The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra

Misha Kapovich, Bernhard Leeb and John J. Millson

October 16, 2002

Abstract

In this paper we apply our results on the geometry of polygons in Cartan subspaces, symmetric spaces and buildings [KLM] to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the computation of the structure constants of the Hecke and representation rings associated with a split reductive algebraic group over ℚ and its complex Langlands’ dual. We give a new proof of the “Saturation Conjecture” for GL(ℓ) as a consequence of our solution of the corresponding “saturation problem” for the Hecke structure constants for all split reductive algebraic groups over ℚ.

1 Introduction

In this paper we will examine and generalize the algebra problems listed immediately below from the point of view of spaces of non-positive curvature. Let α, β and γ be m-tuples of real numbers arranged in decreasing order. In Problem P3 we let 𝕂 be a complete, nonarchimedean valued field. We assume that the valuation v is discrete and takes values in ℤ. We let O ⊂ 𝕂 be the subring of elements with nonnegative valuation.

In order to state Problems P2 and P3 below we recall some definitions from algebra. The singular values of a matrix A are the (positive) square-roots of the eigenvalues of the matrix AA^{∗}. To define the invariant factors of a matrix A with entries in 𝕂 note first that it is easy to see that the double coset

GL(ℓ, O) · A · GL(ℓ, O) ⊂ GL(ℓ, 𝕂)

is represented by a diagonal matrix D := D(A). The invariant factors of A are the integers obtained by applying the valuation v to the diagonal entries of D. If we arrange the invariant factors in decreasing order they are uniquely determined by A.

In the Problem P4 we will assume that α, β and γ are dominant weights of GL(m, ℂ) (i.e. they are vectors in ℤ^{m} with nonincreasing entries) and that V_{α}, V_{β} and V_{γ} are the irreducible representations of GL(ℓ, ℂ) with these highest weights.

Now we can state the four algebra problems that interest us here, following the presentation in [Fu].

• P1. Eigenvalues of a sum. Give necessary and sufficient conditions on α, β and γ in order that there exist Hermitian matrices A, B and C such that the set of eigenvalues (arranged in decreasing order) of A, resp. B, resp. C is α, resp. β, resp. γ and

A + B + C = 0.

• P2. Singular values of a product. Give necessary and sufficient conditions on α, β and γ in order that there exist matrices A, B and C in GL(m, ℂ) the logarithms of whose singular values are α, β and γ, respectively, so that

ABC = 1.

• P3. Invariant factors of a product. Give necessary and sufficient conditions on the integer vectors α, β and γ in order that there exist matrices A, B and C in GL(m, 𝕂) with invariant factors α, β and γ, respectively, so that

ABC = 1.

• P4. Decomposing tensor products. Give necessary and sufficient conditions on α, β and γ such that

(V_{α} ⊗ V_{β} ⊗ V_{γ})^{GL(m,ℂ)} ≠ 0.

These problems have a long history which is described in detail in [Fu]. Their complete solution and the relation between them were established only recently due to the efforts of several people. It turns out that the sets of solutions (α, β, γ) for Problems P1 and P2 form the same polyhedral cone in ℝ^{3m}. This polyhedral cone is given by a finite system of linear homogeneous inequalities involving the Schubert calculus in the Grassmannians G(p, ℂ^{m}). The sets of solutions (α, β, γ) for Problems P3 and P4 are also the same, namely the integral points in the above polyhedral cone. We refer to [Fu], [AW], [AMW], [Kly98], [Kly99] and [Bel] for more details and further developments.

The above description of the set of solutions to Problem P1 using the Schubert calculus was proved by A. Klyachko [Kly98], with an improvement by P. Belkale [Bel] after much classical work. The equivalence of Problems P1 and P2 was proved by Klyachko [Kly99]. The equivalence of Problems P3 and Problem P4 is due to P. Hall, J. Green and his student, T. Klein [K68] and [K69]; see also [Mac, pg. 100] and [K68] for the history of this problem. The description of the solutions to Problem P4 as the set of integral points in the above polyhedral cone is due to A. Knutson and T. Tao, [KT] (see also [DW]). The Knutson-Tao theorem combined with the above equivalence of Problems P3 and P4 establishes that the set of solutions to Problem P3 is also the set of integral points in the above polyhedral cone. In what follows we reverse this path: we first prove directly that the set of solutions to Problem P3 is the set of integral points in the above polyhedral cone, then using the equivalence of Problems P3 and P4 we deduce that this set is also the set of solutions to Problem P4.

The algebra problems above extend naturally to general reductive groups. Let 𝔽 be either the field ℝ or ℂ, and let 𝕂 be a nonarchimedean valued field with discrete valuation ring O and the value group ℤ. For simplicity, let us consider here a split reductive group \underline{G} over ℚ, see chapter 4 for a more general discussion.

• Q1. Eigenvalues of a sum. Set G := \underline{G}(𝔽), let K be a maximal compact subgroup of G. Let 𝔤 be the Lie algebra of G, and let 𝔤 = 𝔨 + 𝔭 be its Cartan decomposition. Give necessary and sufficient conditions on α, β, γ ∈ 𝔭/Ad(K) in order that there exist elements A, B, C ∈ 𝔭 whose projections to 𝔭/Ad(K) are α, β and γ, respectively, so that

A + B + C = 0.

• Q2. Singular values of a product. Let G and K be the same as above. Give necessary and sufficient conditions on α, β, γ ∈ K\G/K in order that there exist elements A, B, C ∈ G whose projections to K\G/K are α, β and γ, respectively, so that

ABC = 1.

• Q3. Invariant factors of a product. Set G := \underline{G}(𝕂) and K := \underline{G}(O). Give necessary and sufficient conditions on α, β, γ ∈ K\G/K in order that there exist elements A, B, C ∈ G whose projections to K\G/K are α, β and γ, respectively, so that

ABC = 1.

Equivalently, if c_{α} , c_{β} and c_{γ} are the characteristic functions of the above double cosets and c_{α}·c_{β} ·c_{γ} = ∑_{δ} m_{α,β,γ}(δ) c_{δ}, is the triple product in the Hecke algebra H_{G} (see chapter 9), give necessary and sufficient conditions on α, β and γ so that

m_{α,β,γ}(1) ≠ 0.

• Q4. Decomposing tensor products. Let \underline{G}^{∨} be the Langlands’ dual group of \underline{G}, see Definition 2.13. Give necessary and sufficient conditions on highest weights α, β, γ of irreducible representations V_{α}, V_{β}, V_{γ} of G^{∨} := \underline{G}^{∨}(ℂ) so that

(V_{α} ⊗ V_{β} ⊗ V_{γ})^{G^{∨}} ≠ 0.

Equivalently, if ch(V_{λ}) is the character of the irreducible representation of G^{∨} with the highest weight λ and ch(V_{α}) · ch(V_{β}) · ch(V_{γ}) = ∑_{δ} n_{α,β,γ}(δ) ch(V_{δ}), is the triple product in the representation ring R(G^{∨}), give necessary and sufficient conditions on α, β and γ so that

n_{α,β,γ}(1) ≠ 0.

Here 1 denotes the trivial character (corresponding to the zero weight vector).

Remark 1.1. The product · in Q3 is the convolution product in the Hecke algebra. Problems Q3 and Q4 are related by the Satake isomorphism S, which is an isomorphism from H_{G} ⊗ ℤ[q^{1/2}, q^{−1/2}] to R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}], where q is the order of the residue field of 𝕂 (assuming that 𝕂 is locally compact and q is finite). However S does not send c_{α} to a multiple of ch(V_{α}) but rather S(c_{α}) and q^{⟨α,ρ⟩}ch(V_{α}) differ by terms of lower order in the dominance order. Here ⟨α, ρ⟩ is a half-integer, see chapter 9.

The quotient spaces 𝔭/Ad(K) and K\G/K in Problems Q1 and Q2 are naturally identified with the Euclidean Weyl chamber Δ inside a Cartan subspace 𝔞 ⊂ 𝔭. Notice that in the case that 𝔽 = ℂ, we have that 𝔭 = i·𝔨 and Problem Q1 can be reformulated as finding necessary and sufficient conditions on triples of Ad(K)-orbits in 𝔨 in order that they contain elements with sum zero.

The double coset space appearing in Problem Q3 is a more subtle discrete object, namely the intersection of Δ with a certain lattice L\underline{G} ⊂ 𝔞. To be more precise let \underline{T} be a maximal split torus of \underline{G}. Let X_{∗}(\underline{T}) be the lattice of cocharacters (i.e. algebraic one-parameter subgroups) of \underline{T}. Then L\underline{G} = X_{∗}(\underline{T}) if \underline{G} is split over ℚ. If the group \underline{G} does not split over ℚ, the extended cocharacter lattice L\underline{G} contains X_{∗}(\underline{T}) as a subgroup of finite index, see section 4.4 for the precise definition of this lattice.

We now explain the appearance of the dual group G^{∨} in Problem Q4. The torus \underline{T} corresponds naturally to a maximal torus \underline{T}^{∨} of \underline{G}^{∨} such that the character group X^{∗}(\underline{T}^{∨}) of \underline{T}^{∨} satisfies the duality

X^{∗}(\underline{T}^{∨}) = X_{∗}(\underline{T}).

Thus 𝔞 = X^{∗}(\underline{T}^{∨}) ⊗ ℝ = X_{∗}(\underline{T}) ⊗ ℝ. The set of highest weights in Problem Q4 is the intersection of the cone Δ ⊂ 𝔞 with the lattice X^{∗}(\underline{T}^{∨}). Thus the parameter space Σ_{inv fact} of the Problem Q3 and the parameter space Σ_{tensor} of Problem Q4 are equal.

In this paper, we reformulate the algebra problems Q1–Q3 as geometric problems which are special cases of a geometric question raised and studied in [KLM]. We fix a Euclidean Coxeter complex (E, W_{aff}) with the Euclidean Weyl chamber Δ ≅ E/W_{sph}, and consider nonpositively curved metric spaces X with geometric structures modelled on (E, W_{aff}). The spaces of this kind which we are interested in are symmetric spaces of nonpositive curvature, their infinitesimal versions (Cartan motion spaces, see section 5.2), and Euclidean buildings. For such spaces X there is a notion of Δlength for oriented geodesic segments which reflects the anisotropy of X. This leads to the following problem:

• GTI: Generalized Triangle Inequalities. Give necessary and sufficient conditions on α, β, γ ∈ Δ in order that there exists a geodesic triangle in X with Δ-side lengths α, β and γ.

As explained in section 5.4, for a symmetric space X, the problem GTI corresponds to the Singular Value Problem Q2. For an infinitesimal symmetric space X, it corresponds to the Eigenvalues of a Sum Problem Q1. The Generalized Triangle Inequalities have been determined in the papers [LM] and [KLM] in full generality.

Theorem 1.2 ([LM, KLM]). Suppose that X is a symmetric space of noncompact type, or an a Cartan motion space or a thick Euclidean building. Then:

1. The set D_{3}(X) ⊂ Δ^{3} of triples (α, β, γ) for which a triangle in the Problem GTI (for X) exists, is a polyhedral cone.

2. D_{3}(X) depends only on the spherical Coxeter complex associated to X. More precisely, suppose that X, X′ are metric spaces (each of which is either a Cartan motion space or a nonpositively curved symmetric space or a Euclidean building) modelled on Euclidean Coxeter complexes (E, W_{aff}), (E′, W′_{aff}) respectively. Then each isometric embedding^{1} f : E → E′ which induces an embedding ϕ of the spherical Coxeter groups W_{sph} → W′_{sph}, induces an embedding D_{3}(X) → D_{3}(X′). In particular, if f and ϕ are also surjective, then the map D_{3}(X) → D_{3}(X′) is a bijection.

Combining this, for instance, with the description of the polyhedron D_{3}(X) for complex semisimple Lie algebras in terms of Schubert calculus [BeSj, LM], one obtains a method for determining the polyhedron D_{3}(X) in general. This completely solves the algebra problems Q1 and Q2 above. The equivalence of Problems Q1 and Q2 for some classical groups (including GL(ℓ)) was proved by Klyachko in [Kly99] and for all complex semisimple groups in [AMW].

The Invariant Factor Problem Q3 corresponds to the case when X is a Euclidean building and in this case the inequalities defining D_{3}(X) give only necessary conditions on (α, β, γ) to solve the algebraic problem Q3. Solutions of the algebraic problem correspond to triangles in X that have vertices with stabilizers conjugate to \underline{G}(O). We thus have to refine the geometric problem taking into account the lack of homogeneity in Euclidean buildings. To do so, we introduce a refined notion of length for oriented geodesic segments \overline{xy}, called refined length σ_{ref}(x, y), which keeps track of the location of endpoints; σ_{ref} takes values in (E × E)/W_{aff}. We thus are lead to the following problem:

• RGTI: Refined generalized triangle “inequalities”. Give necessary and sufficient conditions on α, β, γ ∈ (E × E)/W_{aff} in order that there exists a geodesic triangle in X with refined side lengths α, β and γ.

We have only partial results regarding this problem. We have the following comparison result for the set D_{3}^{ref}(X) ⊂ ((E ×E)/W_{aff})^{3} of triples which can be realized as refined side lengths of a triangle in X.

Theorem 1.3 (Transfer theorem, [KLM]). Let X and X′ be Euclidean buildings modelled on Coxeter complexes (E, W_{aff}) and (E′, W′_{aff}). Then each embedding (E, W_{aff}) ↪ (E′, W′_{aff}) of Coxeter complexes induces an embedding D_{3}^{ref}(X) ↪ D_{3}^{ref}(X′).

In particular, if the Coxeter complexes (E, W_{aff}) and (E′, W′_{aff}) are isomorphic then the isomorphism induces a bijection D_{3}^{ref}(X) → D_{3}^{ref}(X′).

Remark 1.4. Theorems 1.2 and 1.3 are also valid for n-gons.

1An affine embedding would also suffice.

For applications to the algebra problems Q3 and Q4 we have to impose further integrality conditions on the side lengths. Set L := L\underline{G} and let X := X_{G} denote the Bruhat-Tits building associated with the group G = \underline{G}(𝕂). We define the set Δ_{L} of L-integral Δ-lengths as the intersection Δ ∩ L · o, where o is the vertex of E fixed by W_{sph}. We will identify o with the origin 0 ∈ E and thus identify L · o with L. We define the set of L-integral refined lengths as (L × L)/W_{aff}. The Invariant Factor Problem Q3 is then equivalent to determining the set D_{3}^{ref,L}(X) of possible L-integral refined side lengths for triangles.

We have the natural embedding

ι : D_{3}^{ref,L}(X) → L^{3} ∩ D_{3}(X),  (1)

and our goal will be to identify its image. We will show that

ι(D_{3}^{ref,L}(X)) ⊂ D_{3}^{L,0}(X) := {(α, β, γ) ∈ L^{3} ∩ D_{3}(X) : α + β + γ ∈ Q(R^{∨})}.

We remind the reader that ι(D_{3}^{ref,L}(X)) is the set of solutions to Problem Q3, that is:

ι(D_{3}^{ref,L}(X)) = {(α, β, γ) ∈ (L ∩ Δ)^{3} : 1 ∈ Kα(π)Kβ(π)Kγ(π)K}.

The following theorem, a consequence of a recent theorem in logic of M. C. Laskowski [La], building on work of S. Kochen [Ko], reveals the general structure of the image of D_{3}^{ref,L}(X) in D_{3}^{L,0}(X) ⊂ L^{3}. Define a subset of L^{3} to be elementary if it is the set of solutions of a finite set of linear inequalities with integer coefficients and a finite set of linear congruences. Then Laskowski proves the following

Theorem 1.5. (M. C. Laskowski.) There is an integer N = N\underline{G}, depending only on \underline{G}, such that for any nonarchimedean local field 𝕂 with residue characteric greater than N we have

ι(D_{3}^{ref,L}(X)) ⊂ L^{3} is a finite union of elementary sets.

Remark 1.6. It follows from our Transfer Theorem above that Theorem 1.5 holds for all nonarchimedean local fields 𝕂. This is because all the groups \underline{G} as 𝕂 varies have the same Coxeter system. Hence by our Transfer Theorem for a split group the set ι(D_{3}^{ref,L}(X)) does not depend on 𝕂. Hence, once the above statement is true for one of them it is true for all of them.

Problem 1.7. Find the corresponding inequalities and congruences.

We can solve the Problem 1.7 and thus the Problem Q3 for the following groups:

Theorem 1.8. a. Let \underline{G} = SL(ℓ) or \underline{G} = GL(ℓ). Then the embedding (1) is onto.

b. If \underline{G} is covered by SL(ℓ) (and whence L = L\underline{G} ⫆ L_{SL(ℓ)}), then the image of (1) is the subset of the triples (α, β, γ) in Δ_{L}^{3} ∩ D_{3}(X) such that

α + β + γ ∈ L_{SL(ℓ)}.

A similar statement holds for a more general class of groups, see Theorem 8.14. For example, the conclusion of part (a) holds for the groups SL(ℓ, 𝔻) where 𝔻 is a division algebra over 𝕂.

In general, the map (1) is not surjective: counterexamples for groups of type B_{2} = C_{2} (both adjoint and simply-connected) are given in section 7.2. However, we have the following theorem (the solution of the “saturation problem” for the structure constants of the Hecke algebra), see Corollaries 8.8, 8.10:

Theorem 1.9. There exist positive integers k_{inv fact}(G) (depending only on the associated root system R and the lattice L\underline{G}) and k_{R} (depending only on R), such that:

1. For k = k_{inv fact}(G), the image of (1) contains D_{3}(X) ∩ k · Δ_{L}^{3}. Equivalently, if (α, β, γ) ∈ D_{3}(X) ∩ k · Δ_{L}^{3} then

m_{kα,kβ,kλ}(1) ≠ 0.

2. The image of (1) satisfies

k_{R} · D_{3}^{L,0}(X) ⊂ Image(ι) ⊂ D_{3}^{L,0}(X).

