A lightcone embedding of the twin building of a hyperbolic Kac-Moody group

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


Introduction
Buildings were introduced by J. Tits in the 1950's to provide a 'geometric' interpretation of finite dimensional simple algebraic groups and finite groups of Lie type. Following ideas originally developed by F. Klein in his Erlanger program, Tits aimed at understanding a large class of groups, containing simple algebraic groups and finite groups of Lie type, as the automorphism groups of carefully constructed geometric objects called 'buildings'. It turns out that the buildings of Tits correspond to groups which admit an additional structure, called a BN -pair or equivalently a Tits system. This structure also gives rise to a Bruhat decomposition for the corresponding group. Tits's approach was the geometric counterpart to the 'algebraic' construction of these groups by C. Chevalley using automorphisms of Lie algebras [Che55].
In general, the building B of a group G with a BN -pair is an abstract simplicial complex constructed from group theoretical data. Namely, simplices in B are in bijection with the union of all cosets G/P , where P runs through a set of representatives of the conjugacy classes of parabolic subgroups. These simplices satisfy boundary relations which can be phrased in terms of inclusions of these cosets.
For Kac-Moody groups, the closest infinite dimensional analogue of simple algebraic groups, the structure of buildings and BN -pairs is richer. As Kac-Moody groups have two conjugacy classes of Borel subgroups, they admit the definition of two 'opposite' BN -pairs which together form a 'twin BN -pair'. Consequently the geometry associated to a Kac-Moody group G naturally consists of two related components. This object is called a 'twin building' B = B + ∪ B − associated to a twin BN -pair, (B + , B − , N ), where the subgroups B + and B − are the standard Borel subgroups constructed from the positive and negative roots of the Kac-Moody algebra respectively. Thus by construction the building is related to the combinatorial structure of its Kac-Moody groups.
Kac-Moody algebras and groups fall naturally into three types, finite type, affine type and indefinite type. While Kac-Moody groups of finite type (simple Lie groups) and of affine type are well-understood, there are far fewer results known for the indefinite type. The most important subclass of indefinite type is the hyperbolic type, studied since the 1980s by various authors [LM79,Fei80,FF83,KMW88,KM95] but certainly of interest to physicists [Jul85,DHN02,DH09,Wes01].
While the algebraic properties of the hyperbolic Kac-Moody groups and algebras attracted some attention, there have been only a few mathematical results concerning their geometry, for example, the study of homogeneous or symmetric spaces associated to them, and the study of 'polar actions' or isoparametric submanifolds. These classes of objects are very well understood for finite dimensional Lie groups and for affine Kac-Moody groups. It is hoped that the understanding of this geometry, called Kac-Moody geometry, will shed new light on algebraic and structural properties of these groups. As a first step towards the goal of understanding the geometry of hyperbolic Kac-Moody algebras and groups, we show in this work that for hyperbolic Kac-Moody groups G over C, the associated Tits building is not only an abstract simplicial complex admitting an action of G, but that it admits a natural embedding inside the compact real form k of the Kac-Moody algebra g. As a consequence, the structure of the Tits building and of the compact real form of the Kac-Moody algebra are closely related.
Our results generalize work of Quillen and Mitchell in the finite dimensional case, and Kramer and Freyn in the affine case. Mitchell, in a paper based on ideas of Quillen, used embeddings of spherical buildings associated to simple (real or complex) Lie groups into the tangent space of the associated noncompact real symmetric space [Mit88]. In particular, the topological building of a real noncompact Lie group G with maximal compact subgroup K is canonically identified with a space homeomorphic to the unit sphere in the tangent space of the noncompact real symmetric space G/K. For example, the topological building of type A 1 , isomorphic to S 1 , can be embedded into the unit circle in the tangent space of H 2 = SL 2 (R)/SO(2) which is R 2 .
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 [Kra02].
Freyn gave a 2-parameter family ϕ ,r of equivariant embeddings of affine 'twin cities' B = B + ∪B − into the 's-representations' of affine Kac-Moody symmetric spaces [Fre09,Fre11,Fre12b,Fre13c,Fre13b]. A 'twin city' is the natural completion of a twin building; affine twin cities correspond to completions of affine Kac-Moody groups in a similar way as twin buildings correspond to minimal Kac-Moody groups. Twin cities carry a natural topology that is derived from the topology on the corresponding Kac-Moody group.
Denoting by g = k⊕p the Cartan decomposition, and restricting Freyn's result to non-completed affine Kac-Moody algebras, yields an identification of the twin building with the intersection p ,r of the sphere of squared length ∈ R with horospheres parametrized by r d = ±r = 0, where r d 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 r d < 0 respectively r d > 0.
For affine Kac-Moody groups of the compact type, which are symmetric spaces called of 'type II' following Helgason's classification (see [Hel01,Fre13a]), this result includes embeddings of complex buildings into the compact forms of affine Kac-Moody algebras; this case is the affine counterpart to the embeddings constructed in this paper for symmetrizable hyperbolic Kac-Moody algebras. We refer the reader to [Fre09,Fre12a,Fre11,Fre12b] for additional details.
For g a hyperbolic Kac-Moody algebra we are motivated in part by the appearance of g/k and G/K in coset models of certain supergravity theories, where coset spaces of the split real forms occur as parameter spaces for the scalar fields of the theory [Jul85,DHN02,Wes01].
The results in this paper can be extended to include a wider class of Kac-Moody algebras, namely Lorentzian algebras. Nevertheless as soon as one leaves the realm of hyperbolic Kac-Moody algebras, the equivalence between the lightcone and the Tits cone breaks down. Consequently, the geometry of the buildings and their lightcone embeddings becomes more complicated. On the other hand interesting new phenomena occur, for example there are infinite families of rank 3 Lorentzian, non-hyperbolic Kac-Moody algebras whose buildings are isomorphic. Up to isomorphism, the same hyperbolic building will embed into each algebra in this infinite family, but in different ways. An extensive collection of examples is being worked out by A. Tichai [Tic14].
The authors wish to thank Peter Abramenko for his helpful comments on an earlier draft of the manuscript. They would also like to thank Victor Kac for helpful comments in May 2015 at IHÉS. AF would like to thank Kai-Uwe Bux, Max Horn, Tobias Hartnick, Ralf Köhl and Peter Abramenko for helpful discussions at the June 2015 conference on "Generalizations of Symmetric Spaces" in Israel.

Kac-Moody algebras and Kac-Moody groups
2.1. Kac-Moody algebras. A Kac-Moody algebra g F (A) over a field F may be constructed by generators and relations using a collection of data which includes a matrix A = (a ij ) i,j∈I called a generalized Cartan matrix satisfying the following conditions for all i, j ∈ I = {1, . . . , }: a ij ∈ Z, a ii = 2, a ij ≤ 0 if i = j, and a ij = 0 ⇐⇒ a ji = 0.
A generalized Cartan matrix A is indecomposable if there is no partition of the set I = I 1 ∪I 2 into non-empty subsets so that a ij = 0 for i ∈ I 1 and j ∈ I 2 . The matrix A is called symmetrizable if there is an invertible diagonal matrix D = diag(d 1 , · · · , d ) such that DA = (d i a ij ) is symmetric. One distinguishes various types of generalized Cartan matrices: -Finite type: A is positive-definite. In this case A is the Cartan matrix of a finite dimensional semisimple Lie algebra and det(A) > 0. -Affine type: A is positive-semidefinite, but not positive-definite, and all minors are positive definite. In this case det(A) = 0. -Hyperbolic type: A is neither of finite nor affine type, but every proper, indecomposable submatrix is either of finite or of affine type. In this case det(A) < 0.
A complex Kac-Moody algebra g C (A) has at least two real forms, that is, real Lie algebras g R such that g C (A) = C ⊗ g R . The split real form of g C (A) is g R (A) (see [BVBPBMR95]). From this point on, all generalized Cartan matrices in this paper are assumed to be indecomposable, symmetrizable and of hyperbolic type.
Given field F = C or F = R: -a hyperbolic generalized Cartan matrix A = (a ij ) i,j∈I , and -a vector space h over F (which will play the role of a Cartan subalgebra) with dim F (h) = , and basis then there is a set of simple roots Π = {α j | j ∈ I} ⊆ h * such that the pairing ·, · : h * × h → F given by α, h = α(h) satisfies α i (h j ) = a ij for all i, j ∈ I, and the hyperbolic Kac-Moody Lie algebra g = g F (A) is generated by the elements where ad x (y) = [x, y]. For each i ∈ I, let sl i 2 be the Lie subalgebra (isomorphic to sl 2 (F)) with basis {e i , f i , h i }, so that g is generated by these subalgebras. The abelian Lie subalgebra h with basis {h i | i ∈ I} is called the standard Cartan subalgebra of g.
The algebra g = g(A) is infinite dimensional since A is not positive definite, and it admits an invariant symmetric bilinear form ( , ) which is unique up to a global scaling factor ( [Kac90], section II), and which extends the form on h given by 2(h i , h j )/(h j , h j ) = α i (h j ) = a ij . The nondegeneracy of the pairing ·, · between h * and h determines a corresponding form on h * . This means that (α i , α j ) = (h i , h j ) and so that we may take the diagonal matrix D = diag(d 1 , · · · , d ) with d i = 2/(α i , α i ) and the symmetric matrix DA = (d i a ij ) = (2a ij /(α i , α i )). 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, for α = 0 are called root spaces.
