, P±1 = B± tB±w1B
±, and P±2 = B± tB±w2B
±.
As a simplicial complex, the twin building B =
(
B+,B−, δ∗
)
associated to
(
B+, B−, N
)
is a pair
of homogenous P1(C)-trees. The vertices of B are in bijection with the conjugates of P±1 and
P±2 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±1 tG/P±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 P±1 and P±2 :
πi : G/B± −→ G/P±i , i = 1, 2.
If v±i ∈ G/P±i is a vertex, and St
(
v±i
)
= π−1
i (v±i ) is the set of edges with origin v±i , then we may
index St
(
v±i
)
by P±i /B
± ⊆ G/B±, i = 1, 2. It can be seen that P±i /B
± ∼= P1(C) = {∞}∪C. The
W -valued distance and codistance functions defined on the chambers of a twin building can be
converted into N-valued functions by composing with the function |w(n)| = |n|, which coincides
with the length function. So for any two chambers, C± =
(
g1B
±,∆±
)
and D± =
(
g2B
±,∆±
)
in B± we define
d±
(
C±, D±
)
=
∣∣δ±(C,D)
∣∣ and d∗
(
C±, D∓
)
= |δ∗(C,D)|.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 31
α1
α2
-α1
-α2
w2α1
w1α2 w2w1α2
w1w2α1
Φ1 Φ2
-w2α1
-w1α2-w2w1α2
-w1w2α1
-6 -4 -2 2 4 6
-6
-4
-2
2
4
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.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 31
α1
α2
-α1
-α2
w2α1
w1α2 w2w1α2
w1w2α1
Φ1 Φ2
-w2α1
-w1α2-w2w1α2
-w1w2α1
-6 -4 -2 2 4 6
-6
-4
-2
2
4
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.
u u u u u u u u u u u u
u u u u u u u u u u u uC+
0
C−0
C+
1
C−−1
C+
2
C−−2
C+
3
C−−3
C+
4
C−−4
C+
5
C−−5
C+
−1
C−1
C+
−2
C−2
C+
−3
C−3
C+
−4
C−4
C+
−5
C−5
A−
A+
Figure 3. Twin apartment (A+, A−) in twin tree (B+,B−) with chambers C±
n = w(n)C±
0 and vertices
v±
n+ 1
2
= C±
n ∩ C±
n+1.
These distance and codistance functions on chambers may be converted to functions on the
vertices of the twin tree. Ronan and Tits [54] assume that the distance and codistance functions
on vertices exist and obey their axioms. In this description some care must be taken because
a vertex can be in many chambers.
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, C±n = w(n)C±, n ∈ Z, which are
Weyl group translates of the fundamental chamber C± =
(
1B±,∆±∅
)
, which is fixed under the
action of B± ⊃ U±. The boundary vertices of that fundamental chamber are denoted by v±
i− 3
2
=
(
1B±,∆±i
)
for i = 1, 2, corresponding to the maximal parabolic subgroups P±i . By definition,
the vertex v±
i− 3
2
is stabilized by the generator wi ∈W . We can see that chambers C±m and C±n are
adjacent with a common vertex iff |m− n| = 1 because
(
w(m)B±,∆±i
)
∼
(
w(n)B±,∆±i
)
when
w(±(m − n)) = w(m)−1w(n) ∈ P±i iff |m − n| = 1 (w(1) = w2 ∈ P±2 and w(−1) = w1 ∈ P±1 ).
Thus we see the apartment A± as an infinite line:
{
. . . , w1w2w1C±, w1w2C±, w1C±, C±, w2C±, w2w1C±, w2w1w2C±, . . .
}
.
The apartment system A± in B± is the set of apartments obtained from fundamental apartment
A± by the action of G.
Figure 3. Twin apartment
(
A+, A−) in twin tree
(
B+,B−
)
with chambers C±
n = w(n)C±
0 and vertices
v±
n+ 1
2
= C±
n ∩ C±
n+1.
These distance and codistance functions on chambers may be converted to functions on the
vertices of the twin tree. Ronan and Tits [54] assume that the distance and codistance functions
on vertices exist and obey their axioms. In this description some care must be taken because
a vertex can be in many chambers.
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, C±n = w(n)C±, n ∈ Z, which are
Weyl group translates of the fundamental chamber C± =
(
1B±,∆±∅
)
, which is fixed under the
action of B± ⊃ U±. The boundary vertices of that fundamental chamber are denoted by v±
i− 3
2
=
(
1B±,∆±i
)
for i = 1, 2, corresponding to the maximal parabolic subgroups P±i . By definition,
the vertex v±
i− 3
2
is stabilized by the generator wi ∈W . We can see that chambers C±m and C±n are
adjacent with a common vertex iff |m− n| = 1 because
(
w(m)B±,∆±i
)
∼
(
w(n)B±,∆±i
)
when
w(±(m − n)) = w(m)−1w(n) ∈ P±i iff |m − n| = 1 (w(1) = w2 ∈ P±2 and w(−1) = w1 ∈ P±1 ).
Thus we see the apartment A± as an infinite line:
{
. . . , w1w2w1C±, w1w2C±, w1C±, C±, w2C±, w2w1C±, w2w1w2C±, . . .
}
.
The apartment system A± in B± is the set of apartments obtained from fundamental apart-
ment A± by the action of G.
We can describe the action of the real root groups U±(w1w2)mαi , i = 1, 2, m ∈ Z, on the
fundamental apartments A±.
32 L. Carbone, A.J. Feingold and W. Freyn
34 L. Carbone, A.J. Feingold and W. Freyn
u u u u u u u u u u u
u u u u u u u u u u uv+
1/2
v−1/2
v+
3/2
v−−1/2
v+
5/2
v−−3/2
v+
7/2
v−−5/2
v+
9/2
v−−7/2
v+
11/2
v−−9/2
v+
−1/2
v−3/2
v+
−3/2
v−5/2
v+
−5/2
v−7/2
v+
−7/2
v−9/2
v+
−9/2
v−11/2
A−
A+
Figure 4. Twin apartment
(
A+, A−) in twin tree
(
B+,B−
)
with vertices v±
m− 1
2
with the co-distance
function d∗
(
v+
m− 1
2
, v−
n− 1
2
)
= |m− n|.
results should be compared to those of Carbone and Garland [7], where they obtain a spherical
building at infinity, a BN -pair and Bruhat decomposition for complete rank 2 Kac–Moody
groups over finite fields.
A ray in a tree is a sequence of incident 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
a ray. An equivalence class of rays is called an end of the tree, and we denote the end of a ray Xray
by [Xray]. A line in a tree is a sequence of incident 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 is an apartment [X] = [Xray1]∪ [Xray2]
in B±∞ whose two chambers [Xray1] 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 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 .
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 the minimal apartment
system of B± containing A±. (See [1, Section 4.5] for a discussion of complete apartment systems,
and Section 11.8.4 for a discussion of incomplete apartment systems.) A chamber in B±∞ is an
end of an apartment in A±, and Cham±∞ denotes the set of all chambers of B±∞. Of course, this
building is highly degenerate. As a simplicial complex it consists of only 0-simplices, and its
apartments are the subsets of two distinct points corresponding to the two ends of an apartment
in A±. So in this rank 2 case, each apartment is the Coxeter complex consisting of those two
points with order 2 Coxeter group W∞ = W/W even ∼= {±1} ∼= Z/2Z. Since W even acts as
translations in each apartment of B±, it stabilizes the ends, while W odd switches them.
We need to define the distance δ±∞ and codistance δ∗∞. Any two distinct chambers c± =[
X±ray1
]
and d± =
[
X±ray2
]
in Cham±∞ come from a unique line X± = X±ray1∪X±ray2 in B± written
as a union of two rays whose intersection is a vertex X±0 . Let C±1 ∈ X±ray1 and C±2 ∈ X±ray2
Figure 4. Twin apartment
(
A+, A−) in twin tree
(
B+,B−
)
with vertices v±
m− 1
2
with the co-distance
function d∗
(
v+
m− 1
2
, v−
n− 1
2
)
= |m− n|.
Using the notation î = 3− i, we label the roots in Φi by the integers as follows:
Φi(n) =
{
w(n)αi if n = 2m,
w(n)αî if n = 2m+ 1
=
{
(w2w1)mαi if n = 2m,
(w2w1)mw2αî 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:
Φi(n) ∈ Φ+ for
{
n ≤ 0 if i = 1,
n ≥ 0 if i = 2.