Remark 1.10. We may reformulate the above theorem without reference to the Generalized Triangle Inequalities as follows. Suppose there exists N such that

m_{Nα,Nβ,Nγ}(1) ≠ 0.

Then m_{kα,kβ,kλ}(1) ≠ 0.

We have the following explicit formulae for k_{inv fact}(G):

1. For a simply-connected split simple algebraic group \underline{G} over 𝕂 with the associated root system R of rank ℓ, let α_{1}, ..., α_{ℓ} be the simple roots and θ be the highest root:

θ = ∑_{i=1}^{ℓ} m_{i}α_{i}.  (2)

Then k_{inv fact}(G) = k_{R} is the least common multiple (LCM) of m_{1}, ..., m_{ℓ}.

2. In general (for split groups) we have:

Root systemR  G  k_{inv fact}(G) k_{R}
A_{ℓ}  SL(ℓ + 1)  1  1
A_{ℓ}  GL(ℓ + 1)  1  1
A_{ℓ}  P SL(ℓ + 1)  ℓ + 1  1
B_{ℓ}  SO(2ℓ + 1), Spin(2ℓ + 1)  2  2
C_{ℓ}  Sp(2ℓ), P Sp(2ℓ)  2  2
D_{ℓ}  Spin(2ℓ)  2  2
D_{ℓ}  SO(2ℓ)  2  2
D_{ℓ}, ℓ > 4  P SO(2ℓ)  4  2
D_{4}  P SO(8)  2  2
G_{2}  G  6  6
F_{4}  G  12  12
E_{6}  G̃, Ad(G)  6  6
E_{7}  G̃, Ad(G)  12  12
E_{8}  G  60  60

(3)

Here G̃ denotes the simply-connected algebraic group, the symbol Ad(G) denotes the algebraic group of adjoint type, i.e. the quotient of G̃ by its center. In the case of root systems with the index of connection equal to 1, Ad(G) = G̃, so we have used the symbol G to denote the unique connected algebraic group with the given root system. Note that for the non-simply-connected classical groups we always get the order of the fundamental group as the saturation factor (except for the group P SO(8)), we refer the reader to chapter 8 for more details.

We now discuss the relation between the Decomposing Tensor Products Problem Q4 for a reductive complex Lie group G^{∨} = \underline{G}^{∨}(ℂ) and the corresponding problems of more geometric nature (Problems Q1–Q3). As we have pointed out before, the dominant weights α, β, γ of the group G^{∨} belong to the intersection of the lattice L∨ ; \underline{G} with the dual cone Δ^{∨}.

It is well-known that, as in the case of the Invariant Factor Problem, for every solution (α, β, γ) ∈ X^{∗}(\underline{T}^{∨})^{3} to the Problem Q4 for G^{∨}, the triple (α, β, γ) lies in D_{3}(X) where X is the symmetric space G/K (K is a maximal compact subgroup of the complex Lie group G = \underline{G}(ℂ)). The converse in general is false, however it is true up to “saturation depending on α, β, γ.” More precisely, for any α, β, γ ∈ X^{∗}(\underline{T}^{∨})^{3} such that (α, β, γ) ∈ D_{3}(X), there exists a positive integer k = k_{tens}(α, β, γ) such that (V_{kα} ⊗ V_{kβ} ⊗ V_{kγ})^{G^{∨}} ≠ 0. (See the Appendix, chapter 10.) Little is known about the saturation factors k_{tens}. For instance, it is not known whether there is a universal constant depending only on G^{∨}. However, k_{tens}(GL(ℓ)) = 1 due to the solution of the Saturation Conjecture.

As discussed above, the Invariant Factor Problem Q3 is equivalent to a discrete refinement of the Generalized Triangle Inequality Problem for the appropriate Euclidean building. The relationship between the problems Q3 and Q4 is more subtle. For one thing, the triangles with two straight edges and one edge a Lakshmibai-Seshadri path (to be called Littelmann triangles below) used in [Li] to study the decomposition of the tensor products of representations have striking similarities with the “billiard triangles”, which are foldings of triangles in Euclidean buildings into apartments (see section 7.1). We pose

Problem 1.11. Characterize the set of billiard triangles in an apartment of a Euclidean building X that can be unfolded to triangles in X.

Our Theorem 1.13 below suggests the following conjecture along these lines.

Conjecture 1.12. Every Littelmann triangle can be unfolded.

Instead of pursuing this geometric connection between the two problems, we establish a connection between Problems Q3 and Q4 through the Satake isomorphism. We prove the following

Theorem 1.13. Suppose that a triple α, β, γ ∈ X_{∗}(\underline{T}) = X^{∗}(\underline{T}^{∨}) is a solution of Problem Q4. Then it is also a solution of Problem Q3. In other words, existence of an invariant subspace in the triple tensor product for G^{∨} (a “quantuum triangle”) implies the existence of a triangle (with the vertices in the G-orbit of the distinguished vertex o stabilized by K, and Δ-side lengths α, β, γ) in the Euclidean building X_{G} associated with the dual group G = \underline{G}(𝕂).

In section 9.5 we give an example to show that the converse statement is false for the case \underline{G} = SO(5) whence G^{∨} = Sp(4, ℂ) and another to show that it is false for the case \underline{G} = G_{2} and G^{∨} = G_{2}(ℂ). We make the following conjecture concerning the saturation factors k_{tens} for Problem Q4:

Conjecture 1.14. Suppose that a triple α, β, γ ∈ X_{∗}(\underline{T}) = X^{∗}(\underline{T}^{∨}) satisfies the Generalized Triangle Inequalities for the discrete Euclidean building X = X_{G} associated with the group G = \underline{G}(𝕂). Assume that α + β + γ ∈ Q(R^{∨}). Let k = k_{R} be the saturation constant for the root system R (see chapter 3). Then the triple (kα, kβ, kγ) is a solution of Problem Q4 for the group G^{∨}. Equivalently, see the Appendix, we conjecture that if for some N the triple (Nα, Nβ, Nγ) is a solution of Problem Q4 for the group G^{∨} then the triple (kα, kβ, kγ) is a solution as well.

This conjecture is consistent with the conjecture of Shrawan Kumar [Ku], to the effect that there is a saturation factor k_{tens} for Problem Q4 which depends only on G^{∨} and its prime factors are the “bad primes” for the root system associated to G^{∨}. We recall that the bad primes for a root system are the primes that divide the coefficients m_{i} in the equation (2); they are defined and studied in [SpSt].

We conclude by pointing out how our result that the saturation factor for the Invariant Factor Problem Q3 for the group GL(ℓ) is 1 gives a new proof of the Saturation Conjecture for GL(ℓ) (the theorem of Knutson and Tao). Indeed, for the case of \underline{G} = GL(ℓ), Problems P3 and P4 have been known to be equivalent since 1968, due to the work of P. Hall, J. Green and T. Klein [K68] and [K69].

In fact we know from Theorem 1.13 that the implication

(α, β, γ) is a solution of Problem Q4 ⟹ (α, β, γ) is a solution of Problem Q3 is true for all split reductive groups over ℚ.

This was the harder of the two implications for GL(ℓ), proved by T. Klein in [K68] and [K69], see also [Mac, pg. 94–100].

The exceptional (i.e. not true for all split reductive groups) implication

(α, β, γ) is a solution of Problem Q3 ⟹ (α, β, γ) is a solution of Problem Q4

was first proved for GL(ℓ) by Philip Hall but not published. In fact it follows from a beautiful and elementary observation of J. Green, which is set forth and proved in [Mac, pg. 91-92], and which we will explain in §9.6.

Since Problems Q3 and Q4 are equivalent for GL(ℓ) and 1 is a saturation factor for Problem Q3, it follows that 1 is also a saturation factor for Problem Q4 as well.

This paper is organized as follows.

In chapter 2 we review root systems for algebraic reductive groups.

In chapter 3 we discuss Coxeter groups; we are also making some computations with the root systems which will be critical for computation of the saturation factors.

In chapter 4 we set up the general algebra problem R which generalizes the setting of the Problems Q1–Q3. We then discuss in details the parameter spaces for the Problems Q1–Q4 and their mutual relation.

In chapter 5 we first convert the problem R into an abstract geometry problem about existence of polygons with the prescribed generalized side-lengths (Problem 5.1). Next, we introduce a class of metric spaces (metric spaces modelled on Euclidean Coxeter complexes) and restate the abstract geometry problem for three classes of such spaces: Cartan motion spaces (infinitesimal symmetric spaces), nonpositively curved symmetric spaces and Euclidean buildings. We introduce the notion of refined and coarse (the Δ-length) generalized side-lengths for the geodesic polygons in such metric spaces. For the Cartan motion spaces and symmetric spaces, the problem of existence of polygons with the prescribed Δ-side lengths is adequate for solving the corresponding algebra problems (Q1 and Q2), but in the case of buildings it is not.

In chapter 6 we describe the solution (given in [KLM]) to the problem GTI of existence of polygons with the prescribed Δ-side lengths in the above classes of metric spaces X. We also discuss the relation of this solution to symplectic and Mumford quotients. We describe the system of generalized triangle inequalities for X and give an explicit set of inequalities in the case of root system of type C_{2}.

In chapter 7 we show that the geometry problem GTI solved in the previous chapter is not adequate (in the building case) for solving the algebra problem Q3.

In chapter 8 we show that in some cases solution of the unrefined geometry problem GTI given in chapter 6 solves the algebra problem Q3 as well (the case of A-type root systems). In the case of the group GL(ℓ) this results in the new proof of the Saturation Conjecture. More generally, we establish existence of the saturation factors k = k_{inv fact} and compute these numbers; modulo multiplication by k the unrefined geometry problem GTI is equivalent to the algebra problem Q3.

In chapter 9 we compare the algebraic problems Q3 and Q4 and prove the saturation conjecture for GL(m).

Acknowledgments. During the work on this paper the first author was visiting the Max Plank Institute (Bonn), he was also supported by the NSF grants DMS-99-71404 and DMS-02-03045. The first and the third authors were supported by the Mathematics Department of the University of Tübingen during their stay there in May of 2002. The third author was supported by the NSF grant DMS-01-04006. The authors gratefully acknowledge support of these institutions. The authors are grateful to L. Ein, S. Kumar, T. Haines, C. Laskowski, J. Stembridge, E. Vinberg and C. Woodward for useful conversations. We would especially like to thank Steve Kudla for suggesting we use the Satake transform to connect Problems Q3 and Q4 and Jiu-Kang Yu for his help with chapter 9 and in particular for providing us with the proof of Lemma 9.22.

When we told G. Lusztig of our Theorem 1.13 he informed us that although he had not known the result before our message, it was an easy consequence of his work in [Lu83]. We should say that our proof depends in an essential way on Lusztig’s paper [Lu83]. We use his change of basis formulas, Lemma 9.15, and his realization (completed by Kato in [Kato]), that the coefficients in one of those formulas were Kazhdan-Lusztig polynomials for the affine Weyl group, Lemma 9.16.

Finally, this paper is dedicated to M. S. Raghunathan on the occasion of his sixtieth birthday. Some of it was presented by the third author at the conference in his honor held at the Tata Institute in December of 2001. The third author takes pleasure in acknowledging the great impact M. S. Raghunathan had at the beginning of his career, by providing a critical insight in [Mi] and in the collaboration [MR].

Contents

1 Introduction 1

2 The root datum of a reductive group 12 2.1 Split tori over 𝔽 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Roots, coroots and the Langlands’ dual . . . . . . . . . . . . . . . . . 14

3 Root systems and Coxeter complexes 15 3.1 Roots and weights for reductive groups . . . . . . . . . . . . . . . . . 15 3.2 The saturation factors associated to a root system . . . . . . . . . . . 17

4 The first three algebra problems and the parameter spaces Σ for K\\overline{G}/K 21 4.1 The generalized eigenvalues of a sum problem Q1 and the parameter

space Σ of K-double cosets . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 The generalized singular values of a product and the parameter space

Σ of K-double cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 The generalized invariant factor problem and the parameter space Σ

of K-double cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 Comparison of the parameter spaces for the four algebra problems . . 26 5 The existence of polygonal linkages and solutions to the algebra

problems 26 5.1 Setting up the general geometry problem . . . . . . . . . . . . . . . . 27 5.2 Geometries modelled on Coxeter complexes . . . . . . . . . . . . . . . 29 5.3 Bruhat-Tits buildings associated with nonarchimedean reductive Lie

groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.4 Geodesic polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Weighted configurations, stability and the relation to polygons 36 6.1 Gauss maps and associated dynamical systems . . . . . . . . . . . . . 38 6.2 The polyhedron D_{n}(X) . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3 The polyhedron for the root system B_{2} . . . . . . . . . . . . . . . . . 42

7 Polygons in Euclidean buildings and the generalized invariant factor problem 43 7.1 Folding polygons into apartments . . . . . . . . . . . . . . . . . . . . 44 7.2 Counterexamples for buildings associated with P Sp(4) and Spin(5) . 46

8 Existence of fixed vertices in buildings and computation of the saturation factors for reductive groups 51 8.1 Existence of fixed vertices . . . . . . . . . . . . . . . . . . . . . . . . 52 8.2 Saturation factors for reductive groups . . . . . . . . . . . . . . . . . 58

9 The comparison of Problems Q3 and Q4 61 9.1 The Hecke ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 9.2 A geometric interpretation of m_{α,β,γ}(1) . . . . . . . . . . . . . . . . . 63 9.3 The Satake transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 9.4 A solution of Problem Q4 is a solution of Problem Q3 . . . . . . . . . 68 9.5 A solution of Problem Q3 is not necessarily a solution of Problem Q4 72 9.6 The saturation conjecture for GL(ℓ) . . . . . . . . . . . . . . . . . . . 75

10 Appendix: Decomposition of tensor products, Mumford quotients and triangles in symmetric spaces 76

Bibliography 81

2 The root datum of a reductive group

Let \underline{G} be a reductive algebraic group over a field 𝔽 and \underline{T} be a split torus in \underline{G}. Our goal in this section is to describe the root datum associated to the pair (\underline{G}, \underline{T}). The reader will find the definition of root datum in [Sp, §1].

2.1 Split tori over 𝔽

We recall that the algebraic group 𝔾_{m} is the affine algebraic group with coordinate ring 𝔽[S, T](ST − 1) and comultiplication Δ given by Δ(T) = T ⊗ T, Δ(S) = S ⊗ S.

Definition 2.1. An affine algebraic group \underline{T} defined over 𝔽 is a split torus of rank l if it is isomorphic to the product of l copies of 𝔾_{m}.

A character of an algebraic group \underline{T} defined over 𝔽 is a morphism of algebraic groups from \underline{T} to 𝔾_{m}. The product of two characters and the inverse of a character are characters and accordingly the set of characters of \underline{T} is an abelian group denoted by X^{∗}(\underline{T}).

Lemma 2.2. Suppose that \underline{T} is a split torus over 𝔽 of rank l. Then the character group of \underline{T} is a lattice (i.e. free abelian group) of rank l.

Proof: We have 𝔽[\underline{T}] ≅ 𝔽[T_{1}, T_{1}^{−1}, ⋯, T_{l}, T_{l}^{−1}]. A character of \underline{T} corresponds to a Hopf algebra morphism from 𝔽[𝔾_{m}] to 𝔽[T_{1}, T_{1}^{−1}, ⋯, T_{l}, T_{l}^{−1}]. Such a morphism is determined by its value on T. This value is necessarily a grouplike element (this means Δ(f) = f ⊗ f). But the grouplike elements of 𝔽[T_{1}, T_{1}^{−1}, ⋯, T_{l}, T_{l}^{−1}] are the monomials in the T_{i}’s and their inverses. The exponents of the monomial give the point in the lattice.

Corollary 2.3.

Hom(𝔾_{m}, 𝔾_{m}) ≅ ℤ.

We note that the previous isomorphism is realized as follows. Any Hopf-algebra homomorphism ψ of coordinate rings is of the form T → T^{n} for some integer n. Then the above isomorphism sends ψ to n.

Definition 2.4. A cocharacter or a one-parameter (algebraic) subgroup of \underline{T} is a morphism ϕ : 𝔾_{m} → T. The set of cocharacters of \underline{T} will be denoted X_{∗}(\underline{T}).

Lemma 2.5. Suppose that \underline{T} is a split torus over 𝔽 of rank l. Then the cocharacter group of \underline{T} is a free abelian group of rank l.

Proof: We have 𝔽[\underline{T}] ≅ 𝔽[T_{1}, T_{1}^{−1}, ⋯, T_{l}, T_{l}^{−1}]. A cocharacter of \underline{T} corresponds to a Hopf algebra morphism ψ from 𝔽[T_{1}, T_{1}^{−1}, ⋯, T_{l}, T_{l}^{−1}] to 𝔽[𝔾_{m}]. Such a morphism is determined by its value on T_{1}, ⋯, T_{l}. Then ψ corresponds to the lattice vector (m_{1}, ⋯, m_{l}) where T^{m_{i}} = ψ(T_{i}).

We define an integer-valued pairing ⟨ , ⟩ between characters and cocharacters as follows. Suppose ϕ is a cocharacter of \underline{T} and χ is a character. Then χ ∘ ϕ ∈ Hom(𝔾_{m}, 𝔾_{m}) ≅ ℤ. We define ⟨χ, ϕ⟩ to be the integer corresponding to χ ∘ ϕ.

We now describe two homomorphisms that will be useful in what follows. Let T_{e}(\underline{T}) be the Zariski tangent space of \underline{T} at the identity e.

Definition 2.6. We define Φ : X_{∗}(\underline{T}) ⟶ T_{e}(\underline{T}) by

Φ(λ) = λ′(1).

Here 1 is the identity of GL(1, 𝔽) and λ′(1) denotes the derivative at 1.

We also define Φ^{∨} : X^{∗}(\underline{T}) ⟶ T_{e}^{∗}(\underline{T}) by

Φ^{∨}(λ) = dλ|_{e}.