The root system of g is the set Φ = {α ∈ h * | α = 0, g α = 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, g α i = Ce i and g −α i = Cf i , so that ±α i ∈ Φ. In fact, we have Q = Zα 1 ⊕ · · · ⊕ Zα is a free Z-module. Every α ∈ Φ can be written uniquely as α = i=1 k i α i where either all k i ≥ 0, in which case α is called positive, or all k i ≤ 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 α i , i ∈ I = {1, . . . , }, we define the simple root reflection The w i generate a group W = W (A) of orthogonal transformations of h * , called the Weyl group of A. The non-degenerate pairing between h and h * gives a corresponding action of W as orthogonal transformations on h. A root α ∈ Φ is called a real root if there exists w ∈ W such that wα is a simple root. A root α which is not real is called imaginary. We denote by Φ re the set of all real roots and Φ im the set of all imaginary roots.
The multibracket, [e i 1 , e i 2 , · · · , e in ] = ad e i 1 ad e i 2 · · · ad e i n−1 e in is in the root space g α for α = n j=1 α i j ∈ Φ + , while a similar multibracket with each e i j replaced by f i j is in g −α . Therefore, g has a root space decomposition ([Kac90], Theorem 1.2) The standard positive Borel subalgebra b ≡ b + is defined by b + = h ⊕ g + and the standard negative Borel subalgebra by b − = h ⊕ g − . Let g be a symmetrizable Kac-Moody algebra over C, L be a complex vector space and let π : g −→ End(L) be any integrable representation, so that all π(e i ) and π(f i ) are locally nilpotent on L and the linear operators χ π α i (t) = exp(π(te i )) and χ π −α i (t) = exp(π(tf i )), for t ∈ C, are well-defined in GL(L). Let the minimal Kac-Moody group associated to such a representation π be the group generated by these operators, . In fact, for any x ∈ g α , α ∈ Φ re the operator π(x) is locally nilpotent on L, so χ π x = exp(π(x)) is well-defined and these give the real root groups U π α . Now apply the above to the adjoint representation ad : g −→ End(g), which is integrable. This gives the minimal adjoint Kac-Moody group (2) G ad = G ad (C) = χ ad α i (t), χ ad −α i (t) | i ∈ I, t ∈ C ≤ GL(g) and for all x ∈ g α , α ∈ Φ re we have well-defined operators χ ad x = exp(ad x ) ∈ GL(g) giving the real root groups U ad α . These generators act on g as Lie algebra automorphisms, so we have G ad ≤ Aut(g).
Since ad e i and ad f i are locally nilpotent, g is the direct sum of finite dimensional sl 2 (C)-modules for each of the subalgebras sl i 2 . For fixed i ∈ I, on each such summand the exponentials above generate the group SL i 2 isomorphic to SL 2 (C), so G is also generated by the subgroups SL i 2 , i ∈ I.
The operators π(h) ∈ End(L) for h ∈ h are semisimple so they can also be exponentiated to give a commutative group of operators T π C = T π C (G) = {χ π h (t) = exp(π(th)) | h ∈ h, t ∈ C} ≤ G π which is called the standard maximal torus of G π . We also define the standard Borel subgroups (B π ) ± = T π C U π α | α ∈ (Φ re ) ± and the normalizer of T π C denoted by N π C . It is well known that the operators in G ad , generate a subgroup W ad in G ad such that the restriction of w ad i to the standard Cartan subalgebra h equals the simple Weyl group reflection w i and w ad i (e i ) = −f i . It means that W is a homomorphic image of W ad . Note that W ad is a subgroup of N ad C and that N ad 2.3. BN -pair and Tits building of a minimal Kac-Moody group. Our references for this section are [AB08,Ron89]. A Kac-Moody group G over C has subgroups B ± ⊆ G and N ⊆ G, with twin BN -pair (B + , B − , N ) satisfying the Tits axioms: T 2: Define S by the condition W = S and for 1 ≤ i ≤ , w i ∈ S we have w 2 i = 1. T 3: For w i ∈ S and w ∈ W we have For a hyperbolic Kac-Moody group G C (A) we have T C = H. Thus the group T (G) = N ∩B ± is a normal subgroup of N . The group W = N G (T (G))/T (G) is isomorphic to our earlier definition in section 2.1 of the Weyl group W as a group of orthogonal transformations of h * given by formula (1).
We identify W (non-canonically) with a subset (not a subgroup) of N which contains exactly one representative for every element of W . By abuse of notation, this set of representatives will also be called W . We have the (positive and negative) standard Borel subgroups, corresponding to the standard Borel subalgebras, B ± = T (G)U ± where U + is generated by all positive real root groups and U − is generated by all negative real root groups. The BN -pairs (B + , N ) and (B − , N ) have Birkhoff and Bruhat decompositions: A 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 W J = w j | j ∈ J of W and the corresponding subgroups of G, Note that P ± I = G is not parabolic, 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.
A Tits building consists of a simplicial complex B together with a collection of subcomplexes A, each of which is called an apartment, such that (1) each apartment is a Coxeter complex, (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 twin building B consists of a disjoint union B + ∪ B − of buildings, which are 'twinned'. This twinning can be described in various ways, for example via twin apartments. We will not go into details here, but direct the interested reader to [AR98].
From now on let B denote the twin building associated to a Kac-Moody group G. The simplices of B are in bijection with parabolic subgroups in such a way that simplices in B + correspond to positive parabolic subgroups and simplices in B − correspond to negative parabolic subgroups. The vertices (0-simplices) of the twin building B are in bijection with maximal parabolic subgroups in G. Chambers are in bijection with positive and negative Borel subgroups. We will denote these simplices by their corresponding parabolic subgroups. Since parabolic subgroups are self-normalizing, the simplicies in the building can be equivalently indexed by the coset spaces G/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 ± is a parabolic subgroup. Hence B is a simplicial complex of dimension dim(B) = − 1. We note that for Kac-Moody groups, the intersection of a positive parabolic subgroup with a negative parabolic subgroup never contains a Borel subgroup; hence the associated geometry consists of two buildings B + and B − , corresponding to the two opposite BN -pairs (B + , N ) and (B − , N ), yielding a twin building B = B + B − . In B ± the ( − 1)-simplex P ± [1,..., ] = P ± [I] = P ± ∅ = B ± is called the fundamental chamber of B ± . Each fundamental chamber has boundary consisting of the simplices ∆ ± J = P ± J , and has closure ∆ ± = One can give a geometric realization of a building as follows. Let {e 1 , · · · , e } denote the standard orthonormal basis of R . Each r-simplex is identified with a copy of the standard simplex inherits the topology from R r+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 [AB08].
For the buildings associated with the hyperbolic Kac-Moody groups we wish to study, the geometric realization of apartments in the buildings B + and B − can be chosen to be isometric to hyperbolic spaces tesselated by the action of the hyperbolic Weyl group W .
2.4. Compact real forms of Kac-Moody algebras and groups. Let g = g C (A) be a complex Kac-Moody algebra and let h be the standard Cartan subalgebra. The Cartan involution posing ω 0 with complex conjugation, we obtain a conjugate linear involution ω, called the Cartan-Chevalley involution. Then k = {x ∈ g | ω(x) = x} is a Lie algebra over R called the compact real form of g ([Kum02] p. 243).
Note that ω 0 and ω both centralize the Weyl group, so they each induce twin building involutions via formula (4).
We may give generators for the compact real form k as follows. For each j = 1, · · · , , let g j = sl j 2 be the Lie subalgebra of g isomorphic to sl 2 (C) with basis {e j , f j , h j }, so that g is generated by the subalgebras g j and ω(g j ) = g j . Then the real Lie algebra of fixed points of ω on g j , k j = su j 2 has basis with brackets [x j , y j ] = z j , [y j , z j ] = x j , [z j , x j ] = y j , and the compact real form k is generated by all of the subalgebras k j , j = 1, · · · , (see Proposition 1 in [Ber85]). A Cartan subalgebra in the compact real form k is an abelian subalgebra whose complexification is a Cartan subalgebra in g. The standard Cartan subalgebra t = h ∩ k in k has real basis {z j | 1 ≤ j ≤ }.
Let G = G C (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(g α ) = 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 w ad i ∈ W ad . Lemma 2.1. For any α ∈ Φ re , e α ∈ g α and any φ ∈ Aut(g), we have φ • exp(ad eα ) • φ −1 = exp(ad φ(eα) ) so that, in particular, ωU ad α ω −1 = U ad −α and w ad i U ad Proof. For any x ∈ g, since φ is a Lie algebra automorphism, we have The formula with φ = ω gives ωU ad α ω −1 = U ad −α since ω(g α ) = g −α , and with φ = w ad i gives We set K = F ix G (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 K j such that k j = Lie(K j ) for each j = 1, . . . , ([Cap09], [Tit86], [KP85]).
For each v = a + bi ∈ C and 1 ≤ j ≤ we write a generator of T ad C as exp(ad vh j ) = exp(ad ah j )exp(ad bih j ) = exp(ad ah j )exp(ad b2z j ) .
This gives the decomposition T ad It is clear from the two expressions for w ad i that W ad ≤ K, but it is not so obvious that these operators can be expressed as a single exponential w ad i = exp(ad π x i ). This is a special case of the formula w π i = exp(π(π x i )) for any integrable representation π : g → End(V ), where the inner π is in R, proven in [FF16] and first found in certain important cases by [DH09]. This result follows from [Cap09], Proposition 8.1 (iii). See also [KW92], Corollary 5.33 and [KP87], Proposition 3.5.