We also have w1Φ1(n) = Φ2(1− n), w2Φ1(n) = Φ2(−1− n) and −Φ2(n) = Φ1(n+ 1).
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
eΦi(n) =
{
ew(n)αi if n = 2m,
ew(n)αî
if n = 2m+ 1
=
{
w̃(n)eαi if n = 2m,
w̃(n)eαî if n = 2m+ 1,
and then we would have w̃jeα = ewjα for all real roots α and j = 1, 2. L±ray(n) is fixed by U±Φ2(n)
and R±ray(n) is fixed by U±Φ1(n) for all n ∈ Z, but U±Φ2(n)R
±
ray(n− 1) is a distinct family of rays
in B± indexed by C whose intersection with L±ray(n) is a unique vertex, and U±Φ1(n)L
±
ray(n+1) is
a distinct family of rays in B± indexed by C whose intersection with R±ray(n) is a unique vertex.
Proposition 7.1. The action of real root groups U±Φi(k), k ∈ Z, on the chambers C±(n), in the
fundamental apartments A± satisfies the following.
1. The chambers of L±ray(k) =
{
C±(n) |n ≤ k
}
are each fixed by U±Φ2(k) but U±Φ2(k)R
±
ray(k+1)
is a family of distinct rays in B± indexed by C whose intersection with L±ray(k) is a unique
vertex.
2. The chambers of R±ray(k) =
{
C±(n) |n ≥ k
}
are each fixed by U±Φ1(k) but U±Φ1(k)L
±
ray(k−1)
is a family of distinct rays in B± indexed by C whose intersection with R±ray(k) is a unique
vertex.
Proof. (1) We know that C±(n) = w(n)C± and that for real α ∈ Φ, UαC± = C± when α ∈ Φ±.
So UαC±(n) = C±(n) when w(n)−1Uαw(n)C± = C±, that is, when Uw(n)−1αC± = C±. But that
happens when w(n)−1α ∈ Φ±. Recall that
w(n)−1 =
{
w(−n) if n = 2r,
w(n) if n = 2r + 1
and w(n)w(k) =
{
w(n+ k) if n = 2r,
w(n− k) if n = 2r + 1,
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 33
so for fixed k ∈ Z, any n ≤ k, and
α = ±Φ2(k) =
{
±w(k)α2 if k = 2m,
±w(k)α1 if k = 2m+ 1,
we have
w(n)−1α = ±w(n)−1Φ2(k) = ±
{
w(n)−1w(k)α2 if k = 2m,
w(n)−1w(k)α1 if k = 2m+ 1
= ±
w(−n)w(k)α2 if n = 2r, k = 2m,
w(−n)w(k)α1 if n = 2r, k = 2m+ 1,
w(n)w(k)α2 if n = 2r + 1, k = 2m,
w(n)w(k)α1 if n = 2r + 1, k = 2m+ 1
= ±
w(−n+ k)α2 if n = 2r, k = 2m,
w(−n+ k)α1 if n = 2r, k = 2m+ 1,
w(n− k)α2 if n = 2r + 1, k = 2m,
w(n− k)α1 if n = 2r + 1, k = 2m+ 1
= ±
Φ2(k − n) if n = 2r, k = 2m,
Φ2(k − n) if n = 2r, k = 2m+ 1,
Φ1(n− k) if n = 2r + 1, k = 2m,
Φ1(n− k) if n = 2r + 1, k = 2m+ 1
∈ ±Φ+ = Φ±,
since
Φi(s) ∈ Φ+ for
{
s ≤ 0 if i = 1,
s ≥ 0 if i = 2.
For α = ±Φ2(k) and t ∈ C let g(t) = exp(adteα) ∈ Uα and consider the family of rays
UαR
±
ray(k + 1) =
{
g(t)R±ray(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 gw = w−1gw, so we have g(t1)wC± = g(t2)wC± which gives
C± = (g(t1)w)−1g(t2)wC± = g(−t1)wg(t2)wC± = g(−t1 + t2)wC±. Therefore, g(−t1 + t2)w ∈ B±.
But for any t ∈ C, we have g(t)w ∈ Uw−1α. Since
w−1 = w(k + 1)−1 =
{
w(−k − 1) if k is odd,
w(k + 1) if k is even,
we get
w−1α = ±
{
w(−k − 1)w(k)α1 if k is odd,
w(k + 1)w(k)α2 if k is even
= ±
{
w(−1)α1 if k is odd,
w(1)α2 if k is even
= ±
{
w1α1 if k is odd,
w2α2 if k is even
= ±
{
−α1 if k is odd,
−α2 if k is even,
so w−1α ∈ −Φ± which means Uw−1α ≤ 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. �
34 L. Carbone, A.J. Feingold and W. Freyn
8 Embedding the spherical building at infinity in rank 2
Let A be a rank 2 hyperbolic Cartan matrix as in Section 7, but with the additional conditions
that a > 1 and b > 1 because otherwise the real root groups Ui defined below (in Definition 8.4)
may not be abelian (see [8, 49]). Let G = GC(A) be a rank 2 hyperbolic Kac–Moody group with
compact real form K. Let B =
(
B+,B−, δ∗
)
denote its twin building, associated with a twin
BN -pair
(
B+, B−, N
)
, whose simplicial structure is a pair of trees, and whose apartments are
lines which are infinite in both directions. We will now define the spherical twin building at
infinity, B∞ =
(
B+
∞,B−∞, δ∗∞
)
even though we do not obtain a twin BN -pair at infinity. Our
results should be compared to those of Carbone and Garland [7], where they obtain a spherical
building at infinity, a BN -pair and Bruhat decomposition for complete rank 2 Kac–Moody
groups over finite fields.
A ray in a tree is a sequence of incident 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
a ray. An equivalence class of rays is called an end of the tree, and we denote the end of a ray Xray
by [Xray]. A line in a tree is a sequence of incident 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 is an apartment [X] = [Xray1]∪ [Xray2]
in B±∞ whose two chambers [Xray1] 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 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 .
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 the minimal apartment
system of B± containing A±. (See [1, Section 4.5] for a discussion of complete apartment systems,
and Section 11.8.4 for a discussion of incomplete apartment systems.) A chamber in B±∞ is an
end of an apartment in A±, and Cham±∞ denotes the set of all chambers of B±∞. Of course, this
building is highly degenerate. As a simplicial complex it consists of only 0-simplices, and its
apartments are the subsets of two distinct points corresponding to the two ends of an apartment
in A±. So in this rank 2 case, each apartment is the Coxeter complex consisting of those two
points with order 2 Coxeter group W∞ = W/W even ∼= {±1} ∼= Z/2Z. Since W even acts as
translations in each apartment of B±, it stabilizes the ends, while W odd switches them.
We need to define the distance δ±∞ and codistance δ∗∞. Any two distinct chambers c± =[
X±ray1
]
and d± =
[
X±ray2
]
in Cham±∞ come from a unique line X± = X±ray1∪X±ray2 in B± written
as a union of two rays whose intersection is a vertex X±0 . Let C±1 ∈ X±ray1 and C±2 ∈ X±ray2
be the initial chambers in these rays whose intersection is X±0 . Then we define the W∞-valued
distance function by
δ±∞
(
c±, d±
)
= δ±
(
C±1 , C
±
2
)
W even =
{
−1 if c± 6= d±,
1 if c± = d±.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 35
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 35
u u u u u
ux±
y0 y1 y2 y3 yn
· · ·
n n−1 n−2 n−3 · · · 0
B∓
B±
d∗(x±, yi)
Figure 5. Co-distances d∗(x±, yi) = n− i between x± ∈ B± and a path of adjacent vertices in B∓.
distance function by
δ±∞
(
c±, d±
)
= δ±
(
C±1 , C
±
2
)
W even =
{
−1 if c± 6= d±,
1 if c± = d±.
In order to use the codistance function δ∗ in B to get a codistance function δ∗∞, we need the
concept of twin ends.
Let x, y, z be chambers in B±. Then the N-valued distance function defined in Section 7
satisfies the following properties.
1. d±(x, y) = 0 iff x = y.
2. If d±(y, z) = 1 we have d±(x, z) = d(x, y)± ± 1 so d±(x, z) 6= d±(x, y). Also, there is at
most one z with d±(y, z) = 1 and d±(x, z) = d±(x, y)− 1.
3. For any two distinct x, y, there exists z with d±(y, z) = 1 and d±(x, z) = d±(x, y) + 1.
Let x± be a chamber in B± and let y∓ and z∓ be chambers in B∓. Then the N-valued codistance
function defined in Section 7 satisfies the following properties.