Remark 2.7. The character and cocharacter groups X^{∗}(\underline{T}), X_{∗}(\underline{T}) are multiplicative groups, the trivial (co)character will be denoted by 1. However, we will use the embeddings Φ and Φ^{∨} to identify them with additive groups. This will be done for the most part in chapters 4 and 9.

2.2 Roots, coroots and the Langlands’ dual

The reductive group \underline{G} picks out a distinguished finite subset of X^{∗}(\underline{T}), the relative root system R = R_{rel}(\underline{G}, \underline{T}). A character of \underline{T} is a root if it occurs in the restriction of the adjoint representation of \underline{G} to \underline{T}. We let Q(R) denote the subgroup of X^{∗}(\underline{T}) generated by R and define V := Q(R) ⊗ ℝ.

We recall

Definition 2.8. The algebraic group \underline{G} is split over 𝔽 if it has a maximal torus \underline{T} defined over 𝔽, which is split.

From now on we assume \underline{G} is split over 𝔽 and \underline{T} is a maximal torus as in the above definition.

It is proved in [Sp, §2], that R ⊂ V satisfies the axioms of a root system. Moreover in the same section it is proved that to every root α ∈ R there is an associated coroot α^{∨} ∈ X_{∗}(\underline{T}) such that ⟨α, α^{∨}⟩ = 2. We let R^{∨} denote the resulting set of coroots, let Q(R^{∨}) be the subgroup of X_{∗}(\underline{T}) they generate and V^{∨} := Q(R^{∨}) ⊗ ℝ. The root and coroot system R and R^{∨} determine (isomorphic) finite Weyl groups W, W^{∨} which acts on V^{∨} and V respectively. The action of the generators s_{α}, s_{α^{∨}} of the group W, W^{∨} on X^{∗}(\underline{T}) and X_{∗}(\underline{T}) are determined by the formulae:

s_{α}(x) := x − ⟨x, α^{∨}⟩α and s_{α^{∨}}(u) := u − ⟨u, α⟩α^{∨}.

We then have [Sp, §2]:

Proposition 2.9. The quadruple Ψ(\underline{G}, \underline{T}) := (X^{∗}(\underline{T}), R, X_{∗}(\underline{T}), R^{∨}) is a root datum.

Definition 2.10. Let Ψ = (X, R, X^{∨}, R^{∨}) and Ψ′ = (X′, R′, (X′)^{∨}, (R′)^{∨}) be root data. Then an isogeny from Ψ′ to Ψ is a homomorphism ϕ from X′ to X such that ϕ is injective with finite cokernel. Moreover we require that ϕ induces a bijection from R′ to R and the transpose of ϕ induces a bijection of coroots.

Now suppose f : \underline{G} ⟶ \underline{G}′ is a covering of algebraic groups. If \underline{T} is a maximal torus in \underline{G} then its image \underline{T}′ is a maximal torus in \underline{G}′. The induced map on characters gives rise to an isogeny of root data, denoted Ψ(f).

Conversely, we have [Sp, Theorem 2.9]:

Theorem 2.11. 1. For any root datum Ψ with reduced root system there exist a connected split (over 𝔽) reductive group \underline{G} and a maximal split torus \underline{T} such that Ψ = Ψ(\underline{G}, \underline{T}). The pair (\underline{G}, \underline{T}) is unique up to isomorphism.

2. Let Ψ = Ψ(\underline{G}, \underline{T}) and Ψ′ = Ψ(\underline{G}′, \underline{T}′) and ϕ be an isogeny from Ψ′ to Ψ. Then there is a covering f : \underline{G} ⟶ \underline{G}′ with the image of \underline{T} equal to \underline{T}′ such that ϕ = Ψ(f).

Before stating the next definition we need a lemma which we leave to the reader.

Lemma 2.12. If (X, R, X^{∨}, R^{∨}) is a root datum so is (X^{∨}, R^{∨}, X, R).

We now have

Definition 2.13. Let \underline{G} be a (connected) split reductive group over ℚ. Let Ψ = Ψ(\underline{G}, \underline{T}) = (X, R, X^{∨}, R^{∨}) be the root datum of (\underline{G}, \underline{T}). Then the Langlands dual \underline{G}^{∨} of \underline{G} is the unique (up to isomorphism) reductive group over ℚ which has the root datum Ψ′ = (X^{∨}, R^{∨}, X, R).

In fact we will need only the complex points G^{∨} := \underline{G}^{∨}(ℂ) of \underline{G}^{∨} in what follows. We will accordingly abuse notation and frequently refer to G^{∨} as the Langlands dual of \underline{G}. One has

(\underline{G}^{∨})^{∨} ≅ \underline{G}.

3 Root systems and Coxeter complexes

3.1 Roots and weights for reductive groups

In this subsection we review the properties of root systems and Coxeter groups, we refer the reader for a more thorough discussion to [Hum, Section 4.2] and [Bo]. Let R ⊂ V^{∗} be a root system of rank n on a real Euclidean vector space V. We do not assume that n equals the dimension of V. Note that in the case of semisimple Lie algebras, the space V will be a Cartan subalgebra 𝔞 ⊂ 𝔤 with the Killing form.

We will sometimes identify V and V^{∗} using the metric. Let Q(R) ⊂ V^{∗} denote the root lattice, i.e. the integer span of R. This subgroup is a lattice in Span_{ℝ}(R) ⊂ V, it is a discrete free abelian subgroup of rank n. Given a subgroup Λ ⊂ ℝ we define a collection H = H_{R,Λ} of hyperplanes (called walls) in V as the set

H := {H_{α,t} = {v ∈ V : α(v) = t}, t ∈ Λ, α ∈ R}.

In this paper we will be mostly interested in the case when Λ is either ℤ or ℝ, but much of our discussion is more general. We define an affine Coxeter group W_{aff} = W_{R,Λ} as the group generated by the reflections w_{H} in the hyperplanes H ∈ H. The only reflection hyperplanes of the reflections in W_{aff} are the elements of H. The pair (E, W_{aff}) is called a Euclidean Coxeter complex, where E = V is the Euclidean space. The vertices of the Coxeter complex are points which belong to the transversal intersections of n walls in H. (This definition makes sense even if n < dim(V), only in this case there will be continuum of vertices even if Λ = ℤ.) If W_{aff} is trivial, we declare each point of E a vertex. We let E^{(0)} denote the vertex set of the Coxeter complex.

Definition 3.1. An embedding of Euclidean Coxeter complexes is a map (f, ϕ) : (E, W) → (E′, W′), where ϕ : W → W′ is a monomorphism of Coxeter groups and f : E → E′ is a ϕ-equivariant affine embedding.

Let L_{trans} denote the translational part of W_{aff}. If Λ = ℤ then L_{trans} is the coroot lattice Q(R^{∨}) of R. In general, L_{trans} = Q(R^{∨}) ⊗ Λ. The linear part W_{sph} of W_{aff} is a finite Coxeter group acting on V, it is called a spherical Coxeter group. The stabilizer of the origin 0 ∈ E (which we will regard as a base-point o ∈ E) in W_{aff} maps isomorphically onto W_{sph}. Thus W_{aff} = W_{sph} ⋉ L_{trans}. A vertex of the Euclidean Coxeter complex is called special if its stabilizer in W_{aff} is isomorphic to W_{sph}. We let E^{(0),sp} denote the set of special vertices of E.

Remark 3.2. The normalizer N_{aff} of W_{aff} (in the full group V of translations on E) acts transitively on the set of special vertices. The vertex set E^{(0)} of the complex (E, W_{aff}) contains N_{aff} · o, but typically it is strictly larger that. Moreover, in many cases E^{(0)} does not form a group.

We recall that the weight group P (R) and the coweight group P (R^{∨}) are defined by

P (R) ={λ ∈ V^{∗} : λ(v) ∈ ℤ, ∀v ∈ R^{∨}},
P (R^{∨}) ={v ∈ V : α(v) ∈ ℤ, ∀α ∈ R}.

Remark 3.3. In the case when n < dim(V) our definition of weights is different from the one in [Sp].

Again, P (R) and P (R^{∨}) are lattices provided that n = dim(V), otherwise they are nondiscrete abelian subgroups of V. We have the inclusions

Q(R^{∨}) ⊂ P (R^{∨}), Q(R) ⊂ P (R).

The normalizer N_{aff} equals P (R^{∨}) ⊗ Λ.

The spherical Coxeter groups W_{sph} which appear in the above construction act naturally on the sphere at infinity S = ∂_{∞}E; the pair (S, W_{sph}) is called a spherical Coxeter complex. The definitions of walls, vertices, etc., for Euclidean Coxeter complexes generalize verbatum to the spherical complexes. We will use the notation Δ_{sph} ⊂ S for the spherical Weyl chamber, Δ_{sph} is the ideal boundary of the Euclidean Weyl chamber Δ ⊂ E (i.e. a fundamental domain for the action W_{sph} ↷ E, which is bounded by walls).

From our viewpoint, the Euclidean Coxeter complex is a more fundamental object than a root system. Thus, if the root system R was not reduced, we replace it with a reduced root system R′ which has the same group W_{aff}: if α, 2α ∈ R we retain the root 2α and eliminate the root α. We will assume henceforth that the root system R is reduced.

Product decomposition of Euclidean Coxeter complexes. Suppose that (E, W_{aff}) is a Euclidean Coxeter complex associated with the reduced root system R, let R_{1}, ..., R_{s} denote the decomposition of R into irreducible components. Accordingly, the Euclidean space E splits as the metric product

E = E_{0} × ∏_{i=1}^{s} E_{i},

where E_{i} is spanned by R_{i}^{∨}, 1 ≤ i ≤ s. This decomposition is invariant under the group W_{aff} which in turn splits as

W_{aff} = ∏_{i=1}^{s} W_{aff}^{i},

where W_{aff}^{i} = W_{R_{i},Λ} for each i = 1, ..., s; for i = 0 we get the trivial Coxeter group W_{aff}^{0}. The group W_{aff}^{i} is the image of W_{R_{i},Λ} under the natural embedding of affine groups Aff(E_{i}) ⟶ ∏_{i=1}^{s} Aff(E_{i}) = Aff(E).

Analogously, the spherical Coxeter group W_{sph} splits as the direct product

W_{sph}^{0} × ... × W_{sph}^{s}

(where W_{sph}^{0} = {1}). The Weyl chamber Δ of W_{sph} is the direct product of the Weyl chambers Δ_{0} × Δ_{1} × ... × Δ_{s}, where Δ_{i} is a Weyl chamber of W_{sph}^{i} and Δ_{0} = E_{0}. Similarly, the normalizer N_{aff} of W_{aff} splits as

N_{aff} = V_{0} × ∏_{i=1}^{s} N_{aff}^{i},

where V_{0} is the vector space underlying E_{0} and N_{aff}^{i} is the normalizer of W_{aff}^{i} in the group of translations of E_{i}. Note that for each i = 1, ..., s the groups W_{aff}^{i} and N_{aff}^{i} act as lattices on E_{i}. We observe that the vertex set of the complex (E, W_{aff}) equals

E_{0} × E_{1}^{(0)} × ... × E_{s}^{(0)},

where E_{i}^{(0)} is the vertex set of the complex (E_{i}, W_{aff}^{i}). Similarly, the set of special vertices E^{(0),sp} of E equals

E_{0} × E_{1}^{(0),sp} × ... × E_{s}^{(0),sp}.

3.2 The saturation factors associated to a root system

In this section we define and compute saturation factors associated with root systems.

Definition 3.4. Let (E, W_{aff}) be a Euclidean Coxeter complex, W_{aff} = W_{R,ℤ}. We define the saturation factor k_{R} for the root system R to be the least natural number k such that k ·E^{(0)} ⊂ E^{(0),sp} = N_{aff} ·o. The numbers k_{R} for the irreducible root systems are listed in the table (5).

Below we explain how to compute the saturation factors k_{R}. First of all, it is clear that if the root system R is reducible and R_{1}, ..., R_{s} are its irreducible components, then k_{R} = LCM(k_{R_{1}}, ..., k_{R_{s}}), where LCM stands for the least common multiple. Henceforth we can assume that the system R is reduced, irreducible and n = dim(V). Then the affine Coxeter group W_{aff} is discrete, acts cocompactly on E and its fundamental domain (a Weyl alcove) is a simplex.

Let {α_{1}, ..., α_{n}} be the collection of simple roots in R (corresponding to the positive Weyl chamber Δ) and α_{0} := θ be the highest root. Then

θ = ∑_{i=1}^{n} m_{i}α_{i}.  (4)

We can choose as a Weyl alcove a for W_{aff} the simplex bounded by the hyperplanes H_{α_{j},0}, H_{θ,1}, j = 1, ..., n. The vertices of a are: o = x_{0} (the origin) and the points x_{1}, ..., x_{n}. Each x_{i}, i ≠ 0, belongs to the intersection of the hyperplanes H_{α_{0},1}, H_{α_{j},0}, 1 ≤ j ≠ i ≤ n. The set of values (mod ℤ) of the linear functionals α (α ∈ R) on the vertex set E^{(0)} of the Coxeter complex, equals {α_{i}(x_{i}) : i = 1, ..., n}. Note that 1 = θ(x_{i}) = m_{i}α_{i}(x_{i}) where the numbers m_{i} are the ones which appear in the equation (4). Thus α_{i}(x_{i}) = \frac{1}{m_{i}}.

Lemma 3.5. k_{R} = LCM(m_{1}, ..., m_{n}).

Proof: We have: α_{i}(kx_{i}) ∈ ℤ for each i, which in turn implies that α(kx_{i}) ∈ ℤ for all α ∈ R, i = 1, ..., n. Hence α(kE^{(0)}) ⊂ ℤ for each α ∈ R. Since N_{aff} = P (R^{∨}), this proves that kE^{(0)} ⊂ N_{aff} · o. If k ∈ ℕ is such that k · E^{(0)} ⊂ P (R^{∨}), then m_{i} divides k for each i = 1, ..., n.

In our paper we will also need a generalization of the numbers k_{R}, which we discuss for the rest of this section. (The reader who is interested only in the simply-connected groups can ignore this material.) We again consider a general reducible root system R. Suppose that L′ is a subgroup of N_{aff} containing the lattice L_{trans} = Q(R^{∨}); we will assume that L′ acts as a lattice on E (i.e. a discrete cocompact group). Set L := L′ ∩ V_{1} ⊕ ... ⊕ V_{s}, where V_{i} are the vector spaces underlying E_{i}. We note that since L_{trans} ⊂ L and L_{trans} acts as a lattice on E_{1} ⊕ ... ⊕ E_{s}, the discrete group L also acts as a lattice on E_{1} ⊕ ... ⊕ E_{s}.

Let p_{i} denote the orthogonal projections E → E_{i}. Consider the images L_{i} of L under the projections p_{i} (i = 0, ..., s); since L ⊂ N_{aff} and p_{i}(N_{aff}) = N_{aff}^{i}, we have the inclusions

L_{trans}^{i} ⊂ L_{i} ⊂ N_{aff}^{i}, i = 1, ..., s,

where L_{trans}^{i} is the translation subgroup of W_{aff}^{i}.

Example 3.6. Suppose that (E, W_{aff}) is the Coxeter complex associated with the root system of the group GL(n). Then E = ℝ^{n}, L′ = L_{GL(n)} = ℤ^{n} is the cocharacter group of the maximal torus \underline{T} (represented by diagonal matrices) in GL(n). The group L′ is generated by the cocharacters e_{i} = (0, ..., 0, 1, 0...0) (1 is on the i-th place). The coroot lattice Q(R^{∨}) is generated by the simple coroots α_{i}^{∨} = e_{i} − e_{i+1}, i = 1, ..., n − 1. The metric on E is given by the trace of the product of matrices. We have the decomposition E = E_{0} × E_{1} where E_{0} is 1-dimensional and is spanned by the vector e = e_{1} + ... + e_{n}, and the space E_{1} is the kernel of the map

tr : (x_{1}, ..., x_{n}) ↦ ∑_{i=1}^{n} x_{i}.

The projection p_{1} : E → E_{1} is given by p_{1}(u) = u − \frac{1}{m}tr(u)e. The group W_{aff} equals W_{aff}^{1}, which acts on E_{1} as the Euclidean Coxeter group with the extended Dynkin diagram of type Ã_{n−1}.

The intersection L = L′ ∩ E_{1} = Q(R^{∨}), is the coroot lattice of the root system R within E_{1}, that is the root system of SL(n). The projection L_{1} = p_{1}(L′) is the coweight lattice of the root system R within E_{1}.

Consider the group of isometries W̃ generated by elements of W_{aff} and L. Then W̃ is a Euclidean Coxeter group with the linear part W_{sph} and translation part L, W̃ = W_{sph} ⋉ L. Since W̃^{i} = W_{sph}^{i} ⋉ L_{i} normalizes W_{aff}^{i}, for each i we get the induced action of the finite abelian group F_{i} := W̃^{i}/W_{aff}^{i} ≅ L_{i}/L_{trans}^{i} on the Weyl alcove a_{i} of W_{aff}^{i}.

Definition 3.7. A face δ_{i} ⊂ a_{i} of a Weyl alcove a_{i} of W_{aff}^{i}, will be called L_{i}- admissible if there exists an element g ∈ F_{i} which preserves δ_{i} and ⟨g⟩ acts transitively on its vertices (i = 1, ..., s).

Note that in the case L_{i} = L_{trans}^{i}, the only L_{i}-admissible simplices are the vertices of δ_{i}.

Definition 3.8. For each pair of groups (W_{aff}^{i}, L_{i}), define the saturation factor k_{i} = k(W_{aff}^{i}, L_{i}) ∈ ℕ, to be the smallest natural number k_{i} such that for each L_{i}-admissible face c ⊂ a_{i}, the multiple of its barycenter k_{i}b_{c}, belongs to E_{i}^{(0),sp} = N_{aff}^{i} · o. We let k(W_{aff}, L) = k(W_{aff}, L′) denote LCM(k_{1}, ..., k_{s}). In the case L = N_{aff} we will use the notation k_{w} for k(W_{aff}, L).