Tits cone and lightcone of hyperbolic Kac-Moody algebras of compact type
Let g = g C (A) be a Kac-Moody algebra and let h be its standard Cartan subalgebra. The split real form g R = g R (A) of g contains the R-subalgebra h R = h ∩ g R which is just the R-span of {h i | i ∈ I}. We call h R the standard Cartan subalgebra of g R .
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 This same formula restricted to h R gives the action of W and it also determines the action of W on t by These operators are orthogonal with respect to the restriction of the bilinear form (·, ·) to h R and to t, and therefore W preserves each surface of constant square Since we are assuming that the Cartan matrix A is hyperbolic, the form (·, ·) is Lorentzian on h R and on t. The surface where r = 0 is called the nullcone, each surface where r < 0 is called timelike, and each surface where r > 0 is called spacelike. The set of all timelike points has two connected components, one called forward and denoted T L + and the other called backward and denoted T L − . We have T L − = −T L + .
Each of these components is preserved by the linear action of W , which acts consistently on rays, Ray x = {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 The union is called the positive (respectively negative) Tits cone and X = X + ∪ X − is called the Tits cone.
Clearly, we have X − = −X + . For g hyperbolic, X ± ⊇ T L ± , since it is possible that C ± , and therefore X ± , contains rays on the nullcone. This happens for the rank 3 hyperbolic Cartan matrices  corresponding to the hyperbolic Kac-Moody algebra F [FF83] whose Weyl group is the hyperbolic triangle group T (2, 3, ∞), and the "ideal" hyperbolic Kac-Moody algebra I whose Weyl group is the hyperbolic triangle group T (∞, ∞, ∞), respectively. For A strictly hyperbolic, that is, whose principal minors are of finite type, we have X ± = T L ± . Such an example is E 10 .
We have the following description of the closure of the Tits cone (see [Kac90], 5.10.2) We introduce the notations Proposition 3.1. Let g be a complex Kac-Moody algebra, k its compact real form and g R its split real form. Let h, h R , t be the standard Cartan subalgebras of g, g R , k, respectively. Then h R ∼ = R −1,1 has signature ( − 1, 1) and We use a sign convention on k adopted from the theory of finite dimensional Riemannian symmetric spaces [Hel01] and set Note that in the affine case (·, ·) k , this sign convention naturally occurs in the loop group realizations [PS86,Fre09]. With respect to (·, ·) k , the Cartan subalgebra t has Lorentzian signature ( − 1, 1).
Proof. As a vector space, a complex hyperbolic Kac-Moody algebra g has a basis consisting of the Cartan subalgebra generators, h i , i = 1, . . . , , and certain multibrackets [e i 1 , e i 2 , · · · , e in ] and and ω(h i ) = −h i so an R-basis for the compact real form k consists of the compact Cartan basis elements z i = i 2 h i , i = 1, . . . , , and the elements obtained from basis multibrackets above In particular, for n = 1 these include the elements x i = 1 2 (e i − f i ) and y i = i 2 (e i + f i ). Following our sign convention, the Cartan subalgebra t spanned by the elements z i , i = 1, . . . , , has signature ( − 1, 1). Using our sign convention and the characterization of the ad-invariant scalar product in [Kac90], Equation 2.2.1, applied to the basis vectors y i and z i , we find that the bilinear form on each subalgebra k i = su i 2 is positive definite. Furthermore the ad-invariant bilinear form has the following properties on root spaces (see [Kac90], Sections 2.1 and 2.2). The root spaces g α and g −α , which are interchanged by ω, have dual bases As a consequence, for positive roots α, the basis elements of k, Hence it follows from (6) that The Cartan subalgebra has signature ( − 1, 1) and for each positive root α the subspaces This could also be deduced from [RC89], Section 2.1.
Remark 3.3. Note that the bilinear form (·, ·) on g R is indefinite with signature (∞, ∞) because, while the split Cartan h R has the signature ( −1, 1), each pair of dual root vectors {e j α , 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 h R extends C-bilinearly to the complexification h C . In particular, for x ∈ h R we have We introduce notations for certain subsets in k: The ones inside t are related by the W -invariant isometry ϕ to the corresponding subsets defined earlier in h R . Furthermore, we have We call X t the Tits cone of t and note that X t = L t .
Remark 3.4. We will refer to L k as the lightcone of k and L t 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 By Proposition 2.2 all Cartan subalgebras of k are K-conjugate, thus the definition of H is independent of the choice of t.
Proposition 4.1. Let t be any Cartan subalgebra of k and let z = ih ∈ t for h ∈ h R satisfying α(h) = 0 for all α ∈ Φ + . Then the subspace [k, z] of k has basis 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 T z (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, 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 If (·, ·) is Lorentzian and r < 0 then X r is a two-sheeted hyperboloid of constant curvature κ = − 1 r 2 whose connected components we call X + r and X − r . In particular we will use t r ⊂ L 0 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 [Ebe96]. Note that for each r < 0, B ∞ (V ± r ) can be identified with B ∞ (∂L V ) ± . Using this observation we define the lightlike closure of V ± r to be (8) Since the bilinear form (· , ·) k is ad-invariant, the Adjoint action of K on k preserves the surfaces There is an induced action of K on B ∞ (∂L t ) and for each r < 0 on B ∞ (∂L tr ) ± , as well as on t and on t ± r . Hence, for each r < 0, the Adjoint action of K on k induces a well-defined action on the surface: as well as on the lightlike closure H r .
Since all Cartan subalgebras are conjugate, we can define the following surjective map 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 (G ad ) ∩ K be the torus associated to t and let be the normalizer of T in K. For k ∈ K, t ∈ T and u ∈ t we have: Thus the Adjoint action on t is T -invariant and factors to the quotient space K/T yielding a surjective map For any Cartan subalgebra t = ktk −1 of k, L t 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 ∂L t . 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 X t = L 0 t and there is a bijection between each surface (t r ) ± for r < 0 and the set of rays in L 0 t . In this case we do not need to consider the lightlike closure H r 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 ∂L t . So the corresponding points on the surface (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 ) ± ∼ = H 2 are isometric to the Poincaré disk, and the tesselation by the hyperbolic Weyl group T (2, 3, ∞) includes chambers which have an ideal vertex on the boundary. In such an example, the building would have 0-simplices corresponding to those ideal vertices, so to achieve our goal of embedding the twin building of a hyperbolic algebra inside the lightcone of k, we must use H r .
Since W = N/T ≤ K/T , the restriction to t r of the Adjoint action of N on k r coincides with the Weyl group action of W on t r .
Restriction of the second coordinate of the domain of ψ to either fundamental domain C ± r for the action of W on t r gives the surjective maps 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 fixed by the generator w i of W .
The standard Cartan subalgebra is given by t = Rz 1 + · · · + Rz . For 1 ≤ i ≤ and s, t ∈ R, we have exp For each 1 ≤ i ≤ the family of distinct Cartan subalgebras obtained this way can be parameterized by a 2-sphere with antipodes identified, as we will show using the first part of the following theorem.
For any s, t ∈ R such that 0 < r 2 = s 2 + t 2 and for any z ∈ t, we have Proof. We prove only the first relation since the others are analogous. We have Using r 2 = s 2 + t 2 = 0, it is clear that for n ≥ 1 we have where r = √ s 2 + t 2 ≥ 0 and ψ is defined when r > 0 by sin(ψ) = t r and cos(ψ) = s r .
Proof. The Cartan subalgebra t i (s, t) is spanned by L t,i and from theorem 4.4 (1) and the fact that α i (z i ) = 0. Since for 0 ≤ r < π and 0 ≤ ψ < π no two vectors of the above form are colinear, all of the corresponding subspaces t i (s, t) are distinct. But for antipodal points on the unit sphere, (r, ψ) and (π−r, ψ+π), the corresponding subspaces are the same.
4.3. The action of the compact real form on the twin building. To describe the action of K on the twin building, we use the Iwasawa decomposition [DMGH09] which yields where G denotes a complex Kac-Moody group, K denotes the compact real form of G, A ∼ = R rank(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 T C (G) = T A, the decomposition B ± = T C (G)U ± and the Iwasawa decomposition, we have Recalling the description of the building from equation (3) in section 2.3 we obtain an equivalent new description by equation (10): In the description of the twin building B = B + ∪ B − given by equation (11), the natural action of G via left multiplication is apparent, while in the description of B as in equation (12) the G-symmetry is broken to K-symmetry. Note that in equation (11) the cosets of the two opposite buildings B + and B − are defined with respect to different subgroups B + and B − respectively. Hence there are subgroups of G which act differently on B + and B − . For example, for any g ∈ G the coset gB + ∈ G/B + is fixed by all elements in the subgroup B + g = gB + g −1 , but that subgroup acts transitively on But the description of B ± in equation (12) shows that the action of K is the same in both. In particular this means that the group K does not act transitively on any apartment system. More precisely, for A ∈ B ± an apartment, define the orbit A K (A) := K · A. Then for each chamber c ∈ B ± there is exactly one apartment Let c = (f B ± , ∆ ± ∅ ) be any chamber in B ± = (G/B ± × ∆ ± ) / ∼, and let Cham(B ± ) denote the set of all chambers of the building B ± . For i ∈ I, the i-wall of c is (f B ± , ∆ ± i ) and the i-residue of c is defined to be Then we have R i (c) ∼ = P 1 (C). Identifying P 1 (C) with the Riemann sphere C = C ∪ ∞, the action of the subgroup SU (2) i on R i (c) can be identified with the action of SU (2) on C by Möbius transformations. Additional details may be found in [AB08], Chapter 6.