1. If d∓(y∓, z∓) = 1 then d∗(x±, z∓) = d∗(x±, y∓)± 1.
2. Additionally, if d∗(x±, y∓) 6= 0 then there exists a unique z∓ with d∓(y∓, z∓) = 1 and
d∗(x±, z∓) = d∗(x±, y∓) + 1.
Let A± be an apartment in B±, and let Cham±
(
A±
)
be the set of chambers of A±.
(
A+, A−
)
is a twin apartment in the twin tree
(
B+,B−
)
iff for every x± ∈ Cham±(A±) there exists a unique
x∓ ∈ Cham∓
(
A∓
)
such that d∗(x±, x∓) = 0.
Observations: Fix some x± ∈ Cham±
(
A±
)
and let (y0, y1, y2, . . . , yn) be a path of adjacent
chambers in B∓ with d∗(x±, y1) = d∗(x±, y0) − 1 then d∗(x±, yi) = d∗(x±, y0) − i for 0 ≤ i ≤
d∗(x±, y0). (See Fig. 5.) Consequently, for a twin apartment,
(
A+, A−
)
, if x± ∈ Cham±(A±)
and (y0, y1, y2, . . . , yn) is a path of adjacent chambers in A∓ with d∗(x±, y1) = d∗(x±, y0) + 1 for
d∗(x±, y0) > 0 then d∗(x±, yi) = d∗(x±, y0) + i for 0 ≤ i ≤ n.
For x ∈ Cham±(A±) and y ∈ Cham∓
(
A∓
)
with d∗(x, y) = n > 0, there is a twin end
determined as follows. Using x = x± and y = y0 as above, there is a ray r∓ of adjacent chambers
(y0, y1, y2, y3, . . . ) in Cham∓
(
A∓
)
with d∗(x, yi) = n + i for 0 ≤ i, and similarly there is also a
ray r± of adjacent chambers (x0, x1, x2, x3, . . . ) in Cham±
(
A±
)
with x0 = x and d∗(xi, y) = n+i
for 0 ≤ i. These two rays determine two ends, [r±] and [r∓], and we call the pair ([r+], [r−])
the twin end associated with (x, y). In this twin tree case, it is known [53, Chapter 11, twin
buildings] that there exists a twin apartment
(
A+, A−
)
with x ∈ A+ and y ∈ A− and then we
have r± ⊂ A±.
For c± ∈ Cham±∞ and d∓ ∈ Cham∓∞ we define the W∞-valued codistance function by
δ∗∞(c±, d∓) =
{
−1 if c± and d∓ are twin ends
1 otherwise
.
Figure 5. Co-distances d∗(x±, yi) = n− i between x± ∈ B± and a path of adjacent vertices in B∓.
In order to use the codistance function δ∗ in B to get a codistance function δ∗∞, we need the
concept of twin ends.
Let x, y, z be chambers in B±. Then the N-valued distance function defined in Section 7
satisfies the following properties.
1. d±(x, y) = 0 iff x = y.
2. If d±(y, z) = 1 we have d±(x, z) = d(x, y)± ± 1 so d±(x, z) 6= d±(x, y). Also, there is at
most one z with d±(y, z) = 1 and d±(x, z) = d±(x, y)− 1.
3. For any two distinct x, y, there exists z with d±(y, z) = 1 and d±(x, z) = d±(x, y) + 1.
Let x± be a chamber in B± and let y∓ and z∓ be chambers in B∓. Then the N-valued codistance
function defined in Section 7 satisfies the following properties.
1. If d∓(y∓, z∓) = 1 then d∗(x±, z∓) = d∗(x±, y∓)± 1.
2. Additionally, if d∗(x±, y∓) 6= 0 then there exists a unique z∓ with d∓(y∓, z∓) = 1 and
d∗(x±, z∓) = d∗(x±, y∓) + 1.
Let A± be an apartment in B±, and let Cham±
(
A±
)
be the set of chambers of A±.
(
A+, A−
)
is a twin apartment in the twin tree
(
B+,B−
)
iff for every x± ∈ Cham±(A±) there exists a unique
x∓ ∈ Cham∓
(
A∓
)
such that d∗(x±, x∓) = 0.
Observations: Fix some x± ∈ Cham±
(
A±
)
and let (y0, y1, y2, . . . , yn) be a path of adjacent
chambers in B∓ with d∗(x±, y1) = d∗(x±, y0) − 1 then d∗(x±, yi) = d∗(x±, y0) − i for 0 ≤ i ≤
d∗(x±, y0). (See Fig. 5.) Consequently, for a twin apartment,
(
A+, A−
)
, if x± ∈ Cham±(A±)
and (y0, y1, y2, . . . , yn) is a path of adjacent chambers in A∓ with d∗(x±, y1) = d∗(x±, y0) + 1 for
d∗(x±, y0) > 0 then d∗(x±, yi) = d∗(x±, y0) + i for 0 ≤ i ≤ n.
For x ∈ Cham±(A±) and y ∈ Cham∓
(
A∓
)
with d∗(x, y) = n > 0, there is a twin end
determined as follows. Using x = x± and y = y0 as above, there is a ray r∓ of adjacent
chambers (y0, y1, y2, y3, . . . ) in Cham∓
(
A∓
)
with d∗(x, yi) = n + i for 0 ≤ i, and similarly
there is also a ray r± of adjacent chambers (x0, x1, x2, x3, . . . ) in Cham±
(
A±
)
with x0 = x and
d∗(xi, y) = n+i for 0 ≤ i. These two rays determine two ends, [r±] and [r∓], and we call the pair
([r+], [r−]) the twin end associated with (x, y). In this twin tree case, it is known [53, Chapter 11,
twin buildings] that there exists a twin apartment
(
A+, A−
)
with x ∈ A+ and y ∈ A− and then
we have r± ⊂ A±.
For c± ∈ Cham±∞ and d∓ ∈ Cham∓∞ we define the W∞-valued codistance function by
δ∗∞(c±, d∓) =
{
−1 if c± and d∓ are twin ends,
1 otherwise.
This means that for each chamber c ∈ Cham±∞ there is exactly one chamber d ∈ Cham∓∞ with
codistance δ∗(c, d) = −1, while all other chambers d′ ∈ Cham∓∞ have codistance δ∗(c, d′) = 1.
This is consistent with Lemma 2.6.
36 L. Carbone, A.J. Feingold and W. Freyn
We denote all objects associated to the spherical building at infinity, i.e., apartments, cham-
bers or the Weyl group, with the subscript ‘∞’. Thinking of apartments of B∞ as the ends of
apartments of B, our goal now is to prove that the embedding of the twin building
(
B+,B−, δ∗
)
,
given in Theorem 6.1 induces an embedding of the spherical building at infinity
(
B+
∞,B−∞, δ∗∞
)
into a set of rays on the null cones of t0.
In the rank 2 case, the compact real form of the Cartan subalgebra, t, has 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≤0tr 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 L0
t = {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 , x
−
i | i = 1, 2
}
. This corresponds to the “lightlike
closure” B∞(∂Lt) in Definition 4.4. To be more precise, x±i denotes the ray in t±0 such that
x−2 = −x+
1 , x−1 = −x+
2 and
(
x+
1 , α1
)
< 0,
(
x+
1 , α2
)
> 0,
(
x+
2 , α1
)
> 0,
(
x+
2 , α2
)
< 0.
This means that in Fig. 1, x+
1 is the ray going up to the right, and x+
2 is the ray going up to the
left, while x−1 is the ray going down to the right, and x−2 is the ray going down to the left. We
think of
{
x±1 , x
±
2
}
as the ends of the fundamental apartment A± in B±, so as the fundamental
apartment A±∞ in B±∞. As a Coxeter complex for W∞, each apartment consists of just two points
which are exchanged by the nontrivial element of W∞, and the W∞-valued distance function is
obvious.
Based on the labeling of the chambers of the fundamental twin apartment
(
A+, A−
)
shown in
Fig. 3, and the codistance function δ∗, we see that the codistance function on the fundamental
twin apartment (A+
∞, A
−
∞) is
δ∗∞
(
x+
i , x
−
j
)
=
{
−1 if i = j,
1 if i 6= j.
Define the halos, positive and negative, of g to be the union of all K conjugates of
{
x±i | i =
1, 2
}
Ξ±∞ =
{
kx±i k
−1 | k ∈ K, i = 1, 2
}
and let the twin halo of g be Ξ∞ = Ξ+
∞ ∪ Ξ−∞. This will be where we embed the spherical twin
building at infinity B∞ in the next theorem.