We note that in the case L = L_{trans} we get k(W_{aff}, L) = k_{R}. Our next goal is to compute the saturation factors for various irreducible root systems and various lattices L.

We again assume that the root system R is irreducible and that its rank n equals dim(V). Note that the group F = L/L_{trans} acts by automorphisms on the extended Dynkin diagram Γ̃ of the root system R (since F acts on the Weyl alcove a which is uniquely determined by the labelled graph Γ̃, whose nodes correspond to the faces of a). For i = 1, ..., n we mark the i-th node (corresponding to α_{i}) of Γ̃ with the natural number m_{i} which appears the formula for the highest root (4). We mark the 0-th node of Γ̃ (corresponding to θ) with 1. Then the automorphisms of Γ̃ preserve this labelling; the action of the full group N_{aff} on Γ̃ is transitive on the set of all the nodes labelled by 1. Not all automorphisms of Γ̃ can be induced by F even if one takes L as large as possible, i.e. L = N_{aff}. Recall that the action on a comes from the action of W̃ by conjugation on W_{aff}; this action induces inner automorphisms of the spherical Weyl group W_{sph}. Thus, if g is an automorphism of Γ̃ induced by an element of F and g fixes a vertex with the label 1, then g acts trivially on Γ̃. This does not completely determine the image of N_{aff} in Aut(Γ̃) but it will suffice for the computation of the saturation factors.

Here is the procedure for computing the saturation factor k = k(W_{aff}, L). Given g ∈ F (including the identity) consider the orbits of ⟨g⟩ in the vertex set of the graph Γ̃. Here and in what follows ⟨g⟩ denotes the cyclic subgroup of Isom(E) generated by g. Let \overline{O} = \{\overline{x}_{i_{1}}, ..., \overline{x}_{i_{t}}\} be such an orbit. This orbit corresponds to the orbit O = {x_{i_{1}}, ..., x_{i_{t}}} of ⟨g⟩ on the vertex set of the Weyl alcove. Take the barycenter

b(O) = \frac{1}{t} ∑_{j=1}^{t} x_{i_{j}}

of the corresponding vertex set (also denoted O) of the Euclidean Coxeter complex. For the point b = b(O) compute the rational numbers α_{i}(b), i = 0, ..., n.

Then find the LCD (the least common denominator) of the rational numbers α_{i}(b), i = 0, ..., n, call it k_{O}. Finally, let

k := LCM({k_{O}, where O runs through all orbits of all ⟨g⟩ ⊂ F }).

Remark 3.9. Instead of taking all g ∈ F it is enough to consider representatives of their conjugacy classes in W̃ /W_{aff} (under the conjugation by the full automorphism group of Γ̃).

It is clear that the number k computed this way satisfies the required property:

1. For the barycenter b of each L-admissible face of a, the multiple kb belongs to the coweight lattice (which equals N_{aff} · o)

2. The number k is the least natural number with this property. The numbers k_{R} and k_{w} are listed in the table (5) below (the number i in the table is the index of connection). We will verify the computation in the most interesting case, namely for the root system of the A-type.

Lemma 3.10. Suppose that the Dynkin diagram Γ has type A_{n} and that F ≅ ℤ/m. Then k = k(W_{aff}, L) equals m. In particular, if L = N_{aff} then we get k_{w} = n + 1.

Proof: The group F ≅ ℤ/m acts on the graph Γ̃ = Ã_{n} by cyclic permutations. Let g ∈ F be a permutation of order t; note that t divides m. Then for each orbit O of g (in the vertex set of the Weyl alcove a) we get:

b(O) = \frac{1}{t} ∑_{x_{j}∈O} x_{j}.

For each i ≠ 0, α_{i}(b(O)) = 0 if x_{i} ∉ O, and α_{i}(b(O)) = 1/t if x_{i} ∈ O. For the highest root we get: θ(b(O)) = \frac{t−1}{t} if x_{0} ∈ O and θ(b(O)) = 1 if x_{0} ∉ O . In any case, k_{O} = t.

Since all t’s divide the order m of the group F (and for the generator of F, t = m), the LCM of k_{O}’s taken over all orbits and all elements of F, equals m.

Similarly we have

Lemma 3.11. Suppose that the Dynkin diagram Γ has type D_{ℓ} and F ≅ ℤ/2. Then k = k(W_{aff}, L) equals 4 if F permutes at least two roots labelled by 2 and k = 2 if it does not (the latter holds for the orthogonal groups).

We note that for all classical root systems, k_{w} equals the index of connection i and for all exceptional root systems, k_{R} = k_{w} (here we have to regard D_{4} as an exceptional root system).

Root system  θ  i k_{R} k_{w}
A_{ℓ}  α_{1} + ... + α_{ℓ}  ℓ + 1 1 ℓ + 1
B_{ℓ}  α_{1} + 2α_{2} + ... + 2α_{ℓ}  2 2 2
C_{ℓ}  2α_{1} + 2α_{2} + ... + 2α_{ℓ−1} + α_{ℓ}  2 2 2
D_{ℓ}, ℓ > 4  α_{1} + α_{2} + α_{3} + 2α_{4} + ... + 2α_{ℓ}  4 2 4
D_{4}  α_{1} + α_{2} + α_{3} + 2α_{4}  4 2 2
G_{2}  3α_{1} + 2α_{2}  1 6 6
F_{4}  2α_{1} + 3α_{2} + 4α_{3} + 2α_{4}  1 12 12
E_{6}  α_{1} + α_{2} + 2α_{3} + 2α_{4} + 2α_{5} + 3α_{6} 3 6 6
E_{7} α_{1} + 2α_{2} + 2α_{3} + 2α_{4} + 3α_{5}+ 2 12 12
+3α_{6} + 4α_{7}
E_{8} 2α_{1} + 2α_{2} + 3α_{3} + 3α_{4} + 4α_{5}+ 1 60 60
+4α_{6} + 5α_{7} + 6α_{8}

(5)

Remark 3.12. Our discussion of the Coxeter groups was somewhat nongeometric; a more geometric approach would be to start with an affine Coxeter group and from this get root systems, etc.

4 The first three algebra problems and the parameter spaces Σ for K\\overline{G}/K

We will see in this chapter that the problems Q1–Q3 for reductive algebraic groups G can reformulated as special cases of a single algebraic problem as follows. There is a group \overline{G} (closely associated to G) which contains K, a maximal bounded subgroup of G. The conditions of fixing α, β and γ in problems Q1–Q3 will amount to fixing three double cosets in Σ = K\\overline{G}/K. The problems Q1–Q3 will be then reformulated as:

• Problem R(\overline{G}): Find necessary and sufficient conditions on α, β, γ ∈ Σ in order that there exist A, B, C ∈ \overline{G} in the double cosets represented by α, β, γ resp., such that A · B · C = 1.

We will now describe the groups \overline{G} and K for the problems Q1–Q3. The main part of this chapter will then be occupied with describing the double coset spaces Σ = K\\overline{G}/K. In section 4.1 we will also prove that the problem R(\overline{G}) agrees with the Problem Q1 from the Introduction (the equivalence will be clear for two other problems).

1. For the Problem Q1: For 𝔽 = ℝ or ℂ, let \underline{G} be a connected reductive algebraic group over ℝ, G := \underline{G}(𝔽) be a real or complex Lie group with Lie algebra 𝔤. Pick a maximal compact subgroup K of G. Let 𝔨 denote the Lie algebra of K. Then we have the orthogonal decomposition (with respect to the Killing form)

𝔤 = 𝔨 ⊕ 𝔭.

We let \overline{G} be the Cartan motion group \overline{G} = K ⋉ 𝔭.

2. For the problem Q2: \underline{G} and K are the same as in 1, but now we take \overline{G} = G = \underline{G}(𝔽).

3. For the problem Q3: Let 𝕂 be a complete nonarchimedean valued field with a discrete valuation v and the value group ℤ = v(𝕂) ⊂ ℝ. Let O denote the ring of elements in 𝕂 with nonnegative valuation. Let \underline{G} be a connected reductive algebraic group over 𝕂, \overline{G} := G := \underline{G}(𝕂) and K := G(O).

In order to relate Problems Q1–Q3 with geometry we will have to compute the double coset spaces Σ = K\\overline{G}/K. Moreover we will show that the parameter spaces Σ admit canonical linear structures: In the first two problems we will use a Cartan decomposition G = KAK, then Σ will be described as the (positive) Weyl chamber Δ in 𝔞, the Lie algebra of A. In the case of the third problem, Σ will be identified with the intersection L\underline{G} ∩ Δ ⊂ 𝔞 = X_{∗}(\underline{T}) ⊗ ℝ, where \underline{T} is a maximal 𝕂-split torus of \underline{G} and L\underline{G} is extended cocharacter lattice of \underline{T} (see section 4.3).

4.1 The generalized eigenvalues of a sum problem Q1 and the parameter space Σ of K-double cosets

We continue with the notation of previous section, in particular, \overline{G} = K ⋉ 𝔭. We now describe the parameter space Σ = K\\overline{G}/K. We choose a Cartan subspace 𝔞 ⊂ 𝔭 (i.e. a maximal subalgebra of 𝔭, necessarily abelian and reductive). We recall that the positive Euclidean Weyl chamber Δ is the dual cone in 𝔞 to the cone of positive restricted roots in 𝔞^{∗}.

Lemma 4.1. The inclusion ι : Δ → K ⋉ 𝔭 given ι(σ) = (1, σ) induces a bijection onto Σ.

Proof: We first observe that since (k_{1}, x) · (k_{2}, 0) = (k_{1}k_{2}, x), every double coset in K\\overline{G}/K has a representative of the form (1, x). Since (k, 0) · (1, x) · (k, 0)^{−1} = (1, Ad k(x)) the lemma is clear.

Now in case 𝔤 = 𝔤𝔩(m) and 𝔽 = ℂ, the Cartan subaspace 𝔭 is the space of Hermitian m-by-m matrices, 𝔞 is the space of diagonal m-by-m matrices with real entries and the cone Δ ⊂ 𝔞 is the cone in which the diagonal entries are arranged in decreasing order. Thus the parameter space Σ for the double cosets of elements in \overline{G} is the cone of m-tuples of real numbers arranged in decreasing order. This agrees with the parameter space of the Problem P1 in the Introduction.

We complete the proof of equivalence of R(\overline{G}) with Problem Q1 by observing that for g_{i} = (k_{i}, x_{i}), i = 1, 2, 3,

(k_{1}, x_{1}) · (k_{2}, x_{2}) · (k_{3}, x_{3}) = (k_{1}k_{2}k_{3}, x_{1} + Adk_{1}x_{2} + Ad(k_{1}k_{2})x_{3}).  (6)

We see from (6) that if g_{1}g_{2}g_{3} = 1 then putting a = x_{1}, b = Adk_{1}x_{2}, c = Ad(k_{1}k_{2})x_{3} we find

a + b + c = 0.

Thus we find representatives a, b and c in 𝔭 of the orbits Ad(K)(x_{i}), 1 ≤ i ≤ 3 whose sum equals zero (i.e. we have a solution of the Eigenvalues of a Sum Problem). Conversely, if a, b, c ∈ 𝔭 solve the Eigenvalues of a Sum Problem then

g_{1} · g_{2} · g_{3} = (1, a) · (1, b) · (1, c) = (1, 0)

and hence g_{1}, g_{2}, g_{3} solve the double coset problem R(\overline{G}).

Thus we may find A, B and C in \overline{G} in the required K-double cosets with A·B·C = 1 if and only if there exist a, b, c ∈ 𝔭 with a + b + c = 0 in the required Ad K-orbits in 𝔭.

4.2 The generalized singular values of a product and the parameter space Σ of K-double cosets

Let \overline{G} := G = \underline{G}(𝔽) and consider the Cartan decomposition G = KAK where A is the maximal abelian subgroup of \overline{G}, whose Lie algebra is the Cartan subalgebra 𝔞 ⊂ 𝔭 from the previous section. We will identify A and 𝔞 via the exponential map, let A_{Δ} := exp(Δ) ⊂ A. Then we get the refinement of the Cartan decomposition: G = KA_{Δ}K. It is now clear that the inclusion ι : Δ → G given by ι(σ) = exp(σ) induces a bijection onto Σ = K\G/K. In what follows we will parameterize double cosets in Σ by their logarithms, which are vectors in Δ.

In the case G = GL(ℓ, ℂ), we note that K = U(m), the subgroup A consists of diagonal matrices with positive diagonal entries, A_{Δ} consists of matrices D ∈ A with diagonal entries arranged in the decreasing order, and the projection f of g to KgK ∈ Σ = A_{Δ} ≅ Δ is given by first sending g to h = \sqrt{gg^{∗}}, then diagonalizing h, arranging the diagonal values in the decreasing order and then taking their logarithms. Thus f(g) is given by the vector whose components are the logarithms of the singular values of the matrix g. We conclude that Problem P2 in the introduction is a special case of the Problem Q2 in the case G = GL(ℓ, ℂ).

4.3 The generalized invariant factor problem and the parameter space Σ of K-double cosets

The goal of this section is to give an explicit description of the parameter space Σ = Σ_{inv fact} for the Invariant Factor Problem Q3. We start by noticing that the discrete valuation v : 𝕂 → ℤ admits a splitting κ : ℤ → 𝕂: it is given by sending 1 ∈ ℤ to the uniformizer π ∈ 𝕂. Throughout this section, \underline{G} will denote a connected reductive algebraic group over 𝕂, which has the (relative) rank l. Let \underline{T} ⊂ \underline{G} be a maximal 𝕂-split torus, and let X^{∗}(\underline{T}) and X_{∗}(\underline{T}) be the lattices of characters and cocharacters. We denote by \underline{P} ⊂ \underline{G} be a minimal parabolic subgroup such that (\underline{P} , \underline{T}) is a parabolic pair, [C, pg. 127].

This data determines a (relative) root system R_{rel} and hence a (relative) finite Weyl group W_{sph}. Let \underline{Z} ⊂ \underline{G} and \underline{N} ⊂ \underline{G} be respectively the centralizer and normalizer (over 𝕂) of the algebraic group \underline{T}. As usual we let G, N, Z and T denote the groups of 𝕂-points of the corresponding algebraic groups. We let K denote \underline{G}(O) and B := \underline{T} (O) = K ∩ T. We will frequently refer to G as a nonarchimedean reductive Lie group.

As in the previous section, we have to discuss the Cartan decomposition G = K · Γ · K of the group G (where Γ will be an appropriate subset of Z). In the split case this decomposition will have the form G = K · A_{Δ} · K, where A_{Δ} ⊂ A and A will be a subgroup of T so that A × B = T.

We can now describe Σ_{inv fact}(G) = K\G/K, the parameter space for the generalized invariant factors problem for the nonarchimedean reductive Lie group G.

Split Case. We first consider the case when \underline{G} is split over 𝕂 since the description is more transparent in this case. In this case there is an algebraic group defined and split over ℚ and in fact a group scheme over ℤ such that \underline{G} is obtained from it by extension of scalars. We will abuse notation and also denote the above group-scheme over ℤ by \underline{G} as well. Under this assumption the torus \underline{T} is also defined and split over ℚ, hence the group of real points of \underline{T} is well-defined. We thus consider the real torus \underline{T} (ℝ) with the real Lie algebra 𝔞. The map Φ of Definition 2.6 (with 𝔽 = ℝ) gives us an identification

X_{∗}(\underline{T}) ⊗ ℝ ≅ 𝔞.

Then the group X_{∗}(\underline{T}) is identified with a lattice L := L\underline{G} in 𝔞. Similarly we use the map Φ^{∨} to identify X^{∗}(\underline{T}) with a lattice in the dual space 𝔞^{∗}. The root system R sits naturally in the dual space 𝔞^{∗} and the corresponding finite Weyl group W_{sph} acts on 𝔞 and 𝔞^{∗} in the usual way; W_{sph} leaves invariant the lattices X_{∗}(\underline{T}) and X^{∗}(\underline{T}). Let Δ ⊂ E be the (positive) Weyl chamber of the group W_{sph}, determined by the parabolic subgroup \underline{P}. Our goal is to prove that the space K\G/K can be identified with the “cone”

Δ_{L} = Δ ∩ L.

By definition, a cocharacter ϕ is a homomorphism from the group 𝔾_{m}(𝕂) to the group T: note that the map Ψ : X_{∗}(\underline{T}) → T given by Ψ(ϕ) = ϕ(π) is injective (where π is a fixed uniformizer). We define the subgroup A by

A = Ψ(X_{∗}(\underline{T})).

The following lemma is immediate (it suffices to prove it for GL(1)).

Lemma 4.2. The group T is a direct product T = A × B.

Remark 4.3. In the case where G = GL(ℓ, 𝕂) and T is the torus of diagonal matrices with entries in 𝕂, we find that Ψ(ϕ) = ϕ(π) is the diagonal matrix with diagonal entries (π^{m_{1}}, ⋯, π^{m_{l}}), where ϕ corresponds to the lattice vector (m_{1}, ⋯, m_{l}).

We let A_{Δ_{L}} ⊂ A be the image of Δ_{L} = X_{∗}(\underline{T}) ∩ Δ under the map Ψ.

Lemma 4.4. (See [Tits, page 51].) We have the Cartan decomposition

G = K · A_{Δ_{L}} · K,

i.e., each element g ∈ G has a unique representation as a product k_{1} · a · k_{2}, where a ∈ A_{Δ_{L}} and k_{i} ∈ K.

Corollary 4.5. The inclusion ι : A_{Δ_{L}} ⊂ A ⊂ T ⊂ G induces a bijection

A_{Δ_{L}} ⟶^{ι} K\G/K.

Now that we have identified the double coset space

Σ_{inv fact}(G) = K\G/K

with Δ_{L}, it is clear that the generalized invariant factor problem Q3 in the Introduction agrees with the double coset problem R(G). However it may not be clear at this point why these problems agree with Problem P3 in case G = GL(ℓ).