Simplicial structure on H r
In this section we define for each r < 0 a simplicial structure on the set H r ⊂ k r . We begin in the standard Cartan subalgebra with t r ⊂ t. Recall, that t r = t + r ∪ t − r is (up to rescaling) isometric to a pair of hyperbolic spaces (for > 2) which are both tesselated by the action of the Weyl group W . We call an element x ∈ t r singular if it is fixed by a Weyl group element w = 1 and call x regular otherwise. Let t sin r be the set of all singular elements in t r and let t reg r be the set of all regular elements, and similarly we have sets (t sin r ) ± and (t reg r ) ± . Then we have the decomposition t r = t sin r ∪ t reg r . We denote by Comp ± r the set of connected components of t reg r . In each sheet one connected component coincides with the fundamental domain C ± . For each sign ±, the Weyl group acts simply transitively on that set Comp ± r . We use the Weyl group action to index the elements of Comp ± r as follows: Let 1 ∈ W denote the identity element. In each sheet t ± r we index the fundamental chamber C ± with 1. Then we index the connected component c ± = wC ± ∈ Comp ± r by w yielding w denote the closure of the component (t reg r ) ± w and let U ± denote the union so U ± covers t ± r . For any minimal subset J ∈ W such that the intersection is nonempty, we identify the interior of S ± J with a simplex of dimension − |J|. For example, each connected component in Comp ± r is an open subset of t ± r , hence of dimension − 1. It follows that it is identified with a simplex of dimension − 1.
It is straightforward to check that for the Weyl group W . That is, S ± admits a W -action which is simply transitively on simplices of maximal dimension. Thus for any two simplices of maximal dimension, there is a chain of maximal dimensional simplices such that two consecutive ones share a common boundary. Since our compact real Kac-Moody algebra k has rank , all of its Cartan subalgebras, ktk −1 , for k ∈ K, have dimension and each surface (ktk −1 ) r of radius r < 0 has dimension − 1. Thus simplices of maximal dimension have dimension − 1 and their boundaries have dimension − 2.
We may now extend this simplicial structure to all of H r as follows. A similar simplicial structure may be defined on any Cartan subalgebra in H r . We must check that the Weyl group tesselations on different Cartan subalgebras fit together in a well-defined way. Cartan subalgebras intersect exactly in hyperplanes fixed by Weyl group elements (as in section 4.2), hence they intersect in singular elements. Thus each simplex of maximal dimension lies in exactly one Cartan subalgebra. Thus the simplicial structures on different Cartan subalgebras fit together, leading to a tesselation of H r into simplices.
Each simplex of codimension 1 lies in the intersection of a fixed point hyperplane L ktk −1 ,i with the surface (ktk −1 ) r of radius r < 0 as calculated in Section 4.2, while simplices of codimension n lie in the intersection of n such hyperplanes with the surface. Restriction of the Adjoint action to a single Cartan subalgebra coincides with the Weyl group action. This follows easily from the fact that the Weyl group W = N/T is a quotient of the normalizer of T . Hence simplicies of lower dimension are fixed by nontrivial subgroups of W .

The main embedding theorem
An 'embedding' of the twin building B = B + ∪ B − of the Kac-Moody group G with identity element e into the compact real form k is a bijective simplicial map from B onto the simplicial structure defined on H r for each r < 0 defined in section 5.
Theorem 6.1. Let A be a symmetrizable hyperbolic generalized Cartan matrix, g its complex Kac-Moody algebra and G its complex Kac-Moody group and let r < 0. Let K be the compact real form of G and let k be its Lie algebra. Let B = B + ∪ B − be the geometric realization (section 2.3) of the twin building of G over C.
(1) There is a K-equivariant embedding Ψ r : B → H r ⊂ k, that is, the following diagram commutes: where Ad K denotes the Adjoint action and K ⊂ G acts on B by left multiplication.
(3) There is a W -equivariant embedding of the fundamental apartment of B ± into t ± r ⊂ t r . (4) The embedding of the twin building is a simplicial complex isomorphism with respect to the simplicial structure on H r discussed in section 5.
Remark 6.2. The image Im (Ψ r ) is contained in the interior of the lightcone if A is strictly hyperbolic. Examples of this type are the rank 2 hyperbolic Kac-Moody algebras (see section 8).
Otherwise it contains points on the boundary of the lightcone. Examples of this type are the algebras F and I.
Proof. To construct the embedding of the twin building we use the description given in equation 12 B ± = K/T × ∆ ± / ∼ . Our construction is in three steps. We first define the embedding of the fundamental chamber. Then we make use of the K-action on the building and the Adjoint action of K on the Lie algebra. In a third step we establish the properties claimed in the theorem. For r < 0 we choose a connected component (t reg r ) + 1 where 1 ∈ W denotes the identity element and we define (13) (t reg r ) − 1 := −(t reg r ) + 1 . If A is strictly hyperbolic the fundamental domain is contained in the interior of t r ; hence its closure is contained in t r . On the other hand if A is not strictly hyperbolic then it is unbounded and its closure is contained in t r but not in t r . We identify the boundary hyperplanes L t,i of the fundamental chamber with the generators w i , 1 ≤ i ≤ , of the Weyl group W . We also identify intersections of hyperplanes L t,i 1 ∩ · · · ∩ L t,in for distinct indices {i 1 , . . . , i n } ⊂ I with the subset of generators {w i 1 , . . . , w in }. 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 and extend this map to ∆ ± by mapping a point x in the geometric realization of the simplex (1T, ∆ ± J ) with normalized barycentric coordinates x = [λ 1 : · · · : λ ] to the point with the same normalized hyperbolic barycentric coordinates.
Recall the definition of normalized hyperbolic barycentric coordinates: Let ∆ be a simplex in ndimensional hyperbolic space with vertices (v 0 , . . . , v n ). 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 ∆ This yields a simplicial map Ψ r : (1T, ∆ ± ) −→ C ± .
Part 2: Let x ± ∈ ∆ ± . We extend the map defined in Part 1 to a map We have to check that Ψ r is well-defined. Assume we have two equivalent elements (k 1 T, x ± 1 ) ∼ (k 2 T, x ± 2 ), hence x ± 1 = x ± 2 and assuming x ± 1 ∈ ∆ ± J for some subset J I, we have k 1 K J = k 2 K J . Then there is some l ∈ K J such that k 1 = k 2 l and we have Hence Ad l Ψ r (1T, x ± 1 ) = Ψ r (1T, x ± 1 ). Thus, from Ad k 2 l = Ad k 2 Ad l we get Part 3: We have to check that Ψ r satisfies the properties stated in the theorem.
Proof of (4): Remark that Ψ r maps the unique simplex (kT, ∆ ± J ) for J I spanned by vertices (kT, ∆ ± [i] ), i ∈ J, onto the simplex spanned by Ψ r (kT, ∆ ± [i] ), i ∈ J. If two simplices (k 1 T, ∆ ± J ) and (k 2 T, ∆ ± J ) share a common boundary (lT, ∆ ± L ), for J L in B then by definition we have k 1 T ⊂ Ad l K L and k 2 T ⊂ Ad l K L . But then Ψ r (k 1 T, ∆ ± L ) = Ψ r (lT, ∆ ± L ) = Ψ r (k 2 T, ∆ ± L ) is the commonly shared boundary of the simplices Ψ r (k 1 T, ∆ ± J ) and Ψ r (k 2 T, ∆ ± J ). Hence Ψ r preserves the simplicial structure of B and is thus a simplicial complex isomorphism, establishing (4).
Suppose we have an arbitrary element x ± ∈ H ± r . Then there is some group element k ∈ K such that Ad k x ± ∈ t ± r so it is in the closure of some chamber, (t reg r ) ± w , uniquely labeled by an element w ∈ W . The action of the corresponding element w ∈ W ad ≤ K matches the action of w on t. So we have Ad w −1 Ad k x ± ∈ C ± r . Therefore x ± ∈ Ad k −1 Ad w Ψ r (1T, ∆ ± ) = Ψ r (k −1 wT, ∆ ± ) . This shows that in the case when A is strictly hyperbolic, Ψ r : B ± → H ± r is surjective. Otherwise there can be elements x ± ∈ B ∞ (H ± r ) which are in Im(Ψ r ), but some may not. It would be interesting to understand precisely which points in B ∞ (H ± r ) are in Im(Ψ r ). The proof of (3) is now clear. Remark 6.3. Based on our understanding of the examples F and E 10 , we believe that the intersection of the nullcone with Im(Ψ r ) can be characterized as those rays on the nullcone which contain roots of g. Each such ray of null vectors corresponds to a copy of an affine KM subalgebra inside g. Each such ray should be conjugate under the action of the Weyl group W to one ray in the closure of the fundamental chamber, so the number of W orbits is the number of ideal points in that fundamental domain. Classifying those rays then becomes a significant problem involving the arithmetic properties of W , essentially understanding its cusps as a modular group. Such an analysis was carried out for E 10 in Lemma 5.2 of [KPN12].