Before we can state and prove the embedding theorem, we must define the distance func-
tion δ±∞ on Ξ±∞ and the codistance function δ∗∞ on Ξ∞.
For any two chambers k1x
±
i k
−1
1 and k2x
±
j k
−1
2 in Ξ±∞ define the W∞-valued distance function
δ±∞(k1x
±
i k
−1
1 , k2x
±
j k
−1
2 ) =
{
1 if i = j and k1 = k2,
−1 otherwise.
and the codistance function
δ∗∞(k1x
±
i k
−1
1 , k2x
∓
j k
−1
2 ) =
{
−1 if i = j and k1 = k2,
1 otherwise.
Note that
(
Ξ±∞, δ
±
∞
)
is a Tits building with an apartment system determined by δ±∞, and(
Ξ+
∞,Ξ
−
∞, δ
∗
∞
)
is a spherical twin building. We choose x+
1 to be the fundamental chamber
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 37
in A+
∞ and x−2 to be the opposite fundamental chamber in A−∞ with codistance δ∗∞
(
x+
1 , x
−
2
)
= 1,
consistent with the behavior of the codistance δ∗ in B.
Theorem 8.1. There is a K-equivariant bijective twin building map Ψ∞ : B∞ → Ξ∞ respecting
distance and codistance functions δ±∞ and δ∗∞, such that the following diagram commutes:
B∞
g∈K //
Ψ∞
��
B∞
Ψ∞
��
Ξ∞
Adg // Ξ∞
Remark 8.2. 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. We use results of Ronan–Tits [54]
about the stabilizers of twin ends.
Recall that A± denotes the fundamental apartment in B± so it is a line in that tree with
two ends. (See Figs. 3 and 4.) 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 7.1, for each
i = 1, 2, m ∈ Z, the subgroup U(w1w2)mαi fixes the end e±i,∞ if we define
e+
1,∞ = [R+
ray(m)], e+
2,∞ = [L+
ray(m)],
e−1,∞ = [L−ray(m)], e−2,∞ = [R−ray(m)].
Furthermore, we have shown that U−(w1w2)mα1
fixes e±2,∞ and U−(w1w2)mα2
fixes e±1,∞. This
means that ω(e+
1,∞) = e−2,∞ and ω(e+
2,∞) = e−1,∞.
Let us fix some notation, defining the stabilizer of e±i,∞
Definition 8.3. The stabilizer of an end e±i,∞ is the group
B±i,∞ =
{
g ∈ G | g · e±i,∞ = e±i,∞
}
.
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 8.4. For i = 1, 2, we set Ui = 〈Uα |α ∈ Φi〉 and Bi = TW evenUi.
We now have the following result.
Proposition 8.5. For i = 1, 2 we have Bi ≤ B±i,∞.
Proof. For any m ∈ Z, we have established that U(w1w2)mα1
and U−(w1w2)mα2
are in B±1,∞, and
that U(w1w2)mα2
and U−(w1w2)mα1
are in B±2,∞, so Ui ≤ B±i,∞. It is clear that TW even ≤ B±i,∞, so
we get Bi ≤ B±i,∞. �
We would like to make some remarks about why (B1, B2, N) is not a twin BN pair, and
therefore why we have no Bruhat decomposition for the twin building at infinity. There are
three requirements to be a BN -pair, T1–T3, in Definition 2.2, and three more requirements to
be a twin BN -pair, TW1–TW3, in Definition 2.3. The condition T1 requires that G = 〈Bi, N〉,
38 L. Carbone, A.J. Feingold and W. Freyn
38 L. Carbone, A.J. Feingold and W. Freyn
H = Bi ∩ N � N , W∞ = N/H = 〈S〉 for i = 1, 2. Since wi ∈ N and wiBiw
−1
i = B3−i
and G = 〈B1, B2〉, we have G = 〈Bi, N〉. We also have H = TW evenUi ∩ TW = TW even so
N/H = (TW )/(TW even) = W∞. Condition T3 requires that the nontrivial element in W∞,
represented by either w1 or w2, satisfies wiBjw
−1
i 6⊆ Bj , and that is true since B3−j 6⊆ Bj .
However, condition T2 requires that for i, j, k ∈ {1, 2} we have wiBkwj ⊆ BkwiwjBk ∪BkwjBk,
so in particular, T2 requires that B2 = w1B1w1 ⊆ B1∪B1w1B1. If this were true it would mean
that U2 = w1U1w1 ⊆ U1 ∪ U1w1U1 which would imply U1 ⊆ U2 ∪ w1U1U2. Since U1 ∩ U2 = 1,
this would require U1 ⊆ w1U1U2 and then w1U1 ⊆ U1U2 so w1 ∈ U1U2. Certainly we know
from the usual formula that w1 ∈ U1U2U1, but there is no expression for w1 in U1U2. The only
case of concern in TW2 is when s = w1 = w, so that `(sw) = `(1) = 0 < `(w1) = 1, and the
requirement is that B1w1B1w1B2 = B1B2, which is true since w1B1w1 = B2. Concerning TW3,
since we know w1B1 = B2w1 we get that B2w1∩B1 = w1B1∩B1 is the empty set since w1 /∈ B1
so these are distinct left cosets of B1.
We thank Peter Abramenko for his exposition of the following results of Ronan-Tits about
twin trees, and how they can be applied to achieve our goal in this section.
We now discuss real roots and real root groups from the point of view of twin buildings.
Let (B+,B−, d∗) be a twin building with codistance d∗, and fix a fundamental twin apartment
A0 = (A+
0 , A
−
0 ). A root α± of B± is a half-apartment of A±0 , and Φ± is the set of all roots
of A±0 . For each choice of a root α±, denote by −α± the other half-apartment such that
−α ∪ α = A±0 and −α ∩ α is exactly one panel. The use of superscript ± to distinguish roots
(half-apartments) in the two buildings is therefore not always consistent with the choice of a
factor of ±1 according to choice of building. From the Lie algebra point of view, there is just
one set of roots, Φ, but the action of a root group Uα for α ∈ Φ depends on the building B±. For
the twin tree B = (B+,B−, δ∗) we are studying in this section, if α = ±Φ2(k), the corresponding
half-apartment in B± would be L±ray(k) which is fixed by the real root group Uα according to
Proposition 7.1, and −α = ±Φ1(k+1) corresponds to the half-apartment R±ray(k+1) in B± which
is fixed by the real root group U−α. We have L±ray(k) ∪ R±ray(k + 1) = A±0 , and the intersection
L±ray(k) ∩R±ray(k + 1) is the single vertex v±
k+ 1
2
= C±(k) ∩ C±(k + 1).
u u u u
u u u ux+ y+
y− x− z−
z+
C−
C+
A−
A+
Figure 6. Twin apartment (A+, A−) in twin tree (B+,B−) with chambers C+ op C− and ver-
tices x+ op x− and y+ op y−.
As shown in Fig. 6, where C± = (x±, y±) and C+ op C− with d∗(x+, x−) = 0 = d∗(y+, y−),
so d∗(x+, y−) = 1 = d∗(x−, y+). Then there is a vertex z− adjacent to x− with d∗(y+, z−) = 2
and continuing along a ray in that direction the co-distance function d∗(y+, ·) is increasing.
There is also a vertex z+ adjacent to y+ with d∗(x−, z+) = 2 and continuing in a ray in that
direction the co-distance function d∗(x−, ·) is increasing. For any pair of opposite chambers,
C+ op C−, there exists a twin apartment (A+, A−) with C± ∈ A±.
Let V ± = V ±(T ±) be the set of vertices of the tree T ± = B±. Let x ∈ V + and y ∈ V − with
d∗(x, y) = n > 0, so we get rays
r+ of adjacent vertices (x = x0, x1, x2, x3, . . . ) with end e+ and
r− of adjacent vertices (y = y0, y1, y2, y3, . . . ) with end e−
Figure 6. Twin apartment (A+, A−) in twin tree (B+,B−) with chambers C+ op C− and ver-
tices x+ op x− and y+ op y−.
H = Bi ∩ N � N , W∞ = N/H = 〈S〉 for i = 1, 2. Since wi ∈ N and wiBiw
−1
i = B3−i
and G = 〈B1, B2〉, we have G = 〈Bi, N〉. We also have H = TW evenUi ∩ TW = TW even so
N/H = (TW )/(TW even) = W∞. Condition T3 requires that the nontrivial element in W∞,
represented by either w1 or w2, satisfies wiBjw
−1
i 6⊆ Bj , and that is true since B3−j 6⊆ Bj .