To see this we note that we can give a second description of the vector Φ(λ) if we choose coordinates in \underline{T}. Let χ_{1}, ⋯, χ_{n} be a basis for the group X^{∗}(\underline{T}) and define χ : T ⟶ 𝕂^{n} to be the map with components χ_{j}, 1 ≤ j ≤ n. We observe that we may use the derivative of χ_{j} at e to obtain coordinates on T_{e}(\underline{T}) and accordingly we obtain a map χ̇ : 𝔞 ⟶ ℝ^{n}. We leave the following lemma to the reader.

Lemma 4.6. χ̇(Φ(λ) is the (integral) vector in ℝ^{n} obtained by applying the valuation v to the coordinates of χ(λ(π)).

General Case. We now consider the general case when \underline{G} is not necessarily split. Most of the above discussion remains valid, however the Cartan decomposition for G has a slightly different form and for this reason we cannot use the cocharacter lattice as L\underline{G}. We no longer can talk about real points of \underline{T}, so we do not have the interpretation of 𝔞 as a tangent space. We define 𝔞 as X_{∗}(\underline{T}) ⊗ ℝ. We set V := 𝔞.

Let E denote the affine space underlying V; as before, Δ ⊂ V is the positive Weyl chamber for W_{sph}. Note that X^{∗}(\underline{Z}) ⊂ X^{∗}(\underline{T}) is a subgroup of finite index.

Definition 4.7. Define the homomorphism ν : Z → V by the formula (see [Tits])

⟨χ, ν(z)⟩ = −v(χ(z)), ∀z ∈ Z, χ ∈ X^{∗}(\underline{Z}).  (7)

Note that on the left hand side of the above formula, the pairing of χ ∈ X^{∗}(\underline{Z}) ⊂ X^{∗}(\underline{T}) with ν(z) is coming from the pairing ⟨·, ·⟩ between X_{∗}(\underline{T}) and X^{∗}(\underline{T}). On the right hand side of the equation (7), the action of χ on z is coming from the fact that each character χ ∈ X^{∗}(\underline{Z}) defines a character of the group Z = \underline{Z}(𝕂), i.e. we have χ(z) ∈ 𝕂^{×}.

Let Z_{c} denote the kernel of ν and let L\underline{G} denote the image of ν in V. Then Z_{c} is a maximal compact subgroup of Z, see [C, pg. 135].

Definition 4.8. We will refer to the group L := L\underline{G} := ν(Z) as the extended cocharacter lattice of \underline{T}.

We have the inclusions

X_{∗}(\underline{T}) = Hom(X^{∗}(\underline{T}), ℤ) ⊂ L\underline{G} ⊂ Hom(X^{∗}(\underline{Z}), ℤ)

with equality in the case when \underline{G} is split over ℚ. The quotient group N/Z_{c} operates on the affine space E faithfully, through a discrete group W̃. The finite Weyl group W_{sph} is the linear part of W̃ (the stabilizer of the origin) and we have W̃ = W_{sph}⋉L\underline{G}.

We refer to [Tits, section 1.2] for more details. Let Δ_{L} := Δ∩L and Z_{Δ} := ν^{−1}(Δ). Then one has:

Theorem 4.9. [Tits, section 3.3.3] G = KZK and the map K\G/K → V given by

KzK ↦ ν(z), z ∈ Z_{Δ},

descends to a bijection K\G/K → Δ_{L}.

This proves that in the nonsplit case we can also identify Δ_{L} with the parameter space Σ_{inv fact}(G) = K\G/K for the Problem Q3.

4.4 Comparison of the parameter spaces for the four algebra problems

We can now compare the four algebra problems Q1–Q4 stated in the Introduction. We have already compared the parameter spaces for Q3 and Q4 in the Introduction. To compare the first three problems we assume that we are given groups \underline{G}_{1}, \underline{G}_{2}, \underline{G}_{3} which are connected reductive algebraic groups (possibly all the same), where the first two are defined over ℝ, and \underline{G}_{3} is over 𝔽_{(3)} = 𝕂, where 𝕂 is a nonarchimedean valued field with valuation v and value group ℤ. Suppose that the fields 𝔽_{(1)}, 𝔽_{(2)} are either ℝ or ℂ. We assume that all three Lie groups G_{i} = \underline{G}_{i}(𝔽_{(i)}) have the same (relative) rank l and isomorphic (relative) Weyl groups W = W_{sph} acting on the appropriately chosen vector space 𝔞. Let Δ ⊂ 𝔞 be a (positive) Weyl chamber. Let L = L_{\underline{G}_{3}} be the extended cocharacter lattice, see Definition 4.8. Then the parameter spaces Σ_{1}, Σ_{2} for the Problems Q1 and Q2 (for the groups G_{1}, G_{2} resp.) are exactly the same: they are equal to Δ. The parameter space Σ_{inv fact}(G_{3}) is the intersection Δ_{L} of the lattice L with Δ.

Note that in the special case when \underline{G} = \underline{G}_{i}, i = 2, 3, is split over ℚ, L = X_{∗}(\underline{T}) and we get a canonical isomorphism

X_{∗}(\underline{T}) ⊗ ℝ → 𝔞,

described in the previous section (here 𝔞 is the real Lie algebra of \underline{T} (ℝ)). This isomorphism identifies Σ_{inv fact}(G_{3}) with the set of “integer points” Δ_{L} in the Weyl chamber Δ.

5 The existence of polygonal linkages and solutions to the algebra problems

In this chapter we will show that the algebra problems R(\overline{G}) from chapter 4 can be restated geometrically, in terms of the existence of triangles with the prescribed “side-lengths” in three classes of spaces of nonpositive curvature, which are:

1. X is an infinitesimal symmetric space (a Cartan motion space), X = 𝔭.

2. X is a symmetric space of nonpositive curvature.

3. X is an Euclidean building.

5.1 Setting up the general geometry problem

We start with a purely set-theoretic discussion. Let X be a set with a base-point o, and let \overline{G} is a group acting on X. Let K denote the stabilizer of o in G. In general, the \overline{G}-action is not transitive (such examples will appear when X is a Euclidean building), we let Y denote the orbit G · o. For pairs of points (x, x′) ∈ X^{2} we define the invariant

σḠ(x, x′)

as the projection of (x, x′) to \overline{G}\X^{2}. We will regard σḠ as the “generalized \overline{G}-invariant distance” between points in X. Given τ ∈ \overline{G}\X^{2} we regard the set

S(o, τ) = {x ∈ X : σḠ(o, x) = τ},

as the “sphere of radius τ” centered at o.

For pairs of points (x, y) ∈ Y^{2} one can interpret the invariant σḠ(x, y) as follows: Let Σ denote the quotient space

Σ = K\Y = K\\overline{G}/K.

Then for pairs of points (x, y) = (gK, hK) in Y we can identify σḠ(x, y) with the double coset Kg^{−1}hK. In other words, translate the pair (x, y) by g^{−1} to (o, z) = (o, g^{−1}(o)) and then project g^{−1}(o) to the element σḠ(x, y) ∈ K\Y. Then two pairs (x, y) and (x′, y′) in Y^{2} belong to the same \overline{G}-orbit if and only if σḠ(x, y) = σḠ(x′, y′); Hence, \overline{G}\Y × Y can be bijectively identified with Σ = K\\overline{G}/K.

We will use the notation Υ for the map G → Σ. For σ ∈ Σ we let O_{σ} ⊂ \overline{G} denote Υ^{−1}(σ).

It is now clear that the Problem R(\overline{G}) stated in chapter 4 can be restated as a special case (n = 3) of the following:

Problem 5.1. Give conditions on the vector of generalized side-lengths σ^{→}= (σ_{1}, ⋯, σ_{n}) ∈ Σ^{n} that are necessary and sufficient in order that there exist elements g_{1}, g_{2}, ⋯, g_{n} in \overline{G} such that

Υ(g_{i}) = σ_{i}, 1 ≤ i ≤ n, and ∏_{i=1}^{n} g_{i} = id.

We set O = ∏_{i=1}^{n} O_{σ_{i}}. Note that (K ×K)^{n} acts on O by right and left multiplications.

Motivated by the connection between the moduli spaces of n-gons and symplectic quotients when Y is a complete simply-connected 3-dimensional Riemannian manifold of constant curvature (see [KM], [KMT]), [Tr]) we define the “momentum map” μ : O ⟶ \overline{G} by

μ(g_{1}, ..., g_{n}) = g_{1} · g_{2} ⋯ g_{n}.

We define the analogue of the symplectic quotient of O by K, by

O//K = {g ∈ O : μ(g) = id}/K.

Here we divide out by the diagonal action of K, where K acts on each factor by conjugation. This action is the only action of K on O that we will use henceforth.

We let Q_{σ}(\overline{G}) be the subset of g ∈ O such that μ(g) = id. Thus Q_{σ}(\overline{G}) is the set of solutions of the problem 5.1 and O//K is its quotient by K.

Our goal now is to reformulate the problem 5.1 in more geometric terms.

An n-gon (or a closed n-gonal linkage) in X with the vertices x_{1}, ..., x_{n} ∈ X is an n-tuple x^{→}= (x_{1}, ..., x_{n}) ∈ X^{n}, regarded as a map ℤ/nℤ → X. The generalized side-lengths of the polygon x^{→} are the elements

σ_{i} = σḠ(x_{i}, x_{i+1}) ∈ \overline{G}\X^{2}, i ∈ ℤ/nℤ.

Thus for each n-gon we get a vector σ^{→}= (σ_{1}, ..., σ_{n}) of its generalized side-lengths. The group \overline{G} acts naturally on the space Pol_{n}(X) of all n-gons, preserving the generalized side-lengths of the polygons.

Fix σ^{→}= (σ_{1}, ..., σ_{n}) and form the space

Pol_{n,σ}(X) := {P ∈ Pol_{n}(X) : σḠ(x_{i}, x_{i+1}) = σ_{i}},

and its quotient, the moduli space of polygons with the fixed side-lengths in X:

M_{n,σ}(X) := Pol_{n,σ}(X)/\overline{G}.  (8)

In the case when X is a topological space and \overline{G} acts homeomorphically on X we give the space M_{n,σ}(X) the quotient topology.

We note that Pol_{n}(Y) sits naturally in Pol_{n}(X) as the subset of polygons with vertices in Y. The generalized side-lengths of the polygons with vertices in Y are regarded as elements of Σ = K\Y.

We define a map Φ : Q_{σ}(\overline{G}) → Y^{n} by

Φ(g^{→}) = (o, g_{1} · o, g_{1}g_{2} · o, ⋯, g_{1}g_{2} ⋯ g_{n−1} · o).

We see that Φ is K-equivariant (recall that K acts on O by conjugation on each factor).

Lemma 5.2. The map Φ induces a surjection \overline{Φ} from O//K onto the moduli space M_{n,σ}(Y).

Proof: We first verify that each polygon Φ(g^{→}) has the “correct” side-lengths. By the left-invariance of σḠ we have

σḠ(∏_{i=1}^{k} g_{i} · o, ∏_{i=1}^{k+1} g_{i} · o) = σḠ(o, g_{k} · o) = σ_{k}.

To prove surjectivity of the map \overline{Φ} let y^{→}= (y_{1}, ..., y_{n}) ∈ Pol_{n, σ^{→}}(X). Choose g such

that g · y_{1} = o and replace y^{→} by g^{−1}· y^{→}. Since σḠ(y_{1}, y_{2}) = σ_{1} and (now) y_{1} = o we find that y_{2} ∈ S(o, σ_{1}). Hence there exists g_{1} ∈ O_{σ_{1}} such that y_{2} = g_{1} · o. Similarly, there exists g_{2} ∈ O_{σ_{2}} such that g_{1}g_{2} ·o = y_{3}. Continuing in this way we get an n-tuple (g_{1}, …, g_{n−1}, g′_{n}) ∈ O such that

(o, g_{1} · o, g_{1}g_{2} · o, ⋯, g_{1}g_{2} ⋯ g_{n−1} · o, g_{1}g_{2} ⋯ g_{n−1}g′_{n} · o) = (y_{1}, ..., y_{n}, y_{1}),

where y_{1} = o. Thus g_{1}g_{2} ⋯ g_{n−1}g′_{n} = k ∈ K and after replacing g′_{n} with g_{n} = g′_{n}k^{−1} we get an n-tuple g^{→}= (g_{1}, ..., g_{n}) ∈ O such that μ(g^{→}) = id and Φ(g^{→}) =y^{→}.

Thus we find that the problem 5.1 can be solved if and only the moduli space M_{n,σ}(Y) is nonempty.

To prove a version of Lemma 5.2 with a bijective map and to get a better analogy with the symplectic quotients we have to make a further assumption about the action \overline{G} ↷ Y:

Assumption 5.3. Through the rest of this section we assume that \overline{G} contains a subgroup B which acts simply-transitively on Y.

Now, instead of the orbits O_{σ_{i}} and their product O we consider the intersections:

Ô_{σ_{i}} := O_{σ_{i}} ∩ B, and Ô := O ∩ B^{n}.

The group K no longer acts on Ô by conjugations, instead one has the use a dressing action, dress(K), see [KMT] for the definition in the case when X = Y = ℍ^{3}, G = P SL(2, ℂ) and B fixes a point at infinity. In the case of infinitesimal symmetric spaces X = 𝔭 (where 𝔭 ⊕ 𝔨 = 𝔤) one takes B = 𝔭 and the adjoint action of K as the dressing action. We now redefine the symplectic quotient as follows:

Ô//K := (Ô ∩ μ^{−1}(id))/dress(K).

Then one gets an analogue of Lemma 5.2 (which we do not need for the purposes of this paper), we refer the reader to [KMT] for the discussion in the case of X = ℍ^{3}:

Lemma 5.4. The map Φ induces a bijection Φ̂ from Ô//K onto the moduli space M_{n,σ}(Y). Moreover, let Y be a topological space and G ⊂ Homeo(Y). Then the quotient spaces Ô//K and M_{n,σ}(Y) have natural topology and the bijection Φ̂ is a homeomorphism.

The above discussion has been completely formal, our next goal is to describe the spaces X which can be used to analyze the problem 5.1 for various groups \overline{G}.

5.2 Geometries modelled on Coxeter complexes

Fix a spherical or Euclidean Coxeter complex (A, W), where A is a Euclidean space E or a sphere S and W = W_{aff} or W = W_{sph} is a (possibly nondiscrete) Euclidean or a spherical Coxeter group acting on A. If W is discrete then the Coxeter complex (A, W) is called discrete. In the case of Euclidean Coxeter complexes we let L_{trans} ⊂ W denote the translation subgroup of W. Pick a special vertex o ∈ E with stabilizer W_{sph} (see chapter 3) and let Δ ⊂ E be a Weyl chamber of W_{sph}. We will use the notation Δ_{sph} to denote the ideal boundary of Δ; Δ_{sph} is contained in the sphere S which will be regarded as the sphere at infinity of E; thus Δ_{sph} is a fundamental domain for the action W_{sph} ↷ S.

Let Z be a metric space. A geometric structure on Z modelled on (A, W) consists of an atlas of isometric embeddings φ : A ↪ Z satisfying the following compatibility condition: For any two charts φ_{1} and φ_{2}, the transition map φ_{2}^{−1} ∘φ_{1} is the restriction of an isometry in W. The charts and their images, φ(A) = a ⊂ Z, are called apartments. We will sometimes refer to A as the model apartment. We will require that there are plenty of apartments in the sense that any two points in Z lie in a common apartment. All W -invariant notions introduced for the Coxeter complex (A, W), such as walls, singular subspaces, chambers etc., carry over to geometries modelled on (A, W).

One defines the group of automorphisms Aut(A) of the model apartment as the group of isometries of A which normalize the subgroup W. If X is a space modelled on (A, W) then an isometry g : X → X is an automorphism if it sends apartments to apartments and for each pair of apartments (A, φ_{1}), (A, φ_{2}) the composition φ_{2}^{−1}∘g∘φ_{1} is the restriction of an automorphism of A. The group of automorphisms of X is denoted Aut(X).

Examples of the above geometries are provided by symmetric spaces of noncompact type and their infinitesimal analogues (Cartan motion spaces). These are modelled on Euclidean Coxeter complexes with transitive affine Weyl group. In the case of a symmetric space X, the apartments are the maximal flats. The associated Coxeter complex has the form (E, W_{aff}) where E is an apartment and W_{aff} is the group generated by reflections at singular hyperplanes.

Take a real or complex reductive Lie group G, the Lie algebra 𝔤 of G has the decomposition 𝔤 = 𝔭 ⊕ 𝔨, where K ⊂ G is a maximal compact subgroup, let 𝔨 denote the Lie algebra of K. We will identify the Cartan subspace 𝔭 with the tangent space T_{p}X to the symmetric space X = G/K at the point p stabilized by K. The subspaces 𝔭 ⊂ 𝔤, equipped with the affine action of \overline{G} = 𝔭 ⋉ K, are infinitesimal symmetric spaces in the following sense. The (E, W_{aff})-structure on X induces a (E, W_{aff})- structure on the Cartan subspace 𝔭 = T_{p}X, such that the apartments in 𝔭 are the translates of the Cartan subalgebras (i.e. the maximal abelian subalgebras) in 𝔭. The apartments through 0 in 𝔭 are the tangent spaces to the apartments through p in X. The term “infinitesimal symmetric space” may be also justified by noting that there is a one-parameter family of spaces X_{ϵ} parametrized by ϵ ≥ 0 and all isometric to X for ϵ > 0 such that X_{0} = 𝔭, see [KMT, §5].

More abstractly one can define Cartan motion spaces as follows. Let X be a metric space modelled on a Euclidean Coxeter complex (E, W_{aff}), where W_{aff} acts transitively on E. Assume that, as a metric space, X is isometric to a Euclidean space and that it admits an automorphism group \overline{G} which is chamber transitive (i.e. if Δ, Δ′ are two Euclidean Weyl chambers in X then there exists an isometry g ∈ \overline{G} which sends Δ onto Δ′ and sends the tip of Δ to the tip of Δ′).

In this paper we will consider only those Cartan motion spaces which are isomorphic to infinitesimal symmetric spaces.