Thus for each r < 0 we have established an embedding Ψ r of the twin building B of the compact real form k of a strictly hyperbolic Kac-Moody algebra g, and into its lightlike closure, H ± r , otherwise. Remark 6.4. Recall that B ± is contractible since each apartment is contractible (see [Ron89]). Since Ψ r for r ≤ 0 is a simplicial complex isomorphism onto its image, the image Ψ r (B ± ) is also contractible.

This observation yields the corollary
Corollary 6.5. The spaces H ± 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 [Ebe96,Mit88,Hei06,Fre12a]). It can be generalized in these cases to s-representations and relates in this way the (twin) building to the isotropy representations of finite dimensional Riemannian symmetric spaces and affine Kac-Moody symmetric spaces respectively. The existence of hyperbolic Kac-Moody symmetric spaces is conjectural at this stage. We hope to take this up elsewhere.

Cartan involutions and the embedding of the twin building
Recall from section 2.4 that the Cartan involution is the automorphism ω 0 : g −→ g determined by ω 0 (e i ) = −f i , ω 0 (f i ) = −e i and ω 0 (h i ) = −h i , and ω is the conjugate linear automorphism defined as the composition of ω 0 with complex conjugation. This gives Proposition 7.1. The following diagram is commutative: We note that while Ψ r is a simplicial complex isomorphism, the apartment system of B has no direct geometric interpretation in Ψ r (B). Our embedding identifies ω-stable twin apartments in B with Cartan subalgebras in k. The other apartments are hidden because Ψ r is only Kequivariant, but not G-equivariant.

Understanding of the Tits building in rank 2
In this section assume that g = g(A) is a rank 2 hyperbolic Kac-Moody Lie algebra, so that We impose the additional conditions that a > 1 and b > 1 because otherwise the real root groups U i defined below may not be abelian (see [Mor88,CMS15]).
Rank 2 hyperbolic Kac-Moody algebras were studied intensively by Lepowsky and Moody [LM79], by Feingold [Fei80] for the "Fibonacci hyperbolic" (a = b = 3), and by Kang and Melville [KM95]. In these rank 2 cases the hyperboloids t r for r = 0 are hyperbolas, and there is no topological distinction between "one-sheeted" for r > 0 and "two-sheeted" for r < 0, as there is in higher rank. In this section we will try to provide a detailed construction of the twin building, B = B + ∪ B − , whose simplicial structure is a pair of trees related by a twin building involution, and give the apartment system and the action of the complex group G C on each tree. We will also give an explicit description of the smaller apartment system and the action of the compact real form K, which will provide more details of the embedding of the building into the compact real form in this rank 2 case. In the next section we show how this allows the embedding of a spherical building at infinity.
The eigenvalues of A are λ ± = 2 ± √ ab so λ + > 0 and λ − < 0. This means that the signature of the bilinear form determined by A on the split real Cartan subalgebra h R is (1, 1). By Equation (6), the bilinear form on the compact real Cartan subalgebra t = ih R also has signature (1, 1).
The Weyl group of g is the infinite dihedral group which acts on h ⊂ g as well as on h R ⊂ g R and on t ⊂ k. Denote the infinite cyclic subgroup of W by W even = {(w 2 w 1 ) m | m ∈ Z} ∼ = Z, which has index 2 in W , so that for i = 1, 2, is the other coset, and we have the relations With t = w 2 w 1 and r = w i this gives the presentation of W as the infinite dihedral group W = t, r | r 2 = 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: Then we have w 1 w(n) = w(−1 − n) and w 2 w(n) = w(1 − n) for n ∈ Z, and 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 W by the integers in the same way: We define the non-standard partition of the real roots of g, See Figure 1 as an illustration for the "Fibonacci" rank 2 hyperbolic root system, and see Figure  2 for the rank 2 hyperbolic root system coming from the Cartan matrix A given at the beginning of Section 8, with a = 2 and b = 3, which has simple roots of different lengths. The real roots are on the red hyperbolas, and the non-standard partition is according to whether a root is on a left branch or a right branch.
For i = 1, 2 recall that are the lines in t fixed by the simple reflections w i , respectively. Then For i = 1, 2 the line L t,i is the intersection of the family of Cartan subalgebras, parametrized by the 2-sphere with antipodes identified, the complex projective space P 1 (C) (Theorem 4.4). We also have the corresponding statements for the split real form. For i = 1, 2 recall that in the split real Cartan, h R , we have are the lines in h R fixed by the simple reflections w i , respectively. Then In Figures 1 and 2 those fixed lines are the inner green lines.
For i = 1, 2 the line L h R ,i is the intersection of the family of Cartan subalgebras, Let (G, B ± , N ) be a BN -pair for the complex Kac-Moody group G = G C (A). The standard parabolic subgroups P ± J for J {1, 2} are P ± ∅ = B ± , P ± 1 = B ± B ± w 1 B ± , and P ± 2 = B ± B ± w 2 B ± .
As a simplicial complex, the twin building B = B + ∪ B − of (G, B ± , N ) is a pair of homogenous P 1 (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 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 : Apartments in B ± are infinite lines, but not every infinite line in the tree is an apartment. The fundamental apartment A ± in B ± is a union of chambers which are Weyl group translates of the fundamental chamber C ± = (1B ± , ∆ ± ∅ ), which is fixed under the action of B ± , which contains the root group U ± . The boundary vertices of that fundamental chamber are denoted by v ± i = (1B ± , ∆ ± i ) for i = 1, 2, corresponding to the maximal parabolic subgroups P ± i . By definition, the vertex v ± i = (1B ± , ∆ ± i ) is stabilized by the generator w i of the Weyl group W . The apartment system A ± in B ± is the set of apartments obtained from fundamental apartment A ± by the action of G. The question is how precisely can we describe that apartment system.
In t, for each r < 0 the hyperbola t r has two connected components, t ± r , which are each tessellated by the action of W into intervals separated by vertices which are the intersections of t r with the W -images of the two lines L t,i fixed by w i , i = 1, 2. The fundamental domain for the W action on t ± r is C ± , which corresponds to the fundamental chamber of A ± , and the other chambers of A ± correspond to: {. . . , w 1 w 2 w 1 C ± , w 1 w 2 C ± , w 1 C ± , C ± , w 2 C ± , w 2 w 1 C ± , w 2 w 1 w 2 C ± , . . . } so that adjacent chambers in this list are adjacent in the apartment, sharing a common vertex. For example, the common vertex v ± 1 = w 1 C ± ∩C ± corresponds to the maximal parabolic subgroup P ± 1 , and v ± 2 = C ± ∩ w 2 C ± corresponds to the maximal parabolic subgroup P ± 2 . It means that this apartment is a line in the tree, infinite in both directions. Note that according to the geometric pictures shown in Figures 1 and 2, this left-to-right ordering is natural in A + , but would be reversed in A − because the reflecting mirrors of w 1 and w 2 are crossed in the lower lightcone. The order would not be reversed if we think of viewing the lower lightcone from the origin, facing south. We use the integers n ∈ Z to label each chamber in A ± , as we did above to label elements of W , and its boundary vertices but notice that each vertex has two labels: v ± 1 (2m) = v ± 1 (2m − 1) and v ± 2 (2m) = v ± 2 (2m + 1). The chambers with boundary vertices may be written as and [v ± 2 (2m + 1), C ± (2m + 1), v ± 1 (2m + 1)] where we read the order of vertex, chamber, vertex, left to right. When w(2k + 1) acts on one of these, it will reverse the positions of the vertices. This can be seen using the formula w(2k + 1)w(n) = w(2k + 1 − n). We use this notation to describe a (left or right) "ray" of chambers starting with C ± (n) and including all boundary vertices: Note that the action of the Weyl group generators on the chambers of A ± is given by the simple formulas w 1 C ± (n) = C ± (−1 − n) and w 2 C ± (n) = C ± (1 − n) for any n ∈ Z, so that (w 2 w 1 ) m C ± (n) = C ± (n + 2m) for any m, n ∈ Z, and the action on rays is w 1 L ± ray (n) = R ± ray (−1 − n) and w 2 L ± ray (n) = R ± ray (1 − n) so for each m, n ∈ Z we have (w 2 w 1 ) m L ± ray (n) = L ± ray (n + 2m) and (w 2 w 1 ) m R ± ray (n) = R ± ray (n + 2m).
We can describe the action of the real root groups U ±(w 1 w 2 ) m α i , i = 1, 2, m ∈ Z, on the fundamental apartments A ± .
Using the notationî = 3 − i, we label the roots in Φ i by the integers as follows: so that the labels of roots in both branches are the integers in order, negative to positive going in Φ 1 from top to bottom, but going in Φ 2 from bottom to top: We also have w 1 Φ 1 (n) = Φ 2 (1 − n) and w 2 Φ 1 (n) = Φ 2 (−1 − n).
Here is another useful application of the labeling of the real roots by Φ i (n), n ∈ Z, as shown in the paragraph above. A consistent choice of real root vectors given by and then we would have w j e α = e w j α for all real roots α and j = 1, 2. L ± ray (n) is fixed by U ±Φ 2 (n) and R ± ray (n) is fixed by U ±Φ 1 (n) for all n ∈ Z, but U ±Φ 2 (n) R ± ray (n − 1) is a distinct family of rays in B ± indexed by C whose intersection with L ± ray (n) is only the vertex v ± 2 (n) if n is even, v ± 1 (n) if n is odd, and U ±Φ 1 (n) L ± ray (n + 1) is a distinct family of rays in B ± indexed by C whose intersection with R ± ray (n) is only the vertex v ± 1 (n) if n is even, v ± 2 (n) if n is odd. Proposition 8.1. We have the following results about the action of real root groups U ±Φ i (k) , k ∈ Z, on the chambers C ± (n), in the fundamental apartment A ± .