However, condition T2 requires that for i, j, k ∈ {1, 2} we have wiBkwj ⊆ BkwiwjBk ∪BkwjBk,
so in particular, T2 requires that B2 = w1B1w1 ⊆ B1∪B1w1B1. If this were true it would mean
that U2 = w1U1w1 ⊆ U1 ∪ U1w1U1 which would imply U1 ⊆ U2 ∪ w1U1U2. Since U1 ∩ U2 = 1,
this would require U1 ⊆ w1U1U2 and then w1U1 ⊆ U1U2 so w1 ∈ U1U2. Certainly we know
from the usual formula that w1 ∈ U1U2U1, but there is no expression for w1 in U1U2. The only
case of concern in TW2 is when s = w1 = w, so that `(sw) = `(1) = 0 < `(w1) = 1, and the
requirement is that B1w1B1w1B2 = B1B2, which is true since w1B1w1 = B2. Concerning TW3,
since we know w1B1 = B2w1 we get that B2w1∩B1 = w1B1∩B1 is the empty set since w1 /∈ B1
so these are distinct left cosets of B1.
We thank Peter Abramenko for his exposition of the following results of Ronan-Tits about
twin trees, and how they can be applied to achieve our goal in this section.
We now discuss real roots and real root groups from the point of view of twin buildings.
Let
(
B+,B−, d∗
)
be a twin building with codistance d∗, and fix a fundamental twin apartment
A0 =
(
A+
0 , A
−
0
)
. A root α± of B± is a half-apartment of A±0 , and Φ± is the set of all roots
of A±0 . For each choice of a root α±, denote by −α± the other half-apartment such that
−α ∪ α = A±0 and −α ∩ α is exactly one panel. The use of superscript ± to distinguish roots
(half-apartments) in the two buildings is therefore not always consistent with the choice of
a factor of ±1 according to choice of building. From the Lie algebra point of view, there is just
one set of roots, Φ, but the action of a root group Uα for α ∈ Φ depends on the building B±. For
the twin tree B = (B+,B−, δ∗) we are studying in this section, if α = ±Φ2(k), the corresponding
half-apartment in B± would be L±ray(k) which is fixed by the real root group Uα according to
Proposition 7.1, and −α = ±Φ1(k+1) corresponds to the half-apartment R±ray(k+1) in B± which
is fixed by the real root group U−α. We have L±ray(k) ∪ R±ray(k + 1) = A±0 , and the intersection
L±ray(k) ∩R±ray(k + 1) is the single vertex v±
k+ 1
2
= C±(k) ∩ C±(k + 1).
As shown in Fig. 6, where C± = (x±, y±) and C+ op C− with d∗(x+, x−) = 0 = d∗(y+, y−),
so d∗(x+, y−) = 1 = d∗(x−, y+). Then there is a vertex z− adjacent to x− with d∗(y+, z−) = 2
and continuing along a ray in that direction the co-distance function d∗(y+, ·) is increasing.
There is also a vertex z+ adjacent to y+ with d∗(x−, z+) = 2 and continuing in a ray in that
direction the co-distance function d∗(x−, ·) is increasing. For any pair of opposite chambers,
C+ op C−, there exists a twin apartment (A+, A−) with C± ∈ A±.
Let V ± = V ±(T ±) be the set of vertices of the tree T ± = B±. Let x ∈ V + and y ∈ V − with
d∗(x, y) = n > 0, so we get rays
r+ of adjacent vertices (x = x0, x1, x2, x3, . . . ) with end e+ and
r− of adjacent vertices (y = y0, y1, y2, y3, . . . ) with end e−,
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 39
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 39
where d∗(x, yj) = n+ j for all j ≥ 0 and d∗(xi, y) = n+ i for all i ≥ 0 uniquely determines the
vertices xi and yj . This gives d∗(xi, yj) = n + i + j for all i, j ≥ 0. The pair of ends (e+, e−)
determined by this process is called a twin end. Using the N-valued codistance between vertices
in twin trees and the N-valued codistance between chambers (edges), it seems clear that for any
chambers C+
i = [xi, xi+1] in the ray [x, e+) and C−j = [yj , yj+1] in the ray [y, e−), the chamber
codistance δ∗(C+
i , C
−
j ) > 0, so no such pair of chambers is opposite.
The following fact was proved by Ronan–Tits [54, Proposition 3.4].
Proposition 8.6 ([54]). Let x ∈ V + and y ∈ V − with d∗(x, y) = n > 0 determine rays r+ and
r− with ends e+ and e−. Let x′ ∈ V + and y′ ∈ V − be another pair of non-opposite vertices
(so d∗(x′, y′) > 0), with another pair of rays r′+ and r′− with ends e′+ and e′−. If e′+ = e+ then
e′− = e−.
Corollary 8.7. Let Γ be any group acting on the twin tree (T +, T −, d∗) preserving d∗. If we
have a twin end (e+, e−) and if g ∈ Γ with g · e+ = e+ then g · e− = e−.
Proof. With notation as above, let g · (x, y) = (g ·x, g ·y) = (x′, y′). We can apply g to the rays
r+ and r−, to get rays r′+ and r′−, which consist of the vertices g ·xi and g ·yi uniquely determined
by x′ and y′ because g ∈ Γ preserves d∗. But then we must have g · (e+, e−) = (g · e+, g · e−).
We are given that g · e+ = e+, so Proposition 8.6 gives g · e− = e−. �
We have G = T 〈Uα |α ∈ Φ〉 is generated by its (real) root groups, Uα and the torus T =
FixG(A+
0 , A
−
0 ) that fixes (chamber-wise) the fundamental twin apartment A0 = (A+
0 , A
−
0 ) in the
twin tree. In the tree, B±, a root α± means a ray in the fundamental apartment A±0 , and each
root α+ (ray) in A+
0 determines a root α− (ray) in A−0 , giving a twin root α = (α+, α−), a pair
of rays. That root α− is the negative of the root (ray) in A−0 whose edges are the opposites
of the edges of α+. This means that α− is the maximal ray in A−0 such that no edge of α− is
opposite to any edge of a+. (See Fig. 7.)
u u u u
u u u ux+
x− D−C−
C+D+
α− =
[
x−, e1
−
)
α+ =
[
x+, e
1
+
)
e1
−
e1
+
e2
−
e2
+
A−
A+
Figure 7. Twin root α = (α+, α−) in twin apartment (A+, A−) determined by adjacent chambers C+
and D+ with C+ ∩D+ = x+ and C+ ∈ α+, so α− = −opα+. Also showing twin ends (e1+, e
1
−) of α.
Uα acts transitively on the set of chambers (edges) of T + containing x+ different from the
chamber C+ in fixed ray α+.
We have T = ∩α∈ΦNG(Uα), and for i = 1, 2, we have defined ends e±i,∞ as equivalence classes
of rays, and for the following proof we use the notation ei± for e±i,∞, so that
Ui = 〈Uα |α = (α+, α−), α+ represents end ei+〉
and
T ≤ D = TW even = {g ∈ G | g acts by translation on A+
0 } ≤ StabG(A+
0 ) = N.
The following theorem gives a precise description of B±i,∞ = StabG(ei±), but is written only for
the case of i = 1 and ± = +. The result shows that the answer is independent of the choice of
±.
Figure 7. Twin root α = (α+, α−) in twin apartment
(
A+, A−) determined by adjacent chambers C+
and D+ with C+ ∩D+ = x+ and C+ ∈ α+, so α− = −opα+. Also showing twin ends
(
e1+, e
1
−
)
of α.
where d∗(x, yj) = n+ j for all j ≥ 0 and d∗(xi, y) = n+ i for all i ≥ 0 uniquely determines the
vertices xi and yj . This gives d∗(xi, yj) = n + i + j for all i, j ≥ 0. The pair of ends (e+, e−)
determined by this process is called a twin end. Using the N-valued codistance between vertices
in twin trees and the N-valued codistance between chambers (edges), it seems clear that for any
chambers C+
i = [xi, xi+1] in the ray [x, e+) and C−j = [yj , yj+1] in the ray [y, e−), the chamber
codistance δ∗(C+
i , C
−
j ) > 0, so no such pair of chambers is opposite.
The following fact was proved by Ronan–Tits [54, Proposition 3.4].