Example 5.5. Let G and K denote a complex reductive Lie group and its maximal compact subgroup resp., let 𝔤 = 𝔭 ⊕ 𝔨 be the corresponding Cartan decomposition of the Lie algebra of G. Let 𝔭 be an infinitesimal symmetric space modelled on (𝔞, W_{aff}), where W_{aff} = W_{sph} ⋉ 𝔞 and 𝔞 ⊂ 𝔭 is a Cartan subalgebra. Consider the dual space 𝔭^{∗} together with the coadjoint action Ad^{∗}K of K. Then 𝔭^{∗} is isomorphic to 𝔨^{∗} under the map J : v ↦ \sqrt{−1}v. Both 𝔭^{∗} and 𝔨^{∗} have natural structure of Cartan motion spaces modelled on the Euclidean Coxeter complex (𝔞, W_{sph} ⋉ 𝔞), which can be defined as follows: If a ⊂ 𝔭 is an apartment through the origin (i.e., a Cartan subalgebra in 𝔭), then a^{∗} (defined to be the dual vector space of the Cartan subalgebra of 𝔭 corresponding to a) embeds isometrically into 𝔭^{∗}, where we extends each linear functional f ∈ a^{∗} to a functional on 𝔭 by making it zero on the orthogonal complement to a. The embedding a^{∗} ↪ 𝔭^{∗} is an apartment in 𝔭^{∗}. By composing with the isometry J we get an apartment in 𝔨^{∗}. The coadjoint action Ad^{∗}(K) together with parallel translations define chamber-transitive groups of automorphisms of both 𝔭^{∗} and 𝔨^{∗}.

As a Cartan motion spaces, both examples 𝔭^{∗} and 𝔨^{∗} are, of course, isomorphic to (𝔭, W_{sph} ⋉ 𝔞), where the isomorphism is given by the duality 𝔭 → 𝔭^{∗} (induced by the invariant metric).

Question 5.6. Is there a Cartan motion space which is not isomorphic to an infinitesimal symmetric space?

The last kinds of geometries considered in this paper are spherical and Euclidean buildings. We refer the reader to [Ba] for the definitions of CAT (κ) metric spaces.

Definition 5.7. A spherical building is a CAT(1)-space modelled on a spherical Coxeter complex.

We will use the notation ∠_{T its} for the metric in a spherical building. Spherical buildings have a natural structure as polysimplicial piecewise spherical complexes. We prefer the geometric to the combinatorial view point because it appears to be more flexible.

Definition 5.8. A discrete Euclidean building is a CAT(0)-space modelled on a discrete Euclidean Coxeter complex.

In the non-discrete case the definition of Euclidean buildings is more subtle, see [KL, section 4.1.2]. We refer to [KL] for a thorough discussion of buildings from the geometric viewpoint.

A building is called thick if every wall is an intersection of apartments. A nonthick building can always be equipped with a natural structure of a thick building by reducing the Weyl group.

Example 5.9. If X is a symmetric space of noncompact type or a thick Euclidean building modelled on the Coxeter complex (E, W_{aff}), then its ideal boundary ∂_{T its}X is a thick spherical building modelled on (∂_{T its}E, W_{sph}). In the case that X is a building, the spaces of directions Σ_{x}X are spherical buildings modelled on (∂_{T its}E, W_{sph}). The building Σ_{x}X is thick if and only if x is a special vertex of X. We note that in the case when X is a discrete building, Σ_{x}X is just the link of the point x ∈ X.

Let B be a spherical building modelled on a spherical Coxeter complex (S, W_{sph}). We say that two points x, y ∈ B are antipodal, if ∠_{T its}(x, y) = π; equivalently, they are antipodal points in an apartment s ⊂ B containing both x and y. The quotient map S → S/W_{sph} ≅ Δ_{sph} induces a canonical projection θ : B → Δ_{sph} folding the building onto its model Weyl chamber. The θ-image of a point in B is called its type.

Remark 5.10. To define θ(x) pick an apartment s containing x and a chart ϕ : S → s. Then θ(x) is the projection of ϕ^{−1}(x) to S/W_{sph} ≅ Δ_{sph}. We note that this is clearly independent of s and ϕ.

The same definition applies in the case of Euclidean buildings B. The difference however is that the action W_{aff} ↷ E in general is no longer discrete, so we cannot identify the image of B → E/W_{aff} with a simplex. If W_{aff} acts as a lattice on E, then E/W_{aff} can be identified with a fundamental alcove for the action W_{aff} ↷ E.

We now give two properties of the projection θ:

1. If h : a → a′ is an isomorphism of apartments in B (i.e. ϕ′^{−1} ∘ h ∘ ϕ ∈ W) then θ ∘ h = θ.

2. If B is a spherical building, x, x′ ∈ B which belong to apartments a, a′ respectively and −x ∈ a, −x′ ∈ a′ are antipodal to x, x′, then θ(x) = θ(x′) implies θ(−x) = θ(−x′). To prove this pick an isomorphism h : a → a′. Then (since θ(x) = θ(x′)) there exists w ∈ W ↷ a′ such that w(h(x)) = x′. Hence w ∘ h(−x) = −w ∘ (x) = −x′. The claim now follows from 1.

DeRham decomposition of Euclidean buildings.

Suppose that X is a thick Euclidean building modelled on a reducible discrete Coxeter complex (E, W_{aff}). Consider the deRham decomposition of the building X:

X = X_{0} × X_{1} × ... × X_{s},

where each X_{i} is a thick Euclidean building modelled on the Euclidean Coxeter complex (E_{i}, W_{aff}^{i}) and X_{0} ≅ E_{0} is the flat deRham factor. For each i = 1, ..., s the group W_{aff}^{i} acts as an irreducible affine Coxeter group on the Euclidean space E_{i}, for i = 0 we get the trivial Coxeter group W_{aff}^{0}. Accordingly, the Euclidean space E splits (metrically) as the product

E_{0} × ∏_{i=1}^{s} E_{i},

this decomposition is invariant under W_{aff} which in turn splits as

W_{aff} = ∏_{i=0}^{s} W_{aff}^{i},

as explained in Section 3.

5.3 Bruhat-Tits buildings associated with nonarchimedean reductive Lie groups

In what follows we will describe properties of the Euclidean building (the Bruhat-Tits’ building) associated to a reductive nonarchimedean Lie group by Bruhat and Tits. Let 𝕂 be a valued field with valuation v and value group ℤ. Let O denote the subring in 𝕂 which consists of elements with nonnegative valuation. Let \underline{G} be a connected reductive algebraic group over 𝕂 which has relative rank l. The subgroup K := \underline{G}(O) ⊂ G = \underline{G}(𝕂) is a maximal bounded subgroup of G. In this section we review the properties of a Euclidean building X = X\underline{G} attached to the group \underline{G}. We refer the reader to [Tits] and [BT, Chapter 7] for more details.

We begin by recalling the notation from section 4.3. Let \underline{T} ⊂ \underline{G} be a maximal 𝕂split torus, \underline{Z} ⊂ \underline{G} and \underline{N} ⊂ \underline{G} be its centralizer and normalizer (over 𝕂) respectively. We also get the groups N, Z and T of 𝕂-points of the corresponding algebraic groups. Then V := X_{∗}(\underline{T})⊗ℝ, where X_{∗}(\underline{T}) is the group of cocharacters of \underline{T}. The Euclidean space E is the affine space underlying V with appropriately chosen Euclidean metric. Let R_{rel} ⊂ V^{∗} denote the relative root system of the pair (\underline{G}, \underline{T}); then W_{sph} is the finite Coxeter group corresponding to this system.

Definition 4.7 gives us a homomorphism ν : Z → V, with the kernel Z_{c} ⊂ Z and the image equal to the extended cocharacter lattice L\underline{G}. We also get the quotient group N/Z_{c}, which acts on the Euclidean space E discretely and isometrically through a group W̃. Unless \underline{G} is semisimple, W̃ is not an affine Weyl group.

In the case when the group \underline{G} is simply-connected and semisimple, one can take the pair (E, W̃) as the Euclidean Coxeter complex (E, W_{aff}) of the Bruhat-Tits building X\underline{G} attached to the group \underline{G}. In general however it is not the case and one has to do more work to define W_{aff}. The construction of this group will be unimportant for us (we refer to [Tits] for the explicit construction), the important properties of W_{aff} are the following:

1. dim(E) = l, the relative rank of \underline{G} or the dimension of \underline{T}.

2. W_{aff} ⊂ W̃ is a normal subgroup [Tits, section 1.7, page 34], the index |W̃ : W_{aff}| is finite in the semisimple case.

3. W̃ = W_{sph} ⋉ L\underline{G}.

4. W_{aff} is an affine Coxeter group attached to a root system R ⊂ V^{∗}.

5. The root system R in general is not the same as R_{rel} but they have the same rank and the same finite Weyl group W_{sph}, [Tits, section 1.7].

6. W_{aff} = W_{sph} ⋉ L_{trans}, where W_{sph} is the finite Weyl group as above.

7. We have the inclusions

Q(R^{∨}) = L_{trans} ⊂ L\underline{G} ⊂ N_{aff} = P (R^{∨}).

8. If \underline{G} is a simply-connected semisimple group then W_{aff} = W̃, [Tits, section 1.13]. If \underline{G} is split over an unramified extension of 𝕂 then L\underline{G} = X_{∗}(\underline{T}), [Tits, section 1.3].

9. If \underline{G} is split then the reduced (Bruhat-Tits) root system associated with W_{aff} is the usual reduced root system R_{rel} of the group \underline{G} (note that R_{rel} is also the absolute root system since we are in the split case).

Here as usual, L_{trans} is the translation subgroup of W_{aff} and N_{aff} is the normalizer of W_{aff} in V. When we are dealing with the root system associated with \underline{G}, as in 4 or 5 above, we will refer to R, and not to R_{rel}.

Definition 5.11. We will call R the Bruhat-Tits root system associated with the algebraic group \underline{G}.

We note that the root systems R and R_{rel} are not very different since they have the same finite Coxeter group. Thus, in the irreducible case, both root systems are either isomorphic or one of them is of type B_{l} and the other of type C_{l}. In particular, the saturation factors for R and R_{rel} are exactly the same (see chapter 3).

Having described the properties of the Coxeter complex (E, W_{aff}), we will describe the properties of the Bruhat-Tits building X = X\underline{G} associated with the group \underline{G}:

List 5.12. 1. X is modelled on the Euclidean Coxeter complex (E, W_{aff}) described above.

2. The group G acts on X by (isometric) automorphisms.

3. The subgroup K is the stabilizer of a special vertex o in X.

4. G acts transitively on the set of apartments in X.

5. For each apartment a ⊂ X let G_{a} be the stabilizer of a in G. Then the image of G_{a} in Aut(a) is the group W̃ (containing W_{aff} as a normal subgroup).

6. The building X is thick (see [BT, Prop. 7.4.5]).

7. The group W_{aff} acts discretely and (in the semisimple case) cocompactly on E.

Remark 5.13. The properties in the List 5.12, except the last one, hold in the case of symmetric spaces and their infinitesimal analogues.

We note that the affine space E has the distinguished point o, the origin (corresponding to the trivial cocharacter). The corresponding vertex in X is also denoted by o, it is stabilized by K = \underline{G}(O).

5.4 Geodesic polygons

In this paper we will be considering polygons in the metric spaces modelled on the Coxeter complexes, which were discussed in section 5.2. With the exception of the rank zero spherical buildings, all such metric spaces X are geodesic and thus we define a geodesic polygon in X as a polygon with the vertices z_{1} … z_{n} together with a choice of sides \overline{z_{i}z_{i+1}}, i.e. the geodesic segments connecting z_{i} to z_{i+1}. We note that in the case of metric spaces modelled on Euclidean Coxeter complexes the sides are uniquely determined by the polygon z_{1} … z_{n}. We recall that n-gons in X are regarded as maps ℤ/nℤ → X.

Let (A, W) be a spherical or Euclidean Coxeter complex. The complete invariant of a pair of points (x, y) ∈ A^{2} with respect to the action W ↷ A, is its image σ_{W} (x, y) under the canonical projection to A×A/W. We define the refined length of a geodesic segment \overline{xy} as σ_{ref}(x, y) := σ_{W} (x, y). This notion carries over to geometries modelled on the Coxeter complex (A, W): For a pair of points (x, y) pick an apartment a containing x, y and, after identifying a with the model apartment A, let σ_{ref}(x, y) be the projection to A × A/W.

In the case of Euclidean Coxeter complexes there are extra structures associated with the concept of refined length. Given a Euclidean Coxeter complex (E, W_{aff}), pick a special vertex o ∈ E. Then we can regard E as a vector space V, with the origin 0 = o. Let Δ ⊂ E denote a Weyl chamber of W_{sph}, the tip of Δ is at o.

Suppose that L is a subgroup of the group V of all translations of E so that:

L_{trans} ⊂ L ⊂ N_{aff},

where N_{aff} ⊂ V is the normalizer of W_{aff}. Since o is the origin in E, we will identify the orbits L · o and L_{trans} · o with L and L_{trans} respectively. We define the set of refined L-integral lengths as the subset

(L × L)/W_{aff} ⊂ (E × E)/W_{aff}.

If L = L_{trans} then we have a natural bijection

(L_{trans} × L_{trans})/W_{aff} ≅ Δ ∩ L_{trans}.

Since (E, W_{aff}) is a Euclidean Coxeter complex, there is also a coarser notion of Δ-length obtained from composing σ_{ref} with the natural forgetful map

E × E/W_{aff} → E/W_{sph} ≅ Δ.

To compute the Δ-length σ(x, y) we regard the oriented geodesic segment \overline{xy} as a vector in E and project it to Δ.

Again, the concepts of Δ-length, L-integral Δ-lengths, etc., carry over to the geometries modelled on (E, W_{aff}). Note that Δ-length and refined length coincide for symmetric spaces and their infinitesimal analogues because the affine Weyl group acts transitively. We define the set of L-integral Δ-lengths as the subset Δ ∩ L ⊂ Δ. A segment with L-integral Δ-length has L-integral refined length iff its vertices lie in the distinguished orbit L · o (identified with L). For segments with endpoints of type W_{aff} · o, the notions of Δ-length and refined length are equivalent.

Given a collection τ = (τ_{1}, ..., τ_{n}) ∈ Δ^{n} of Δ-lengths we define the moduli space M_{n,τ}(X) as the quotient of the collection of geodesic polygons in X with the Δ-side-lengths τ by the action of the group G. We give M_{n,τ}(X) the quotient topology.

We are now ready to state the questions which will be (for n = 3) the geometric counterparts to the algebra questions in chapter 4:

Problem 5.14. Let X be a symmetric space of noncompact type or a Cartan motion space, or a thick Euclidean building. Describe the set D_{n}(X) ⊂ Δ^{n} of Δ-side lengths which occur for geodesic n-gons in X.

Problem 5.15. Let X be a thick Euclidean building. Describe the set

D_{n}^{ref,L}(X) ⊂ (W \L × L)^{n} ⊂ (W \A × A)^{n}

of refined L-integral side-lengths which occur for n-gons in X.

Below we explain how given an algebraic problem Q1–Q3 one finds a metric space X modelled on a Euclidean Coxeter complex, so that the geometric problem 5.15 (for n = 3) is equivalent to the corresponding algebraic problem.

As we explained in the beginning of chapter 4, with each problem Qi, i=1, 2, 3, we can associate a pair of groups K ⊂ \overline{G}.

1. For the Problem Q1 we take \overline{G} = K ⋉ 𝔭 and let X := 𝔭. Then X is a metric space modelled on the Euclidean Coxeter complex (A, W) = (E, W_{aff}).

2. For the Problem Q2 we take \overline{G} = G and let X be the symmetric space X = G/K.

In both cases \overline{G} acts transitively on X and we apply Lemma 5.2 to the transitive action \overline{G} ↷ X, to conclude that Q1 and Q2 are equivalent to Problem 5.14 for the corresponding space X.

Example 5.16. Let G = GL_{m}(ℂ), K = U(m). Then the symmetric space X associated with G is G/K, which is the space ℙ_{m} of positive-definite Hermitian m × m matrices. The model apartment A in ℙ_{m} consists of diagonal matrices with positive diagonal entries. The finite Weyl W_{sph} = S_{m} group acts on A by permutations of the diagonal elements; the affine Weyl group W_{aff} is isomorphic to W_{sph} ⋉ ℝ^{m−1}. Thus we get a geometric model for analyzing the Problem P2 from the Introduction.

3. For the Problem Q3 we assign to the group G = \underline{G}(𝕂) the Euclidean (Bruhat-Tits) building X as it was explained in the previous section. We let Y be the G-orbit of the special vertex o ∈ X.

Although, unlike in the previous two examples, the group G does not act transitively on the building X, this group acts transitively on the subset Y ⊂ X and we apply Lemma 5.2 to the action G ↷ Y to see that the problem Q3 (or, equivalently, R(G)) is equivalent to finding necessary and sufficient conditions for existence of geodesic triangles in X whose vertices are in Y and whose Δ-lengths are the prescribed elements of Δ_{L}, where L = L\underline{G}. More generally, there is a surjective map

pr : W \A × A → G\X × X,

and by applying the Transfer Theorem 1.3 we get:

Proposition 5.17. Suppose that there exists a polygon P in X whose σ_{G}-side-lengths are (σ_{1}, ..., σ_{n}). Then for any choice of σ̃_{i} ∈ pr^{−1}(σ_{i}), there exists a polygon P̃ in X whose σ_{ref}-side-lengths are σ̃_{1}, ..., σ̃_{n}.

Therefore we will stick to the notion of the refined side-length σ_{ref} through the rest of the paper.

6 Weighted configurations, stability and the relation to polygons

Let X be a symmetric space of nonpositive curvature or a Euclidean building. Recall that the ideal boundary ∂_{∞}X has the structure of a spherical building B = ∂_{T its}X, the metric on B is denoted by ∠_{T its}. Given a Weyl chamber Δ in X, we get a spherical Weyl chamber Δ_{sph} = ∂_{∞}Δ ⊂ ∂_{T its}X. We will identify Δ_{sph} with the unit vectors in Δ. Recall that there is a canonical projection θ : ∂_{T its}X → Δ_{sph}, see section 5.2.