(1) The chambers of L ± ray (k) = {C ± (n) | n ≤ k} are each fixed by U ±Φ 2 (k) but U ±Φ 2 (k) R ± ray (k + 1) is a family of distinct rays in B ± indexed by C whose intersection with L ± ray (k) is only the vertex v ± 2 (k).
(2) The chambers of R ± ray (k) = {C ± (n) | n ≥ k} are each fixed by U ±Φ 1 (k) but U ±Φ 1 (k) L ± ray (k − 1) is a family of distinct rays in B ± indexed by C whose intersection with R ± ray (k) is only the vertex v ± 1 (k). Proof.
The proof of (2) is similar.
9. Embedding the spherical building at infinity in rank 2 Let G = G C (A) be a rank 2 hyperbolic Kac-Moody group with compact real form K. Let B = B + ∪ B − denote its twin building, whose simplicial structure must be a pair of trees. We define the twin spherical building at infinity, B ∞ = B + ∞ ∪ B − ∞ , by doing so for each tree B ± as follows. A ray in a tree is a sequence of connected vertices and edges (v 1 , e 1 , v 2 , e 2 , · · · ) with an initial vertex, v 1 , infinite in only one direction. Two rays are called equivalent if their intersection is infinite, that is, they have a common tail. An equivalence class of rays is called an end of the tree. A line in a tree is a sequence of connected vertices and edges infinite in both directions, so it can be expressed as the union of two rays having a finite nonempty intersection, which can always be taken to be exactly one vertex. So each line has two ends, the classes of those two rays. For each apartment, X = X ray1 ∪ X ray2 in B ± there an apartment [X] = [X ray1 ] ∪ [X ray2 ] in B ± ∞ whose two chambers [X ray1 ] and [X ray2 ] uniquely determine the line X as follows. Without loss of generality, we may assume that X ray1 ∩ X ray2 = {X 0 } is exactly one vertex. Suppose another apartment Y = Y ray1 ∪ Y ray2 with Y ray1 ∩ Y ray2 = {Y 0 } also exactly one vertex, has [X ray1 ] = [Y ray1 ] and [X ray2 ] = [Y ray2 ], so the intersections I 1 = X ray1 ∩ Y ray1 and I 2 = X ray2 ∩ Y ray2 are both infinite rays. The intersection I 1 ∩ I 2 ⊆ X ray1 ∩ X ray2 is either one vertex {X 0 } or empty, and I 1 ∩ I 2 ⊆ Y ray1 ∩ Y ray2 is either one vertex {Y 0 } or empty. So if I 1 ∩ I 2 is non-empty, then it is one vertex and that vertex is X 0 = Y 0 , so X = Y . Assume now that I 1 ∩ I 2 is empty and write I 1 = (v 1 , e 1 , v 2 , e 2 , · · · ) and I 2 = (w 1 , f 1 , w 2 , f 2 , · · · ) so v 1 is a vertex in X ray1 ∩ Y ray1 and w 1 is a vertex in X ray2 ∩ Y ray2 but the connected path in X from v 1 to w 1 goes through X 0 while the connected path in Y from v 1 to w 1 goes through Y 0 . If those two paths were distinct, there would be a loop in the tree, so they must be identical, giving X = Y .
For any apartment X in B ± and for any vertex v in X, we may write X = X 1 (v) ∪ X 2 (v) where X i (v), i = 1, 2, are the rays starting from v and going towards the two ends of X, which have been labeled 1 and 2. Let us suppress the ± for the moment to simplify the notation. Vertex v determines a partition of the tree into two "rooted" trees, T ree i (v), i = 1, 2, such that There can be no other vertices or edges in common between these two trees, since otherwise there would be a loop passing through that intersection and v. Sometimes the phrase "Nile Delta" is used to describe X 1 (v) ∪ T ree 2 (v) and X 2 (v) ∪ T ree 1 (v), since they look like a long river splitting into tributaries at the point v. We may use this terminology later to help understand the action of the group G on the tree.
Roughly speaking, a spherical building at infinity of a tree B ± consists of a 'sufficiently large' set of ends. The minimal spherical building at infinity may be constructed as follows. For each sign ± let A ± be the fundamental apartment of B ± with fundamental chamber C ± , and let A ± = {g · A ± | g ∈ G} be the set of all G-translates of A ± . Then A ± is an apartment system (see [AB08]). 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 chamber complex it consists of a set of points, and its apartments are exactly subsets of two distinct points corresponding to the two ends of a line. It is straightforward to show that G acts doubly transitively on Cham + ∞ . (See Lemma 9.1.) So in this rank 2 case, using this doubly transitive action, the existence of a spherical BN -pair with Weyl group W ∞ = {±1} ∼ = Z/2Z follows, thus relating the spherical building at infinity to the structure of G. We denote all objects associated to the spherical building at infinity, i.e. apartments, chambers or the Weyl group, with the subscript '∞'. Our construction of the spherical BN-pair follows the approach used in [CG03] to define spherical BN-pairs for rank 2 hyperbolic Kac-Moody groups over finite fields, which are then used to construct lattices in trees.
Lemma 9.1. The action of G on B ± ∞ is doubly transitive.
By construction of B ± ∞ there exist apartments A ± c and A ± d , such that c ± i are ends of A ± c and d ± i are ends of A ± d . As the action of G is transitive on the apartment system of B ± , there is a group element g 0 ∈ G such that g 0 · A ± c = A ± d . If g 0 · c ± i = d ± i we are done. Otherwise, g 0 ·c ± 1 = d ± 2 and g 0 ·c ± 2 = d ± 1 so we define g 0 = w 1 g 0 and see that g 0 ·c ± 1 = w 1 ·d ± 2 = d ± 1 and g 0 · c ± 2 = w 1 · d ± 1 = d ± 2 so g = g 0 = w 1 g 0 is the desired element of G.
In this section, for G of rank 2, we extend the embedding of the twin building B ± , given in Theorem 6.1 to the spherical building at infinity.
In the rank 2 case, the split real form of the Cartan subalgebra, h R , and the compact real form of the Cartan subalgebra, t, have a bilinear form with signature (1, 1). So for each 0 = r ∈ R, t r is a hyperbola with two connected components (branches) t ± r , and t 0 = ∂L t is a pair of lines (the asymptotes). We have L t = ∪ r≤0 t r and the real roots of g are on the hyperbolas t (α i ,α i ) 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 L 0 t = {x ∈ t | (x, x) < 0}, which is the Tits cone. We define an equivalence relation on the nonzero vectors in t 0 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 t 0 , which we denote by . This corresponds to the "lightlike closure" B ∞ (∂L t ) in definition 4.3. 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 , α 1 ) < 0, (x + 1 , α 2 ) > 0, (x + 2 , α 1 ) > 0, (x + 2 , α 2 ) < 0.
We will show that {x + 1 , x + 2 } corresponds to the fundamental apartment A + and {x − 1 , x − 2 } corresponds to the fundamental apartment A − . 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 twin spherical building at infinity B ∞ in the next theorem.
Theorem 9.2. There is a K-equivariant bijective map Ψ ∞ : B ∞ → Ξ ∞ such that the following diagram commutes: . Such a spherical building at infinity exists only in the highly degenerate case of rank 2. The analogous construction for higher rank hyperbolic Kac-Moody groups would have to involve a new kind of structure beyond the theory of buildings, perhaps replacing Coxeter groups with some other class of groups.
Our strategy for the proof of this result is to follow the proof of Theorem 6.1, but since chambers consist only of points, there are some simplifications. On the other hand, we need a more precise characterization of the Borel subgroups stabilizing chambers in Cham ± ∞ . Thus we first study the existence of a spherical BN -pair with Weyl group W ∞ .
Recall that A ± denotes the fundamental apartment in B ± so it is a line in that tree with two ends. Let e ± i,∞ , i = 1, 2, denote the two ends of that line, that is, the equivalence classes of certain rays defined as follows. As shown in Proposition 8.1, for each i = 1, 2, m ∈ Z, the subgroup U (w 1 w 2 ) m α i fixes the end e ± i,∞ if we define e + 1,∞ = [R + ray (m)], e + 2,∞ = [L + ray (m)], ]. Furthermore, we have shown that U −(w 1 w 2 ) m α 1 fixes e ± 2,∞ and U −(w 1 w 2 ) m α 2 fixes e ± 1,∞ . This means that ω(e + 1,∞ ) = e − 2,∞ and ω(e + 2,∞ ) = e − 1,∞ . We now define the stabilizer of e ± i,∞ and its complement Then for any m ∈ Z, we have established that U (w 1 w 2 ) m α 1 and U −(w 1 w 2 ) m α 2 are in B ± 1,∞ , and that U (w 1 w 2 ) m α 2 and U −(w 1 w 2 ) m α 1 are in B ± 2,∞ . Recall the non-standard partition of the real roots of g, Definition 9.4. For i = 1, 2, we set We have just shown the following result.