Proposition 8.6 ([54]). Let x ∈ V + and y ∈ V − with d∗(x, y) = n > 0 determine rays r+
and r− with ends e+ and e−. Let x′ ∈ V + and y′ ∈ V − be another pair of non-opposite vertices
(so d∗(x′, y′) > 0), with another pair of rays r′+ and r′− with ends e′+ and e′−. If e′+ = e+ then
e′− = e−.
Corollary 8.7. Let Γ be any group acting on the twin tree (T +, T −, d∗) preserving d∗. If we
have a twin end (e+, e−) and if g ∈ Γ with g · e+ = e+ then g · e− = e−.
Proof. With notation as above, let g·(x, y) = (g·x, g·y) = (x′, y′). We can apply g to the rays r+
and r−, to get rays r′+ and r′−, which consist of the vertices g · xi and g · yi uniquely determined
by x′ and y′ because g ∈ Γ preserves d∗. But then we must have g · (e+, e−) = (g · e+, g · e−).
We are given that g · e+ = e+, so Proposition 8.6 gives g · e− = e−. �
We have G = T 〈Uα |α ∈ Φ〉 is generated by its (real) root groups, Uα and the torus T =
FixG
(
A+
0 , A
−
0
)
that fixes (chamber-wise) the fundamental twin apartment A0 =
(
A+
0 , A
−
0
)
in the
twin tree. In the tree, B±, a root α± means a ray in the fundamental apartment A±0 , and each
root α+ (ray) in A+
0 determines a root α− (ray) in A−0 , giving a twin root α = (α+, α−), a pair
of rays. That root α− is the negative of the root (ray) in A−0 whose edges are the opposites
of the edges of α+. This means that α− is the maximal ray in A−0 such that no edge of α− is
opposite to any edge of a+. (See Fig. 7.)
Uα acts transitively on the set of chambers (edges) of T + containing x+ different from the
chamber C+ in fixed ray α+.
We have T = ∩α∈ΦNG(Uα), and for i = 1, 2, we have defined ends e±i,∞ as equivalence classes
of rays, and for the following proof we use the notation ei± for e±i,∞, so that
Ui = 〈Uα |α = (α+, α−), α+ represents end ei+〉
and
T ≤ D = TW even =
{
g ∈ G | g acts by translation on A+
0
}
≤ StabG
(
A+
0
)
= N.
The following theorem gives a precise description of B±i,∞ = StabG
(
ei±
)
, but is written only for
the case of i = 1 and ± = +. The result shows that the answer is independent of the choice
of ±.
40 L. Carbone, A.J. Feingold and W. Freyn
Theorem 8.8. We have StabG
(
e1
+
)
= DU1 = B1.
Proof. We already have the containment DU1 ≤ StabG
(
e1
+
)
. Let g ∈ G with g
(
e1
+
)
= e1
+, so
for any ray r+ in A+
0 with end e1
+, the ray g(r+) also has end e1
+. It means there is a subray
r′+ ⊂ r+ such that g(r′+) ⊆ r+ ⊂ A+
0 . If necessary, we may adjust the choice of r+ by adding or
deleting one chamber at its beginning, so that there is a translation d ∈ D with dg(r′+) = r′+.
If we replace r+ by r′+ and replace g by g′ = dg then g′ fixes r′+ pointwise. Without loss of
generality, we now assume that g ∈ StabG
(
e1
+
)
and we have a ray r+ =
[
y+, e
1
+
)
in A+
0 such
that g(x) = x for every vertex x ∈ r+. That means the end e1
+ of r+ is fixed by g so g
(
e1
−
)
= e1
−
by Corollary 8.7 since g preserves the twin structure. This now implies the existence of a ray
r− = [z−, e1
−) in A−0 with end e1
− which is pointwise fixed by g. If r− were not pointwise fixed
by g, then z− 6= g(z−) ∈ r− so δ∗(y+, g(z−)) > δ∗(y+, z−), but δ∗(y+, z−) = δ∗(g(y+), g(z−)) =
δ∗(y+, g(z−)) since g(y+) = y+.
Case (1): The pair (r+, r−) is not contained in any twin root (α+, α−) of A0 =
(
A+
0 , A
−
0
)
so
there are chambers C± ∈ r± with C+ op C−. Then we have g
(
C+
)
= C+ and g
(
C−
)
= C−
which implies that g fixes the unique twin apartment
(
A+
0 , A
−
0
)
containing both C+ and C−,
and that implies g ∈ FixG
(
A+
0 , A
−
0
)
= T .
Case (2): The pair (r+, r−) is contained in some twin root α = (α+, α−) of A0 =
(
A+
0 , A
−
0
)
.
Let x± be the starting point of ray (root) α± where x+ op x− (that is, d∗(x+, x−) = 0). We
have r+ ⊆ α+ and r− ⊆ α−. Let y+ be the starting point of ray r+ and suppose y− op y+. Then
we have the twin root β = (β+, β−) where β+ = r+ and β− has starting point y− so r− ⊆ β−.
Here we have C− is the first chamber of α− with starting vertex x−, and C+ is the opposite
chamber which is adjacent to the first chamber of α+ with starting vertex x+. (See Fig. 8.)
Our goal is to modify g to g′ by multiplying with elements from U1 such that g′ · C− = C−
and g′ · C+ = C+, which would imply that g′ ∈ T . Let (C1, C2, . . . , Cm−1, Cm) be a sequence
of adjacent edges in A+
0 with Ci = [xi−1, xi] for 1 ≤ i ≤ m and x0 = x+, forming the interval
[x+, y+] so that Cm ∩ r+ = y+ = xm. Then (g · C1, g · C2, . . . , g · Cm−1, g · Cm) is a sequence
of adjacent edges in T + and g · Cm is attached to vertex y+ = xm since g fixes r+ pointwise.
For 0 ≤ i ≤ m define the ray (root) βi =
[
xi, e
1
+
)
so that Uβi ≤ U1 fixes βi pointwise and acts
transitively on the set of all edges attached to xi. There is an element um ∈ Uβm such that
(umg) · Cm = um · (g · Cm) = Cm so umg fixes the ray βm−1 = Cm ∪ r+ pointwise. Then there
is an element um−1 ∈ Uβm−1 such that (um−1umg) · Cm−1 = Cm−1 so um−1umg fixes the ray
βm−2 = Cm−1∪Cm∪r+ pointwise. Continuing this way, we find a sequence of elements ui ∈ Uβi
for 1 ≤ i ≤ m such that g′ = u1u2 · · ·umg fixes the ray α+ = C1 ∪ C2 ∪ · · · ∪ Cm−1 ∪ Cm ∪ r+
pointwise. Since G acts on the twin tree preserving d∗, it preserves the opposition relation, so
α− = − op α+ means g′ also fixes the ray α− pointwise. Thus, g′ fixes the twin end (e1
+, e
1
−)
and by Corollary 8.7, g′ fixes the other twin end
(
e2
+, e
2
−
)
of the twin apartment A0 =
(
A+
0 , A
−
0
)
.
That implies that g′ stabilizes A0, so g′ ∈ T ≤ D and then g ∈ DU1. �
Theorem 8.8 gives a complete characterization of the stabilizers of an end, yielding
B±i,∞ = TW evenUi = Biq for i = 1, 2
is independent of the choice of ±.
Let Bi,K = Bi ∩ K. In our next step towards the proof of Theorem 8.1, we show that
B1,K = B2,K = BK is independent of the choice of i, and establish a K-equivariant bijection
between G/Bi and K/BK .
Lemma 8.9. For i = 1, 2 we have Ui · h = h⊕⊕α∈Φi
gα.
Proof. With our assumptions on the rank 2 Cartan matrix, Ui is abelian since [eαj , eαk ] = 0 for
any two roots αj , αk ∈ Φi and for any root vectors eαj ∈ gαj and eαk ∈ gαk . A general element
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 41A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 41
u u u
u u u uu
u
@
@@
g ·Cm
uuHHH
g ·Cm−1
u· · ·uu g ·C2g ·C1
CmCm−1uu
u
x+
x−C
−
C1C+ C2 · · ·
r+ =
[
y+, e
1
+
)
y+
r− =
[
z−, e1
−
)
z−
α− =
[
x−, e1
−
)
α+ =
[
x+, e
1
+
)
e1
−
e1
+
e2
−
e2
+
A−
A+
Figure 8. Diagram for Case (2) in proof of Theorem 8.8 showing: Twin root α = (α+, α−) determined
by adjacent chambers C+ and C1. Also showing pair of rays (r+, r−) contained in α, where r+ =
[
y+, e
1
+
)
and r− =
[
z−, e1−
)
. The path of edges (C1, C2, . . . , Cm−1, Cm) forms the interval [x+, y+] which contains
the sequence of adjacent vertices (x+ = x0, x1, . . . , xm−1, xm = y+) so Ci = [xi−1, xi] for 1 ≤ i ≤ m. The
path of edges (g · C1, g · C2, . . . , g · Cm−1, g · Cm) forming the interval [g(x+), g(y+) = y+] is also shown.
in Ui can be written as u =
r∏
j=1
exp(adeαj ), so for any h ∈ h we have
u(h) = h−
r∑
j=1
αj(h)eαj ∈ h⊕
r⊕
j=1
gαj .