Take a collection of weights m_{1}, ..., m_{n} ≥ 0 and define a finite measure space (ℤ/nℤ, ν) where the measure ν on ℤ/nℤ is given by ν(i) = m_{i}. An n-tuple of ideal points (ξ_{1}, ..., ξ_{n}) ∈ B^{n} together with (ℤ/nℤ, ν) determine a weighted configuration at infinity, which is a map

ψ : (ℤ/nℤ, ν) → ∂_{∞}X.

The type τ(ψ) = (τ_{1}, …, τ_{n}) ∈ Δ^{n} of the weighted configuration ψ is given by τ_{i} = m_{i} · θ(ξ_{i}). Let μ = ψ_{∗}(ν) be the pushed forward measure on B. We define the slope of a measure μ on B with finite total mass |μ| as

slope_{μ}(η) = − ∫_{B} cos ∠_{T its}(ξ, η) dμ(ξ).

In this paper we will consider only measures with finite support.

Definition 6.1 (Stability). A measure μ on B is called semistable if slope_{μ}(η) ≥ 0 and stable if slope_{μ}(η) > 0 for all η ∈ B.

There is a refinement of the notion of semistability motivated by the corresponding concept in geometric invariant theory.

Definition 6.2 (Nice semistability). A measure μ on B is called nice semistable if μ is semistable and {slope_{μ} = 0} is a subbuilding or empty. In particular, stable measures are nice semistable.

A weighted configuration ψ on B is called stable, semistable or nice semistable, respectively, if the corresponding measure ψ_{∗}ν has this property.

Remark 6.3. The definitions given above are not the definitions given in [LM]. In [LM] the definitions are in terms of the behaviour of a certain convex function on X, the weighted Busemann function. The above definitions are consequences of the definitions in [LM]. They are equivalent to those of [LM] in the case X is a symmetric space. The above definitions have the advantage that they involve only the geometry of the Tits boundary ∂_{T its}X and lead more quickly to the results we want. They correspond to the criteria for semistability based on Mumford’s numerical function in the classical Geometric Invariant Theory.

For the purposes of this paper, i.e. the study of polygons, nice semistability plays a role in the case of symmetric spaces and infinitesimal symmetric spaces only. We note however that for these spaces, existence of a semistable configuration ψ on ∂_{T its}X implies existence of a nice semistable configuration on ∂_{T its}X, which has the same type as ψ, see [LM].

Example 6.4. (i) Let B be a spherical building of rank 0. Then a measure μ on B is stable iff it contains no atoms of mass ≥ \frac{1}{2}|μ|, semistable iff it contains no atoms of mass > \frac{1}{2}|μ|, and nice semistable iff it is either stable or consists of two atoms of equal mass.

(ii) Suppose that B is a unit sphere and regard it as the ideal boundary of a Euclidean space E, B = ∂_{T its}E. Each semistable measure μ has slope zero everywhere.

Define the subset Δ_{ss}^{n}(B) ⊂ Δ^{n} consisting of those n-tuples τ ∈ Δ^{n} for which there exists a weighted semistable configuration on B of type τ.

Suppose now that G is a reductive complex Lie group, K ⊂ G is a maximal compact subgroup, X = G/K is the associated symmetric space. Then the spaces of weighted configurations in ∂_{T its}X of the given type τ ∈ Δ^{n} can be identified with products

F = F_{1} × ... × F_{n}

where F_{i}’s are smooth complex algebraic varieties (generalized flag varieties) on which the group G acts transitively. Hence G acts on F diagonally.

In case X is the symmetric space associated to a complex Lie group, the notions of stability (semistability, etc.) introduced above coincide with corresponding notions from symplectic geometry, and, in the case where the weights τ_{i}’s are L-integral (i.e., belong to L = L\underline{G}) they also coincide with the concepts of stability (semistability, etc.) used in the Geometric Invariant Theory; for a proof of this see [LM].

6.1 Gauss maps and associated dynamical systems

We now relate polygons in X (where X is a metric space modelled on a Euclidean Coxeter complex) and weighted configurations on the ideal boundary B = ∂_{∞}X, which is, in a sense, the heart of [LM] and [KLM]. If X is an infinitesimal symmetric space 𝔭 we identify the visual boundary of X with the Tits boundary ∂_{T its}X′ of the corresponding symmetric space X′ via the exponential map 𝔭 → X′.

Consider a (closed) polygon P = x_{1}x_{2} … x_{n} in X, i.e. a map ℤ/nℤ → X. The distances m_{i} = d(x_{i}, x_{i+1}) determine a finite measure ν on ℤ/nℤ by ν(i) = m_{i}. The polygon P gives rise to a collection Gauss(P) of Gauss maps

ψ : ℤ/nℤ ⟶ ∂_{∞}X  (9)

by assigning to i an ideal point ξ_{i} ∈ ∂_{∞}X so that the geodesic ray \overline{x_{i}ξ_{i}} (originating at x_{i} and asymptotic to ξ_{i}) passes through x_{i+1}. This construction, in the case of the hyperbolic plane, already appears in the letter of Gauss to Bolyai, [G]. Taking into account the measure ν, we view the maps ψ : (ℤ/nℤ, ν) → ∂_{∞}X as weighted configurations of points on ∂_{∞}X. Note that if X is a Riemannian symmetric space and the m_{i}’s are all non-zero, there is a unique Gauss map due to the unique extendability of geodesics. On the other hand, if X is a Euclidean building then, due to the branching of geodesics, there are in general infinitely many Gauss maps. However, the corresponding weighted configurations are of the same type, i.e. they project to the same weighted configuration on Δ_{sph}.

The following crucial observation explains why the notion of semistability is important for studying closed polygons.

Lemma 6.5 ([LM], [KLM]). For each Gauss map ψ the pushed forward measures μ = ψ_{∗}ν are semistable. If X is a symmetric space or an infinitesimal symmetric space then the measure μ is nice semistable.

We are now interested in finding polygons with prescribed Gauss map. Such polygons will correspond to the fixed points of a certain dynamical system. For ξ ∈ ∂_{∞}X and t ≥ 0, we define the map ϕ := ϕ_{ξ,t} : X → X by sending x to the point at distance t from x on the geodesic ray \overline{xξ}. Since X is nonpositively curved, the map ϕ is 1-Lipschitz. Fix now a weighted configuration ψ : (ℤ/nℤ, ν) → ∂_{∞}X with non-zero total mass. We define the 1-Lipschitz self-map

Φ = Φ_{ψ} : X ⟶ X

as the composition Φ_{n} ∘⋯∘Φ_{1} of the maps Φ_{i} = ϕ_{ξ_{i},m_{i}}. The fixed points of Φ are the first vertices of closed polygons P = x_{1} … x_{n} so that ψ is a Gauss map for P. Since the map Φ is 1-Lipschitz, and the space in question is complete and has nonpositive curvature, the existence of a fixed point for Φ reduces (see [KLM]) to the existence of a bounded orbit for the dynamical system (Φ^{n})_{n∈ℕ} formed by the iterations of Φ. Of course, in general, there is no reason to expect that (Φ^{n})_{n∈ℕ} has a bounded orbit: for instance, if the support of the measure μ = ψ_{∗}(ν) is a single point, all orbits are unbounded.

One of the main results of [KLM] is that under nice semi-stability assumption on ψ and local compactness assumption on X, the system (Φ^{n})_{n∈ℕ} has a bounded orbit:

Theorem 6.6. Suppose that X is either a symmetric space or a locally compact Euclidean building or a (thick) Euclidean building with one vertex. Suppose that ψ is a nice semistable weighted configuration on ∂_{T its}X (in the symmetric space case) or a semistable configuration (in the building case). Then Φ_{ψ} has a fixed point.

Combining this with the Transfer Theorem 1.3 we get:

Theorem 6.7 ([KLM]). Suppose that X is a symmetric space of nonpositive curvature or an infinitesimal symmetric space or a Euclidean building with a model Weyl chamber Δ. Then D_{n}(X) = Δ_{ss}^{n}(∂_{T its}X).

The equivalence of the Problems Q1 and Q2 (and consequently P1 and P2) in the Introduction follows immediately from the above theorem since for an infinitesimal symmetric space 𝔭 and the corresponding symmetric space X the Tits boundaries are the same. As another corollary of the combination of Theorems 1.3 and 6.7 we get:

Theorem 6.8 ([KLM]). Let X be either a thick Euclidean building, a symmetric space or an infinitesimal symmetric space. Then D_{n}(X) depends only on the associated spherical Coxeter complex and not on the type of the geometry.

In the case of symmetric spaces X = G/K (and their infinitesimal analogues 𝔭) one actually gets a more precise result. Given a type τ (so that for each i the vector τ_{i} is nonzero), the set of nice semistable configurations Conf_{τ,nsst}(B) of type τ on B = ∂_{T its}X has natural topological structure. Define the moduli space M_{τ,nsst}(B) of nice semi-stable configurations on B of type τ as the quotient Conf_{τ,nsst}(B)/G. We note that the moduli space (see (8)) M_{n,τ}(X) of n-gons in X with the given side-lengths τ ∈ Δ^{n}, is also a topological space.

Theorem 6.9. The moduli space M_{τ,nsst}(B) is Hausdorff, [LM]. The map P ↦ Gauss(P) defines a natural homeomorphism M_{n,τ}(X) → M_{τ,nsst}(B), see [LM], [KLM].

Remark 6.10. There is a closely related moduli space M_{τ,ss}(B) of extended orbit equivalence classes of semistable weighted configurations on B. This space is more natural from the algebro-geometric viewpoint. At present we have not proved that the spaces M_{τ,ss}(B) are Hausdorff. They are known to be Hausdorff if G is a complex Lie group, see [HL] and [Sj], Thus in the case of complex Lie groups we also have a homeomorphism M_{n,τ}(X) → M_{τ,ss}(B).

Below are few more details concerning the symplectic nature of the moduli space M_{n,τ}(X) ≅ M_{τ,ss}(B) in the case when G is a complex Lie group. Recall that 𝔤 = 𝔭 ⊕ 𝔨 and 𝔭 = \sqrt{−1}𝔨. For τ ∈ Δ^{n} ⊂ 𝔞^{n} ⊂ 𝔭^{n} we identify each τ_{i} with the element α_{i} := \sqrt{−1}τ_{i} of the Lie algebra 𝔨. We consider the product M := ∏_{i=1}^{n} O_{i} of the orbits O_{i} := Ad(K)(α_{i}), i = 1, ..., n. The manifold M carries a natural symplectic structure which is invariant under the diagonal adjoint action of the group K. Let f : M → 𝔨 be the momentum mapping of this action. By Lemma 10.7, the map f is given by the formula

f : (β_{1}, .., β_{n}) ↦ ∑_{i=1}^{n} β_{i}.

and we obtain an identification of the moduli space M_{n,τ}(𝔭) of polygons in 𝔭 with the symplectic quotient:

M_{τ,ss}(B) = Conf_{τ}(B)//G := M//G := {ξ : f(ξ) = 0}/K.

In the case when all τ_{i} belong to the cocharacter lattice L, this quotient is the same as the Mumford quotient of the projective variety Conf_{τ} by the group G.

6.2 The polyhedron D_{n}(X)

One of the main results of [LM] is a description of D_{n}(X) (where X is an infinitesimal symmetric space) in terms of the Schubert calculus in the Grassmannians associated to complex and real Lie groups G (i.e. the quotients G/P where P is a maximal parabolic subgroup of G). Below we describe D_{n}(X) for the three classes of metric spaces considered in the present paper. We first do it in the context of symmetric spaces of noncompact type (i.e. their deRham decomposition contains no Euclidean factor^{2}) since the description in this class is more natural.

Let X be a symmetric space of noncompact type and G the identity component of its isometry group. The ideal boundary ∂_{T its}X is a spherical building modelled on a spherical Coxeter complex (S, W) with model spherical Weyl chamber Δ_{sph} ⊂ S. We identify S with an apartment in ∂_{T its}X. Let Δ denote the Euclidean Weyl chamber of X. We identify Δ_{sph} with ∂_{T its}Δ.

Let B be the stabilizer of Δ_{sph} in G. For each vertex ζ of ∂_{T its}X one defines the generalized Grassmannian Grass_{ζ} = Gζ = G/P. (Here P is the maximal parabolic subgroup of G stabilizing ζ.) It is a compact homogeneous space stratified into Borbits called Schubert cells. Every Schubert cell is of the form C_{η} = Bη for a unique vertex η ∈ W ζ ⊂ S^{(0)} of the spherical Coxeter complex. The closures \overline{C_{η}} are called Schubert cycles. They are unions of Schubert cells and represent well defined elements in the homology H_{∗}(Grass_{ζ}, ℤ_{2}).

2In the context of Lie groups it corresponds to the case of semisimple algebraic groups.

For each vertex ζ of Δ_{sph} and each n-tuple η^{→}= (η_{1}, …, η_{n}) of vertices in W ζ consider the following homogeneous linear inequality for ξ ∈ Δ^{n}:

∑_{i} ξ_{i} · η_{i} ≤ 0  (∗_{ζ; η^{→}})

Here we identify the η_{i}’s with unit vectors in Δ.

Let I_{ℤ_{2}}(G) be the subset consisting of all data (ζ, η^{→}) such that the intersection of the Schubert classes [\overline{C}_{η_{1}}], …, [\overline{C}_{η_{n}}] in H_{∗}(Grass_{ζ}, ℤ_{2}) equals [pt].

Theorem 6.11 ([LM]). Δ_{ss}^{n}(∂_{T its}X) ⊂ Δ^{n} consists of all solutions ξ to the system of inequalities (∗_{ζ; η^{→}}) where (ζ, η^{→}) runs through I_{ℤ_{2}}(G).

Remark 6.12. This system of inequalities depends on the Schubert calculus for the generalized Grassmannians G/P associated to the group G. It is one of the results of [KLM] that the set of solutions depends only on the spherical Coxeter complex.

Typically, the system of inequalities in Theorem 6.11 is redundant. If G is a complex Lie group one can use the complex structure to obtain a smaller system of inequalities. In this case, the homogeneous spaces Grass_{ζ} are complex manifolds and the Schubert cycles are complex subvarieties and hence represent classes in integral homology. Let I_{ℤ}(G) ⊂ I_{ℤ_{2}}(G) be the subset consisting of all data (ζ, η^{→}) such that the intersection of the Schubert classes [\overline{C}_{η_{1}}], …, [\overline{C}_{η_{n}}] in H_{∗}(Grass_{ζ}, ℤ) equals [pt]. We have the analogous result

Theorem 6.13 (Stability inequalities, see [BeSj], [LM]). Δ_{ss}^{n}(∂_{T its}X) consists of all solutions ξ to the system of inequalities (∗_{ζ; η^{→}}) where (ζ, η^{→}) runs through I_{ℤ}(G).

Remark 6.14. The above theorem was proven independently and by completely different methods in [BeSj], [LM] in the context of infinitesimal symmetric spaces associated with semisimple complex Lie groups.

We now consider the general case when X is a symmetric space or a Euclidean building which splits as X_{0}×X_{1}, where X_{0} is the flat deRham factor of X. In the case when X = 𝔭 is a Cartan motion space we consider the decomposition X = X_{0} × X_{1} corresponding to the orthogonal decomposition 𝔭 = 𝔭_{0} ⊕ 𝔭_{1}, where 𝔭_{0} is the Lie algebra of the split part of the of the central torus of G (where 𝔤 is the Lie algebra of G). In this case we again refer to 𝔭_{0} := X_{0} as the Euclidean deRham factor of X. If 𝔭 = T_{p}(X′), where X′ is a nonpositively curved symmetric space, then the above decomposition of 𝔭 is the infinitesimal version of the splitting off the deRham factor of X′.

Let E = E_{0} × E_{1} be the corresponding decomposition of the Euclidean Coxeter complex, the Weyl chamber Δ ⊂ E splits as Δ_{0} × Δ_{1}, where Δ_{1} ⊂ E_{1} is the Weyl chamber for the action W_{sph} ↷ E_{1} and Δ_{0} = E_{0}. Let D_{n}(X_{i}) ⊂ (Δ_{i})^{n} denote the side-lengths polyhedron for the space X_{1}. It is clear that

D_{n}(X_{0}) = {(σ_{1}, ..., σ_{n}) : ∑_{i=1}^{n} σ_{i} = 0}.

Then we get:

D_{n}(X) = D_{n}(X_{0}) × D_{n}(X_{1}).  (10)

We refer the reader to Proposition 8.6 for the explanation.

Combining our results one obtains the following recipe for determining the polytope D_{n}(X) for any of the spaces X as in Theorem 6.8 (i.e., infinitesimal symmetric spaces, symmetric spaces and Euclidean buildings): Given X, first of all, split X as X_{0} × X_{1}, where X_{0} is the Euclidean deRham factor. In view of the formula (10) it suffices to describe D_{n}(X_{1}); so we let X := X_{1}. Then find a complex semisimple Lie group G of noncompact type, whose spherical Coxeter complex is isomorphic to the one of X. Let X′ be the symmetric space G/K associated with G. Using Schubert calculus for the Grassmannians G/P associated to G as in Theorem 6.13 above, compute the system of stability inequalities describing Δ_{ss}^{n}(∂_{∞}X′) = D_{n}(X′). The polytopes D_{n}(X) and D_{n}(X′) are equal. We note that although the polyhedron D_{n}(X) depends only on the spherical Coxeter group, the system of stability inequalities describing D_{n}(X) depends on the root system. In fact in [LM] it is shown that the systems obtained for the root systems B_{3} and C_{3} are different (even though they have the same number of inequalities).