Lemma 9.6. We have the decomposition Q ± i,∞ = B ± i,∞ (−1)B ± i,∞ or equivalently a Bruhat decomposition Proof. The action of G on B ± i,∞ is doubly transitive by lemma 9.1, and by definition B ± i,∞ is the stabilizer of e ± i,∞ . We continue the proof for the case i = 1, since the other case is similar. We will show that for any pair of chambers c ± 1 , c ± 2 ∈ Cham ± ∞ there is a g ∈ Q ± 1,∞ mapping c ± 1 onto the end e ± 1,∞ of the fundamental apartment A ± and c ± 2 onto the other end e ± 2,∞ of A ± . To do this, we chose a (uniquely defined) apartment A ± c ∈ B ± , whose ends are c ± 1 and c ± 2 . We chose first b 1 ∈ B ± 1,∞ such that b 1 c ± 1 = e ± 2,∞ . Then we apply the Weyl group element −1 ∈ W ∞ , so (−1)b 1 c ± 1 = e ± 1,∞ . Transitivity of B ± 1,∞ on ends other than e ± 1,∞ assures the existence of an Applying that description to the spherical BN -pair at infinity in Theorem 9.7, we have ∞ is the fundamental chamber in the fundamental apartment and ∼ is the equivalence relation which would only be non-trivial on simplices in the boundary of ∆ ± ∞ , so that relation is trivial. The fundamental apartment of B ± ∞ is the two-element set A ± ∞ = {e ± 1,∞ , e ± 2,∞ }, corresponding to the ends of the fundamental apartment A ± in B ± . Either of these two points could be chosen as the fundamental chamber. The apartment system of B ± ∞ is the collection of twoelement sets consisting of the ends of apartments in the apartment system A ± = {gA ± | g ∈ G} of B ± . The action of −1 ∈ W ∞ on each apartment A ± c = {c ± 1 , c ± 2 } interchanges those two chambers, (−1) · c ± 1 = c ± 2 . So the set of all chambers of B ± ∞ is the G orbit of any one chamber, ,∞ e ± 2,∞ . Another interpretation of this result is that the set of ends of apartments in B ± consists of one end of fundamental apartment A ± (the class of any right ray in A + , any left ray in A − ), and the images of the opposite end of A ± under the stabilizer of the first end. To get all apartments in the apartment system of B ± ∞ we apply the same reasoning to pairs, starting with the fundamental apartment A ± ∞ = [e ± 1,∞ , e ± 2,∞ ]. Its orbit under G consists of two parts, NOTE: These statements are still under construction.
Recall the even subgroup of the Weyl group W has been defined to be W even = {t m = (w 1 w 2 ) m | m ∈ Z} ∼ = Z so that W = W even ∪ W odd where W odd = w 1 W even . The action of W even on an apartment in B ± , {C(n) | n ∈ Z}, with chambers C(n), (a line in the tree) is by translation of chambers, t m C(n) = C(n + 2m), leaving the two ends fixed. In particular we have W even ⊂ B ± i,∞ for i = 1, 2.
We define As B ± consists of all elements in G fixing the fundamental chamber C ± ⊂ A ± , its conjugate wB ± w −1 fixes the chamber wC ± ∈ A ± . Hence B ± W can be characterized as follows: Recall the definition of U ± from section 4.3. As we defined B ± W , we similarly define the group U ± W = w∈W wU ± w −1 and since B ± = T C U ± , we have the relation (17) B ± W = T C U ± W . Remark 9.8. Each element w ∈ W has a length, the minimum number of simple reflections in an expression for w. But that number is also the number of positive real roots sent by w to Φ re− . Since wU α w −1 = U wα , as longer elements are applied, more positive root generators in U + are eliminated from the intersection (they are not in U + ), and in the complete intersection all positive root generators are eliminated. But that means that what is left must correspond to the imaginary root spaces, which can only be the result of complicated commutators of real root generators. Hidden inside this intersection may be the structures giving the imaginary root multiplicities.
Lemma 9.9. Each group U i , i = 1, 2, is abelian and is normalized by W even , so U i W even = W even U i . Furthermore, U i W odd = W odd Uî.
To see that W even normalizes each group U i , note that for any w ∈ W and for any α ∈ Φ i , we have w exp(ad sxα ) w −1 = exp(ad sw(xα) ) ∈ U w(α) ≤ U i since W even Φ i = Φ i . On the other hand, W odd Φ i = Φî, so we get the final statement.
Lemma 9.10. W normalizes B ± W and U ± W , so Proof. Obvious.
Corollary 9.11. We have W even normalizes B ± W and U ± W , so W even B ± W = B ± W W even = B ± 1,∞ ∩ B ± 2,∞ and W even U ± W = U ± W W even .
Proof. By definition B ± W is the pointwise stabilizer of the standard apartment A ± , while W even contains the translations of the standard apartment. Hence the group B ± W W even contains all elements in G which preserve both ends of the standard apartments A ± in B ± . Hence we have Proposition 9.12. We have U 2 ∼ = {f : Z → C | supp(f ) < ∞} under pointwise addition.
Proof. We assume a choice of real root vectors {e α ∈ g α | α ∈ Φ re } has been made as follows. We begin by letting e α i = e i be the simple root generators of g, and then set e −α i = w i (e i ) = −f i . Application of additional generators w j fills out the rest of the real root vectors, and in this way we have arranged it so that we have w i (e α ) = e w i α for i = 1, 2 for all α ∈ Φ re . Each element of u ∈ U 2 is a finite product of elements exp(ad zαeα ) for α ∈ Φ 2 , z α ∈ C, but we have a labeling by the integers of Φ 2 , so u can be written as where only a finite number of the z n are non-zero, and the order of the factors can be decreasing since U 2 is abelian. We associate u with the function f u : Z → C such that f u (n) = z n , so supp(f u ) < ∞. Conversely, each function f : Z → C with supp(f ) < ∞ defines an element The pointwise addition of two such functions corresponds to the product in U 2 , that is, for any From now on we use the functional notation f u for u ∈ U 2 , and introduce some new notations for supp(f u ) in order to express its action on We call these indices the hinges of u and for each hinge m j , we call f u (m j ) the direction of that hinge, so that this information determines the apartment uA ± with end e ± 2,∞ as a deformation of A ± . For f u for u ∈ U 2 , note that L ± ray (m 1 ) ⊂ L ± ray (m 2 ) ⊂ · · · ⊂ L ± ray (m r ) and U ±Φ 2 (m j ) fixes L ± ray (m j ), so u fixes L ± ray (m 1 ) and moves the segments of chambers (m 1 , m 2 ], (m 2 , m 3 ], · · · , (m r−1 , m r ], (m r , e 1 ), whose union is the ray R ± ray (m 1 + 1). So uA ± = L ± ray (m 1 ) ∪ u(m 1 , m 2 ] ∪ u(m 2 , m 3 ] ∪ · · · ∪ u(m r−1 , m r ] ∪ uR ± ray (m 1 + 1) and at each hinge m j we have a direction f u (m j ) = z j ∈ C. Thus, uA ± is the apartment with ends e 2 and ue 1 . We have used the abbreviations (m j , m j+1 ] for a sequence of connected chambers with labels corresponding to that interval of labels, and e i for e ± i,∞ . See Figure 4 for an illustration. : Apartments A = A ± and uA for u ∈ U 2 with supp(f u ) = {m 1 > m 2 > · · · > m r } and f u (m j ) = z j .
We could have associated a function f : Z → C with finite support with an element of U 1 or U 2 , but that would have caused some confusion about the choice. We know that w i , i = 1, 2, each switch the ends of A ± , their action on the chambers is by w 1 C ± (n) = C ± (1 − n) and w 2 C ± (n) = C ± (−1 − n), and for any n ∈ Z we have allows us to write elements of U 1 as w 1 u f w −1 1 . There is a similar formula using w 2 : We wish to use these descriptions of U 1 and U 2 , and their actions on the tree, to understand what are all possible ends of apartments in B ± . The problem is that G = U 1 , U 2 so that in principle, we have to consider all finite sequences of products U 1 U 2 U 1 U 2 · · · U 1 e i to get all possible ends. These divide into those coming from U 1 e 2 and those coming from U 2 e 1 . The situation is symmetric so it would be enough to understand whether U 1 U 2 e 1 reaches more ends than U 2 e 1 . It seems unlikely that U 1 U 2 e 1 = U 2 e 1 , that is, the ends that can be reached by the action of U 2 on e 1 are all the ends of the building other than e 2 . The ends {ue 1 | u ∈ U 2 } must all be distinct, since otherwise there would be a loop in the tree, and none of them can be e 2 since ue 2 = e 2 = ue 1 would give e 1 = e 2 . Similarly, the ends {ue 2 | u ∈ U 1 } must all be distinct, and none of them can be e 1 . Thus it appears that the set of all ends of the building B ± , denoted by Ends(B ± ), includes at least {e ± 2,∞ } ∪ {u 2 e ± 1,∞ | u 2 ∈ U 2 } and {e ± 1,∞ } ∪ {u 1 e ± 2,∞ | u 1 ∈ U 1 }.
Conjecture 9.13. Let A ± g = g · A ± for some g ∈ B ± i,∞ . Then there exists some u ∈ U i such that u · A ± g = A ± .
Attempt to prove conjecture: We will do the case A + g = g · A + for some g ∈ B + 2,∞ since the other cases are similar. We write more briefly A for A + and we write its ends as e i = e + i,∞ for i = 1, 2. We wish to show there exists some u ∈ U 2 such that uA g = A. This is trivially true with u = 1 when g = 1. Assume g = 1 and that the result has been established for all shorter elements g = g i 2 · · · g in ∈ B + 2,∞ . Using the correspondence between lines in the tree B + and pairs of ends in B + ∞ , we write A = [e 2 , e 1 ] and A g = gA = [ge 2 , ge 1 ] = [e 2 , ge 1 ] because g ∈ B + 2,∞ fixes the end e 2 .