�
Lemma 8.10. We have Bi,K = TW even = BK is independent of i = 1, 2.
Proof. By Theorem 8.8 we have Bi = TW evenUi. It is clear that TW even ⊂ K. 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
for any α ∈ Φ, we get ω ◦ Uα ◦ ω−1 = U−α from Lemma 2.9. Suppose 1 6= u =
r∏
j=1
exp(adeαj ) ∈
(Ui)K for αj ∈ Φi, 1 ≤ j ≤ r, where eαj ∈ gαj . Then u = ω(u) =
r∏
j=1
exp(ade−αj ) for some
basis vector e−αj ∈ g−αj . Let h ∈ h be an element such that αj(h) 6= 0 for 1 ≤ j ≤ r.
From Lemma 8.9, we get u(h) = h−
r∑
j=1
αj(h)eαj . Using the formula above for ω(u) we similarly
calculate ω(u)(h) = h+
r∑
j=1
αj(h)e−αj , but then u(h) = ω(u)(h) gives the non-trivial dependence
relation −
r∑
j=1
αj(h)eαj =
r∑
j=1
αj(h)e−αj among the independent basis vectors from the root
spaces ⊕rj=1gαj and ⊕rj=1g−αj . This contradiction proves that u = 1 so (Ui)K = {1}. �
Corollary 8.11. The maps Ch± : K/BK → Cham(B±∞) given by Ch±(kBK) = k(e±i,∞) are
bijective for either choice of i = 1, 2 and either choice of ±.
Proof. For each choice of ±, K acts transitively on Cham(B±∞), which is the K orbit of e±i,∞,
with stabilizer Bi ∩K = BK , so the orbit-stabilizer theorem gives the result. �
Lemma 8.12. For Ξ∞ = Ξ+
∞∪Ξ−∞ the twin halo of g, each element the set {x±i | 1 = 1, 2} ⊂ Ξ±∞
is fixed by BK , which is the pointwise stabilizer in K 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)
Figure 8. Diagram for Case (2) in proof of Theorem 8.8 showing: Twin root α = (α+, α−) determined
by adjacent chambers C+ and C1. Also showing pair of rays (r+, r−) contained in α, where r+ =
[
y+, e
1
+
)
and r− =
[
z−, e1−
)
. The path of edges (C1, C2, . . . , Cm−1, Cm) forms the interval [x+, y+] which contains
the sequence of adjacent vertices (x+ = x0, x1, . . . , xm−1, xm = y+) so Ci = [xi−1, xi] for 1 ≤ i ≤ m. The
path of edges (g · C1, g · C2, . . . , g · Cm−1, g · Cm) forming the interval [g(x+), g(y+) = y+] is also shown.
in Ui can be written as u =
r∏
j=1
exp(adeαj ), so for any h ∈ h we have
u(h) = h−
r∑
j=1
αj(h)eαj ∈ h⊕
r⊕
j=1
gαj . �
Lemma 8.10. We have Bi,K = TW even = BK is independent of i = 1, 2.
Proof. By Theorem 8.8 we have Bi = TW evenUi. It is clear that TW even ⊂ K. 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
for any α ∈ Φ, we get ω ◦ Uα ◦ ω−1 = U−α from Lemma 2.9. Suppose 1 6= u =
r∏
j=1
exp(adeαj ) ∈
(Ui)K for αj ∈ Φi, 1 ≤ j ≤ r, where eαj ∈ gαj . Then u = ω(u) =
r∏
j=1
exp(ade−αj ) for some
basis vector e−αj ∈ g−αj . Let h ∈ h be an element such that αj(h) 6= 0 for 1 ≤ j ≤ r.
From Lemma 8.9, we get u(h) = h−
r∑
j=1
αj(h)eαj . Using the formula above for ω(u) we similarly
calculate ω(u)(h) = h+
r∑
j=1
αj(h)e−αj , but then u(h) = ω(u)(h) gives the non-trivial dependence
relation −
r∑
j=1
αj(h)eαj =
r∑
j=1
αj(h)e−αj among the independent basis vectors from the root
spaces ⊕rj=1gαj and ⊕rj=1g−αj . This contradiction proves that u = 1 so (Ui)K = {1}. �
Corollary 8.11. The maps Ch± : K/BK → Cham
(
B±∞
)
given by Ch±(kBK) = k
(
e±i,∞
)
are
bijective for either choice of i = 1, 2 and either choice of ±.
Proof. For each choice of ±, K acts transitively on Cham
(
B±∞
)
, which is the K orbit of e±i,∞,
with stabilizer Bi ∩K = BK , so the orbit-stabilizer theorem gives the result. �
Lemma 8.12. For Ξ∞ = Ξ+
∞∪Ξ−∞ the twin halo of g, each element the set
{
x±i | 1 = 1, 2
}
⊂ Ξ±∞
is fixed by BK , which is the pointwise stabilizer in K 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
42 L. Carbone, A.J. Feingold and W. Freyn
independent points on the two lines of ∂Lt to linearly independent points, a basis for t, so k
normalizes t. But NK(t) = TW , and the elements in the set TW odd exchange x±1 and x±2 , so
k ∈ TW even = BK . �
We now have all technical ingredients to prove Theorem 8.1.
Proof of Theorem 8.1. We begin to define Ψ∞ by setting
Ψ∞
(
e+
1,∞
)
= x+
1 and Ψ∞
(
e−1,∞
)
= x−1
and we extend this to all of B∞ by
Ψ∞
(
(kBK)e±1,∞
)
= kΨ∞
(
e±1,∞
)
k−1
for any k ∈ K. This map is well-defined and injective by Lemma 8.12. It is bijective by
Corollary 8.11, and is clearly K-equivariant by its definition. �
The results in this section suggest that on the Lie algebra level one could study a non-standard
Cartan decomposition
g = h⊕
⊕
α∈P1
gα
⊕
α∈P2
gα
based on a non-standard partition Φ = P1 ∪ P2 such that Φi ⊂ Pi for i = 1, 2, and such that
n1 =
⊕
α∈P1
gα and n2 =
⊕
α∈P2
gα
are subalgebras. Then bi = h ⊕ ni, i = 1, 2, would be non-standard Borel subalgebras corre-
sponding to the subgroups Bi. That would be accomplished if each subset Pi were 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 L0
hR
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 determined by x+
2 the partition would be P1 = Φ1∪
(
Φim
)+
and P2 = Φ2∪
(
Φim
)−
, while for
the line determined by x+
1 the partition would be P1 = Φ1 ∪
(
Φim
)−
and P2 = Φ2 ∪
(
Φim
)+
. We
have found only two distinct non-standard Borels at infinity, Bi, i = 1, 2 which may correspond
to these two partitions, but we have not yet seen a family of non-standard Borels corresponding
to other choices of partitions.
Having found a non-standard Cartan decomposition as above, one gets a corresponding non-
standard decomposition of the universal enveloping algebra U(g) = U(n1)U(h)U(n2) and can
construct induced Verma modules of two types, Vermai(λ) = U(ni)v
i
λ, i = 1, 2, where viλ are
vectors such that h · viλ = λ(h)viλ for any h ∈ h, n1 · v2
λ = 0 and n2 · v1
λ = 0. The quotient of such
a Verma module by its maximal proper submodule would be an irreducible module, Irredi(λ)
generated by viλ. These would be integrable g-modules for λ = n1λ1 +n2λ2 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 [5, Section 6]. Such modules have been
found to occur in the decomposition of the rank 3 hyperbolic algebra, F , with respect to its
rank 2 hyperbolic Fibonacci subalgebra, recently studied by Penta [50].
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 43
9 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 [19, 20,
22, 23, 25, 26].
Recalling the well-known finite-dimensional theory (see for example [16, 32]), 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 TpM .
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 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 repre-
sentation 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 Rie-
mannian symmetric spaces [10]. Similar observations were made in the affine case [28, 31]. 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.
A A new formula for the Weyl group generators
on any integrable module
It is remarkable that the proof of the following theorem, starting from a simple calculation in
the two-dimensional representation of sl(2,C)i, generalizes to a result in any integrable module
for any Kac–Moody algebra g. Special cases of this formula appeared in some physics papers,
for example, in [12], where Damour and Hillmann found it in a representation of K(E10).
It is known [35] that for any integrable representation φ : g→ End(V ), the group W̃ generated
by the operators
w̃φi = exp(φ(ei)) exp(φ(−fi) exp(φ(ei)), 1 ≤ i ≤ n
is a subgroup of the Kac–Moody group G that acts on V as follows. If Vµ is the µ-weight space
of V , then w̃φi (Vµ) = Vwi(µ), where wi is the reflection with respect to the simple root αi in the
dual space, h∗, of the standard Cartan subalgebra, h. The Weyl group, W , is the Coxeter group
generated by those simple reflections, and is also defined on h by the formula wi(h) = h−αi(h)hi.
There is a surjection from W̃ onto W whose kernel is given by Remark 3.8 in [35]. In the case
when φ is the adjoint representation, the restriction of each w̃φi to h equals wi.
44 L. Carbone, A.J. Feingold and W. Freyn
For any integrable representation φ : g→ End(V ), define the “compact” operators from Gθ,
sφi = exp(φ(πxi)) = exp(φ(π(ei − fi)/2)), 1 ≤ i ≤ n.
We prove that w̃φi = sφi . This provides a new description of W̃ as generated by operators from
the real compact form. This generalizes a result discovered from a physics point of view by
Damour and Hillmann in a representation of K(E10).
Theorem A.1. Let g be any Kac–Moody algebra of rank n with the usual Chevalley generators
ei, fi, hi and let φ : g→ End(V ) be any integrable representation. Then for 1 ≤ i ≤ n, we have
exp(φ(ei)) exp(φ(−fi) exp(φ(ei)) = exp(φ(π(ei − fi)/2)).
Proof. Using the notations as above, for 1 ≤ i ≤ n, let gi = sl(2,C)i be the Lie subalgebra of g
with basis {ei, fi, hi}. Then V has a direct sum decomposition into irreducible gi-modules
V =
⊕
j∈J
Vj(m),
where dim(Vj(m)) = m+1 and the index set J includes information about the gi highest weight
vector in Vj(m) that locates it in V . It is clear that w̃φi = sφi on V if and only if their restrictions
to each Vj(m) are equal. They are certainly equal for all trivial one-dimensional modules, Vj(0).
On any two-dimensional module, Vj(1), the computation comes down to the simple fact that
exp
([
0 1
0 0
])
exp
([
0 0
−1 0
])
exp
([
0 1
0 0
])
=
[
1 1
0 1
] [
1 0
−1 1
] [
1 1
0 1
]
=
[
0 1
−1 0
]
matches
exp
([
0 t
−t 0
])
=
[
cos(t) sin(t)
− sin(t) cos(t)
]
,
when t = π/2. The well-known tensor product decomposition of irreducible sl(2,C)-modules
says that for any integer m ≥ 1, we have
V (m)⊗ V (1) = V (m+ 1)⊕ V (m− 1).
So if the operators are equal on modules V (m) and V (1) then they are equal on the tensor
product, so they are equal on the component V (m+ 1). This proves by induction that they are
equal on any Vj(m) that occurs in V . �
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant
No. 1002477. The first author was supported in part by the Simons Foundation, Mathematics
and Physical Sciences-Collaboration Grants for Mathematicians, Award Number: 422182. All
authors wish to thank the IHÉS for support during various visits during 2013–2019. The second
and third authors wish to thank the Max-Planck Institute for Gravitational Physics (Albert
Einstein Institute), Potsdam, Germany, for support during various visits during 2013–2019.
The authors wish to thank Peter Abramenko for his helpful comments on an earlier draft of
the manuscript, and for more recent comments on twin tree structures. They would also like to
thank Victor Kac for helpful comments in May 2015 at IHÉS. The second author 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.
Finally, the authors wish to express their thanks to the anonymous referees for many valuable
suggestions to improve this paper.
A Lightcone Embedding of the Twin Building of a Hyperbolic Kac–Moody Group 45
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1 Introduction
1.1 Summary of results
2 Kac–Moody algebras and Kac–Moody groups
2.1 Kac–Moody algebras
2.2 Kac–Moody groups
2.3 Twin BN-pair and twin Tits building of a minimal Kac–Moody group
2.4 Compact real forms of Kac–Moody algebras and groups
3 Tits cone and lightcone of hyperbolic Kac–Moody algebras of compact type
4 Group actions of the compact real form K
4.1 The adjoint action of K on k
4.2 The local structure of the adjoint action
4.3 The action of the compact real form on the twin building
5 Simplicial complex, distance and codistance on H"0362Hr
6 The main embedding theorem
7 Special results for the twin building in rank 2
8 Embedding the spherical building at infinity in rank 2
9 Conclusion and further directions
A A new formula for the Weyl group generators on any integrable module
References
|
| id | nasplib_isofts_kiev_ua-123456789-210705 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:32Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Carbone, Lisa Feingold, Alex J. Freyn, Walter 2025-12-15T15:25:15Z 2020 A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group. Lisa Carbone, Alex J. Feingold and Walter Freyn. SIGMA 16 (2020), 045, 47 pages 1815-0659 2020 Mathematics Subject Classification: 20G44; 20E42; 20F05; 51E24 arXiv:1606.05638 https://nasplib.isofts.kiev.ua/handle/123456789/210705 https://doi.org/10.3842/SIGMA.2020.045 Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra 𝖌 = 𝖌(A) and (adjoint) Kac-Moody group 𝐺 = 𝐺(A) = ⟨exp(ad(teᵢ)),exp(ad(tfᵢ)) | t ∈ ℂ⟩ where ei and fi are the simple root vectors. Let (B⁺, B⁻, N) be the twin BN-pair naturally associated to 𝐺 and let (𝓑⁺, 𝓑⁻) be the corresponding twin building with Weyl group W and natural 𝐺-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building (𝓑⁺, 𝓑⁻) of 𝐺 and the Kac-Moody algebra 𝖌 = 𝖌(A) in a new geometrical way. The Cartan-Chevalley involution, ω, of 𝖌 has a fixed point real subalgebra, 𝔨, the 'compact' (unitary) real form of 𝖌, and 𝔨 contains the compact Cartan t = 𝔨 ∩ h. We show that a real bilinear form (⋅,⋅) is Lorentzian with signatures (1,∞) on 𝔨, and (1, n−1) on t. We define {k ∈ 𝔨 | (k,k) ≤0 } to be the lightcone of 𝔨, and similarly for t. Let K be the compact (unitary) real form of 𝐺, that is, the fixed point subgroup of the lifting of ω to 𝐺. We construct a K-equivariant embedding of the twin building of 𝐺 into the lightcone of the compact real form 𝔨 of 𝖌. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an SU(2)-orbit of chambers stabilized by U(1), which is thus parametrized by a Riemann sphere SU(2)/U(1) ≅ S². For n = 2, the twin building is a twin tree. In this case, we construct our embedding explicitly, and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone. This material is based upon work supported by the National Science Foundation under Grant No.1002477. The first author was supported in part by the Simons Foundation, Mathematics and Physical Sciences-Collaboration Grants for Mathematicians, Award Number: 422182. All authors wish to thank the IHES for support during various visits from 2013 to 2019. The second and third authors wish to thank the Max-Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany, for support during various visits during 2013-2019. The authors wish to thank Peter Abramenko for his helpful comments on an earlier draft of the manuscript and for his more recent comments on twin tree structures. They would also like to thank Victor Kac for helpful comments in May 2015 at IHES. The second author would like to thank Kai-Uwe Bux, Max Horn, Tobias Hartnick, Ralf Kohl, and Peter Abramenko for helpful discussions at the June 2015 conference on "Generalizations of Symmetric Spaces" in Israel. Finally, the authors wish to express their thanks to the anonymous referees for their many valuable suggestions to improve this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group Article published earlier |
| spellingShingle | A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group Carbone, Lisa Feingold, Alex J. Freyn, Walter |
| title | A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_full | A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_fullStr | A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_full_unstemmed | A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_short | A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group |
| title_sort | lightcone embedding of the twin building of a hyperbolic kac-moody group |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210705 |
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