If g i 1 ∈ U 2 ⊂ B + 2,∞ then g −1 i 1 g = g i 2 · · · g in ∈ B + 2,∞ so by induction there is an element u 2 ∈ U 2 such that u 2 g i 2 · · · g in A = A. Then u = u 2 g −1 i 1 ∈ U 2 satisfies uA g = A. Now assume g i 1 ∈ U 1 ⊂ B + 1,∞ then g −1 i 1 g = g i 2 · · · g in ∈ U 1 B + 2,∞ . The problem is that we have too little information about g i 2 · · · g in so we cannot apply the inductive hypothesis to it.
If we could prove the conjecture above, we would have the following two results: Conjecture 9.14. For i = 1, 2 we have the decomposition Proof.
(1) We have to establish the stated decomposition of B ± i,∞ into the product of the subgroups B ± W = T C U ± W , U i and W even . Hence there is some bw ∈ B ± W W even such that u n . . . u 1 g = bw. Thus we find the product expression g = u −1 1 . . . u −1 n bw giving the decomposition (2) Using lemmas 9.9 and 9.10 yields the product decompositions To do this use the notation of part 1. Assume g ∈ B ± i,∞ has a decomposition g = ubw such that u ∈ U i , b ∈ B ± W and w ∈ W even . We start with the apartment A ± g −1 = g −1 ·A ± . Clearly g · A ± g −1 = A ± . The argument in step 1 shows that there is some u ∈ U i such that ug −1 · A ± = u · A ± g −1 = A ± . Hence there is some element wb ∈ W even B ± W such that The use of lemmas 9.9 and 9.10 establishes the other decompositions.
For 1 ≤ i ≤ 2 we define U ± i,∞ = U ± W U i . Conjecture 9.15. The hyperbolic Kac-Moody group G admits a spherical Iwasawa decomposition Proof. We know that K acts transitively on chambers of B ± so it acts transitively on B ± ∞ . Let g ∈ G. This gives chambers c ± i = g · e ± i,∞ , i = 1, 2. Then by transitivity of K on B ± ∞ we find elements k i ∈ K such that k i ·c ± i = e ± i,∞ , so that (k i g)·e ± i,∞ = e ± i,∞ . Thus k i g ∈ B ± i,∞ , establishing (18) From B ± i,∞ = W even T C U ± i,∞ = W even T T ad R U ± i,∞ we get K ∩ B ± i,∞ T W even and in consequence a decomposition Remark 9.16. A spherical Iwasawa decomposition was established in Proposition 11.99 in [AB08] for Euclidean buildings and follows from a spherical Bruhat decomposition. For our Kac-Moody group G over C, we proved a spherical Bruhat decomposition (lemma 9.6). In rank 2, we may use the fact that the hyperbolic and affine spherical buildings coincide. Hence we may apply Proposition 11.99 in [AB08] to deduce the theorem.
Let B ± i,K = B ± i,∞ ∩ K. In our next step towards the proof of Theorem 9.2, we show that B + 1,K = B − 1,K = B + 2,K = B − 2,K = B K is independent of the choice of sign ± and of the choice of i, and establish a K-equivariant bijection between G/B ± i,∞ and K/B K .
Lemma 9.17. We have B ± i,K = T W even , so B K = B ± i,K is independent of the choice of sign ± and of i = 1, 2.
Proof. By lemma ?? we have B ± i,∞ = B ± W U i W even . We calculate the intersection of each component with K. Recall that B ± ∩ K = T . As T ⊂ B ± W ⊂ B ± , we see that B ± W ∩ K = T . For i = 1, 2, set (U i ) K = U i ∩ K = {u ∈ U i | ω(u) = u} where ω denotes the Cartan involution.
Corollary 9.18. The chambers of B ± ∞ are in bijection with the elements of K/B K , hence from equation (16) we get B ± ∞ = K/B K × {e ± i,∞ } for either choice of i = 1, 2.
Proof. The group K acts transitively on chambers which are the points of of B ± ∞ . Hence chambers correspond to the quotient of K modulo the subgroup B K fixing the chamber e ± i,∞ for either choice of i = 1, 2.
Proof. The action of the even Weyl group W even fixes the four rays of ∂L t , 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 B K , which is contained in the pointwise stabilizer of this set. Now suppose that k ∈ K stabilizes the set pointwise. Then k sends two linearly independent points on the two lines of ∂L t to linearly independent points, a basis for t, so k normalizes t. But N K (t) = T W , and the elements in the set T W odd exchange x ± 1 and x ± 2 , so k ∈ T W even = B K .
We now have all technical ingredients to prove Theorem 9.2.
The results in this section suggest that on the Lie algebra level one could study a non-standard Cartan decomposition g = h ⊕ α∈P 1 g α α∈P 2 g α based on a non-standard partition Φ = P 1 ∪ P 2 such that Φ i ⊂ P i for i = 1, 2, and such that n 1 = α∈P 1 g α and n 2 = α∈P 2 g α are subalgebras. Then b i = h ⊕ n i , i = 1, 2, would be non-standard Borel subalgebras related to the Borel subalgebras B ± i,∞ . That would be accomplished if each subset P i is closed under addition. Since Φ is a subset of the dual of the split real Cartan subalgebra, h * R , one might consider using a timelike line in the interior of the lightcone L 0 h R to partition all of Φ. If such a line does not contain any roots, every root is on one side or the other, and Φ 1 and Φ 2 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 P i 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 P 1 = Φ 1 ∪ (Φ im ) + and P 2 = Φ 2 ∪ (Φ im ) − , while for the line determined by x + 1 the partition would be P 1 = Φ 1 ∪ (Φ im ) − and P 2 = Φ 2 ∪ (Φ im ) + . Associated to each of the four non-standard Borels at infinity, B ± i,∞ , is a choice of one of these four half-partitions.
Having found a non-standard Cartan decomposition as above, one gets a corresponding nonstandard decomposition of the universal enveloping algebra U(g) = U(n 1 )U(h)U(n 2 ) and can construct induced Verma modules of two types, V erma i (λ) = U(n i )v i λ , i = 1, 2, where v i λ are vectors such that h · v i λ = λ(h)v i λ for any h ∈ h, n 1 · v 2 λ = 0 and n 2 · v 1 λ = 0. The quotient of such a Verma module by its maximal proper submodule would be an irreducible module, Irred i (λ) generated by v i λ . These would be integrable g-modules for λ = n 1 λ 1 + n 2 λ 2 in the weight lattice of g (λ 1 , λ 2 are the fundamental weights of g such that λ i (h j ) = δ ij ) but outside of the lightcone, so that n 1 and n 2 have opposite signs. Examples of such integrable modules, in addition to the adjoint representation, were mentioned by Borcherds in Section 6 of [Bor86]. There is evidence that they occur in the decomposition of the rank 3 hyperbolic algebra, F, with respect to its rank 2 hyperbolic subalgebras. That decomposition is currently under study [Pen16].

Conclusion and further directions
Our lightcone embedding of the twin building of a hyperbolic Kac-Moody group is motivated by the conjectural existence of hyperbolic Kac-Moody symmetric spaces. There have been some efforts recently to develop the geometry of hyperbolic Kac-Moody symmetric spaces, building on work of the third author on the construction of affine Kac-Moody symmetric spaces [Fre09,Fre11,Fre12b,Fre13c,Fre13b].
Recalling the well-known finite dimensional theory (see for example [Hel01,Ebe96]), the boundary of a symmetric space M of non-compact type IV corresponding to a complex simple Lie group G, can be identified with the building over C associated to G. Via the isotropy representation at a point p ∈ M , the building can be embedded into the unit sphere of the tangent space T p M . In this way, points in the building get identified with directions in the tangent space of the symmetric space. Via the duality between the compact type and non-compact types, we can identify spaces of type IV with spaces of type II and thus also obtain an embedding of the building into the tangent space of a compact symmetric space.
In the absence of hyperbolic Kac-Moody symmetric spaces, important properties of the local geometry are captured via the embedding of the building into the Lie algebra. Our embedding of the building into the Lie algebra gives local pictures of the tangent spaces of conjectural hyperbolic Kac-Moody symmetric spaces of types II and IV . We note however, that since our twin building embeds into the lightcone of the compact form of the Lie algebra, it captures only the timelike directions in the tangent space.
Via Proposition 4.2, we have also obtained a hyperbolic analog of the notion of a polar representation by the group K on the K-conjugacy class of a Cartan subalgebra. Recall that a group representation G : V −→ V on a vector space V is called polar if there exists a subspace Σ ⊂ V , called a section, such that each orbit G · v for v ∈ V intersects Σ orthogonally. Finite dimensional polar representations are orbit equivalent to isotropy representations of finite dimensonal Riemannian symmetric spaces [Dad85]. Similar observations were made in the affine case [HPTT95,Gro00]. Our Proposition 4.2 shows that the action of K on H is a polar action with section t.
Further questions about polar representations for hyperbolic Kac-Moody groups remain open and the full differential geometry need to develop hyperbolic Kac-Moody symmetric spaces remains elusive. We hope to take this up elsewhere.