Example 6.15. Suppose that G is a reductive algebraic over ℚ and G^{∨} is its Langlands’ dual. Let 𝔤 and 𝔤^{∨} be the Lie algebras of G and G^{∨} and let X = G/K, X^{∨} be the corresponding symmetric spaces. Let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition. Take a Cartan subalgebra 𝔞 ⊂ 𝔭. The pairing ⟨ , ⟩ induces an isomorphism 𝔞^{∗} ≅ 𝔞^{∨}, where 𝔞^{∨} is a Cartan subalgera of 𝔤^{∨}. We then identify 𝔞 with 𝔞^{∗} using the invariant metric. This gives us an isometry f : 𝔞 → 𝔞^{∨} which conjugates the spherical Coxeter group W_{sph} of 𝔤 to the spherical Coxeter group W_{sph}^{∨} of 𝔤^{∨}. This isometry also carries a Euclidean Weyl chamber Δ of W_{sph} ↷ 𝔞 onto a Euclidean Weyl chamber for the action W_{sph}^{∨} ↷ 𝔞^{∨}. Thus f induces a bijection

f^{n} : D_{n}(X) → D_{n}(X^{∨}).

For instance, if G and G^{∨} are simple complex Lie groups of type B_{ℓ} and C_{ℓ} respectively, then the above construction provides an isometry D_{n}(X) ≅ D_{n}(X^{∨}).

In the next subsection we will write down the inequalities 6.13 for the root system B_{2} in the case n = 3.

6.3 The polyhedron for the root system B_{2}

In [LM] the polyhedra D_{3}(X) were computed for all X of rank 2 or 3. We now give the example of the polyhedron for the root system B_{2}. This example will be useful to us later.

Since all symmetric spaces with the same root system give rise to the same polyhedron we may take X = SO(5, ℂ)/SO(5). The Weyl chamber Δ is given by

Δ = {(x, y) : x > y > 0}.

Thus the Δ-lengths σ_{i} are vectors (x, y) in Δ. Since the root system has rank 2, Δ_{sph} has exactly two vertices ζ_{1}, ζ_{2}. Thus we get two generalized Grassmannians, Grass_{ζ_{i}}, i = 1, 2. These Grassmannians are the spaces of isotropic lines and isotropic planes in ℂ^{4}. Thus the set of stability inequalities breaks into two subsystems, one for each ζ_{i}. This gives a system of 19 inequalities in addition to the inequalities defining the chamber Δ. Below we have dropped one of these 19 inequalities, which was implied by the inequalities defining the chamber.

The first subsystem of stability inequalities (corresponding to the Grassmannian of isotropic lines) is given by

x_{i} ≤ x_{j} + x_{k}, {i, j, k} = {1, 2, 3}
y_{i} ≤ y_{j} + x_{k}, {i, j, k} = {1, 2, 3}.

In order to describe the second system we let S be the sum of all the coordinates of the side-lengths, so

S = x_{1} + y_{1} + x_{2} + y_{2} + x_{3} + y_{3}.

The second subsystem (corresponding to the Grassmannian of isotropic planes) is then given by

x_{i} + y_{j} ≤ S/2, 1 ≤ i, j ≤ 3.

To the above system of 18 inequalities we also have to add the “chamber inequalities”:

x_{i} ≥ y_{i}, i = 1, 2, 3 and y_{i} ≥ 0, i = 1, 2, 3.

In total we get 24 inequalities. Thus we have:

Corollary 6.16. Suppose that X is a symmetric space with the (finite) Weyl group of type B_{2}. Then there exists a triangle in X whose Δ-side lengths are vectors σ_{i} = (x_{i}, y_{i}) ∈ Δ, i = 1, 2, 3 if and only if σ_{i}’s satisfy the above system of 18 inequalities.

Remark 6.17. Using a computer we have verified that the above system is minimal, the polyhedron D_{3} is a cone over a compact polytope with 15 vertices and 24 topdimensional faces.

7 Polygons in Euclidean buildings and the generalized invariant factor problem

Let X be a (thick) Euclidean building modelled on a discrete Coxeter complex (E, W_{aff}). As in section 5.4, let L be a lattice in E which contains the translation subgroup L_{trans} of W_{aff} and which normalizes W_{aff}. Note that L acts by automorphisms on the Coxeter complex (E_{,}W_{aff}). For the algebraic applications we would like to determine for which τ^{→}∈ D_{n}(X) ∩ L^{n}, there exists a polygon in X with Lintegral side-lengths τ^{→} and the first vertex at a distinguished special vertex o ∈ X. In other words, we are interested in the image of the map

ι : D_{n}^{ref,L}(X) → D_{n}(X) ∩ L^{n}.  (11)

In this chapter we will show that in general the map (11) is not onto (section 7.2): the counterexamples are based on the idea of folding triangles into apartments (see section 7.1). In the subsequent chapter 8 we will prove “positive results”: some of them guarantee that the map (11) is onto for certain pairs (W_{aff}, L), the other results establish sufficient conditions for elements of D_{n}(X) ∩ L^{n} to belong to the image of ι.

7.1 Folding polygons into apartments

In this section we describe a construction which produces from triangles in a building billiard triangles in an apartment.

Suppose that Δ(x, y, z) is a triangle in a (Euclidean or spherical) building B. In general it is not contained in an apartment. However (see [KLM]) there exists a finite subdivision of the edge \overline{xy} by points x_{0} = x, x_{1}, ..., x_{k−1}, x_{k} = y such that each geodesic triangle Δ(z, x_{i}, x_{i+1}) is contained in an apartment in B.

Remark 7.1. One can actually show that there is a uniform upper bound on the number k which depends only on the Coxeter group W.

For each i, let a_{i} ⊂ B be an apartment containing Δ(z, x_{i}, x_{i+1}). We will identify a_{0} with the model apartment A. We will produce points x′_{i} in the first apartment a_{0} such that the triangles Δ(z, x′_{i}, x′_{i+1}) are congruent to the triangles Δ(z, x_{i}, x_{i+1}) via apartment isomorphisms α_{i} : a_{i} → A. This is done inductively as follows. We start with x′_{0} = x_{0} and x′_{1} = x_{1}. Suppose that x′_{i} has been constructed. To find x′_{i+1}, choose α_{i} : a_{i} → a so that it carries \overline{zx_{i}} to \overline{zx′_{i}}. Put x′_{i+1} = α_{i}(x_{i+1}). The procedure yields a billiard triangle in the apartment A consisting of two geodesic sides \overline{zx′_{0}} and \overline{zx′_{k}} with the same refined lengths as the corresponding sides of the original triangle Δ(x, y, z), and one piecewise geodesic path x′_{0}x′_{1} … x′_{k}. The points x′_{1}, ..., x′_{k−1} are the break points of the broken side of this billiard triangle.

Consider the spherical building Y_{i} := Σ_{x_{i}}(B) whose Coxeter complex is (Σ_{x′_{i}}A, W_{i}), where the group W_{i} is the stabilizer of x_{i} in W. Let θ_{i} be the canonical projection of Y_{i} to the Weyl chamber in this building.

Remark 7.2. Notice that W_{i} may be smaller than W_{sph}.

Since the refined lengths of \overline{x_{i}x_{i+1}} and \overline{x′_{i}x′_{i+1}} are equal, it follows that θ_{i}(\overrightarrow{x_{i}x_{i−1}}) = θ_{i}(\overrightarrow{x′_{i}x′_{i−1}}), θ_{i}(\overrightarrow{x_{i}x_{i+1}}) = θ_{i}(\overrightarrow{x′_{i}x′_{i+1}}). Because the directions \overrightarrow{x_{i}x_{i−1}} and \overrightarrow{x_{i}x_{i+1}} in the spherical building Y_{i} = Σ_{x_{i}}(B) are antipodal, the properties of the canonical projection θ_{i} (see section 5.2) imply that the directions \overrightarrow{x′_{i}x′_{i−1}} and \overrightarrow{x′_{i}x′_{i+1}} in the spherical Coxeter complex (Σ_{x′_{i}}A, W_{i}) are antipodal modulo the action of W_{i}.

Definition 7.3. A broken triangle T ⊂ A with two geodesic sides \overline{zx′_{0}} and \overline{zx′_{k}} and one piecewise geodesic path x′_{0}x′_{1} … x′_{k} is a billiard triangle if at every break point x′_{i}, the directions \overrightarrow{x′_{i}x′_{i−1}} and \overrightarrow{x′_{i}x′_{i+1}} in the spherical Coxeter complex Σ_{x′_{i}}A are antipodal modulo the action of the stabilizer W_{i} of x′_{i} in the Coxeter group of A.

We note that each broken side x′_{0}x′_{1} … x′_{k} of a billiard triangle can be straightened in the model apartment A, i.e. there exists a geodesic segment \overline{x′′_{0}x′′_{k}} ⊂ A (a straightening of x′_{0}x′_{1} … x′_{k}) such that x′′_{0} = x′_{0}, the metric length of \overline{x′′_{0}x′′_{k}} is the same as of x′_{0}x′_{1} … x′_{k}, and the direction of \overline{x′′_{0}x′′_{k}} at x′_{0} is the same as the direction of \overline{x′_{0}x′_{1}}.

A billiard triangle T in the model apartment can be unfolded to a geodesic triangle in the building if and only if, for each break point x′_{i}, 0 < i < k, the following holds. Let ξ′_{i}, η′_{i}, ζ′_{i} ∈ Σ_{x′_{i}}A be the directions towards x′_{i−1}, x′_{i+1}, z. The necessary and sufficient condition is:

Figure 1: A billiard triangle.

Condition 7.4. For each i there exists a triangle Δ(ξ_{i}, η_{i}, ζ_{i}) in the spherical building Y_{i} = Σ_{x′_{i}}B so that ∠(ξ_{i}, η_{i}) = π and the refined lengths of \overline{ξ_{i}ζ_{i}} and \overline{η_{i}ζ_{i}} are the same as for \overline{ξ′_{i}ζ′_{i}} and \overline{η′_{i}ζ′_{i}} respecively.

Remark 7.5. A necessary condition for existence of a triangle Δ(ξ_{i}, η_{i}, ζ_{i}) is that ∠(ξ′_{i}, ζ′_{i}) + ∠(η′_{i}, ζ′_{i}) ≥ π, which is just the usual (metric) triangle inequality in the spherical building Y_{i}.

Lemma 7.6. Suppose that we have an apartment a′ in a Euclidean building X and a billiard triangle T′ ⊂ a′, T′ has geodesic sides \overline{z′x′_{0}}, \overline{z′x′_{k}} and the broken side x′_{0}x′_{1} … x′_{k}. Suppose that for each break point x′_{i}, i = 1, ..., k − 1, there is a wall H_{i} ⊂ a′ through x′_{i} which weakly separates^{3} {x′_{i−1}, x′_{i+1}} from the vertex z′. Assume that the reflection w in the wall H_{i} carries the direction \overrightarrow{x′_{i}x′_{i+1}} to the direction antipodal to \overrightarrow{x′_{i}x′_{i−1}}. Then there exists a geodesic triangle T = Δ(x, y, z) ⊂ X so that

σ_{ref}(\overline{z′x′}) = σ_{ref}(\overline{zx_{0}}), σ_{ref}(\overline{z′y′}) = σ_{ref}(\overline{zx_{k}})

and the refined side-length of \overline{xy} is the same as for the straightening of the broken side x′_{0}x′_{1} … x′_{k}.

Proof: We prove the lemma by verifying Condition 7.4 for each i = 1, ..., k − 1. The spherical building Y := Σ_{x′_{i}}X contains the apartment s′ = Σ_{x′_{i}}(a′) and the directions ξ′, ζ′, η′ of the geodesic segments

\overline{x′_{i}x′_{i−1}}, \overline{x′_{i}x′_{i+1}}, \overline{x′_{i}z′}.

The wall F := Σ_{x′_{i}}(H_{i}) in the spherical apartment s′ separates s′ into half-apartments s′_{+}, s′_{−}, so that cl(s′_{+}) contains ζ′ and cl(s′_{−}) contains the directions ξ′, η′. Because the building Y is thick, there exists an apartment s ⊂ Y which intersects s′ along the halfapartment s′_{+}. Let s_{−} := cl(s\s′_{+}). There exists an isomorphism of Coxeter complexes ι : s′ → s which restricts to the identity on s_{+}. Set η := ι(η′), ζ := ζ′, ξ := ξ′. Clearly,

σ_{ref}(\overline{ζξ}) = σ_{ref}(\overline{ζ′ξ′}), σ_{ref}(\overline{ζη}) = σ_{ref}(\overline{ζ′η′}).

3I.e., H_{i} separates a′ into half-apartments a′_{+}, a−′ such that z is in the closure of a′_{+} and x′_{i−1}, x′_{i+1} are in the closure of a′_{−}.

It remains to verify that ∠(ξ, η) = π, i.e. that the points ξ and η are antipodal. Note that we also have the third apartment s′′ = s′_{−} ∪ s_{−} and the isomorphism of Coxeter complexes

j : s′ → s′′

which is the identity on s′_{−}, see Figure 2. The restriction of j to s′_{+} equals ι ∘ w. Thus j carries w(η′) (which is antipodal to ξ′ by assumption) to the point η, and j(ξ′) = ξ. Thus the points η, ξ are antipodal.

Figure 2: Unfolding.

Remark 7.7. It is clear that the same argument as above works if instead of a single reflection w which carries \overrightarrow{x′_{i}x′_{i+1}} to the direction antipodal to \overrightarrow{x′_{i}x′_{i−1}}, we have an element g of the spherical Weyl group, which is the composition of reflections g = w_{m} ∘ ... ∘ w_{1}, where each w_{j} is a reflection in a wall H_{j} ⊂ a′ through x′_{i}, weakly separating w_{j−1} ∘ ... ∘ w_{1}(x′_{i+1}) and x′_{i−1} from z′.

7.2 Counterexamples for buildings associated with P Sp(4) and Spin(5)

In this section we construct an example of a discrete thick Euclidean building X (modelled on discrete Euclidean Coxeter complex with the root system R of type C_{2}) and a triangle P ⊂ X with the Δ-lengths τ_{1}, τ_{2}, τ_{3} ∈ L = P (R^{∨}) so that:

1. The vertices of P are at vertices of X.

2. There is no triangle P′ ⊂ X with vertices at special vertices of X and the Δ-lengths τ_{1}, τ_{2}, τ_{3}.

In other words, we show that the map (11) in general is not surjective (where we are taking L = P (R^{∨})). We then give another example, when R is the root system of type B_{2} and L = Q(R^{∨}). Algebraically, these examples will show that certain triples of vectors α, β, γ in Δ ∩ L do not yield solutions of the Problem Q3 for the groups P Sp(4) and Spin(5), despite the fact that there is a triangle in X with the Δ-side lengths α, β, γ. We then explain how to generalize one of these examples to the root systems C_{ℓ}, ℓ ≥ 2 and L = Q(R^{∨}). (Algebraically, this corresponds to the simply-connected groups Sp(2ℓ), ℓ ≥ 2.)

Example 7.8. Let X be a thick Euclidean building with associated discrete Euclidean Coxeter complex (E, W_{aff}) of type C_{2}. Then the map (11) is not surjective for n = 3 and L = P (R^{∨}).

Proof: Recall that the Coxeter complex (E, W_{aff}) is two-dimensional, and the Weyl alcove is an isosceles right-angled triangle. The vertex stabilizers are isomorphic to either D_{4} or ℤ_{2} × ℤ_{2}. The vertices of the former type are special. There are two W_{aff}-orbits of special vertices and one orbit of non-special ones.

Consider the billiard triangle T shown in Figure 3. It has geodesic sides \overline{zx}, \overline{zy} and the broken side \overline{xuy}. Its Δ-side lengths are L_{trans}-integral and it has non-special vertices.

Figure 3: A billiard triangle.

We first check that this triangle can be unfolded to a geodesic triangle Δ(z, x, y′) in X in the sense of section 7.1. This can be done as follows. We note that the geodesic l (weakly) separates z from x and y. Hence we can unfold the billiard triangle T to a geodesic triangle in X using Lemma 7.6.

Now we prove that there exists no geodesic triangle Δ(x′′, y′′, z′′) in X with special vertices and the same Δ-side lengths as Δ(x, y′, z). Suppose that such a triangle exists. As described in section 7.1, we subdivide the side \overline{x′′y′} and fold Δ(x′′, y′′, z′′) to a billiard triangle in an apartment a. Notice that the side \overline{x′′y′′} can be folded only at its midpoint u′′. Since u′′ is a non-special vertex, y′ is folded onto x′′, or the folded triangle remains geodesic. The former case is impossible because Δ(x′′, y′′, z′′) is not isosceles. The latter case is ruled out by inspection: We leave it to the reader to verify that there are no geodesic triangles contained in an apartment which have the same Δ-side lengths as Δ(x′′, y′′, z′′).

Recall that there are coordinates (x, y) on the Cartan subalgebra of Sp(4) such that the simple roots for the root system C_{2} are (1, −1) and (0, 2). In the dual coordinates, the simple coroots are (1, −1) and (0, 1) and the coweight lattice is generated by the vectors (1, 0) and (\frac{1}{2}, \frac{1}{2}). In these coordinates, the Δ-lengths of the segments \overline{zx}, \overline{xy} and \overline{yz} are α = (1, 1), β = (1, 0) and γ = (\frac{3}{2}, \frac{1}{2}) respectively. Thus the first part (unfolding) of the previous argument could be replaced by checking that the side-length vectors α, β, γ satisfy the inequalities of Section 6.3. We note that the first two of the above vectors are in the coroot lattice of Sp(4), but the vector γ = (\frac{3}{2}, \frac{1}{2}) is not. The vector γ belongs to the coweight lattice of Sp(4). Hence, from the group-theoretic viewpoint, this is an example for the adjoint group P Sp(4), but not for Sp(4). Observe that the sum α+β +γ = (1, 1)+(1, 0)+(\frac{3}{2}, \frac{1}{2}) does not belong to Q(R^{∨}); therefore the second part of the argument can be replaced by appealing to Part 4 of Theorem 8.2.

We now give an example of a triangle where all three side-length vectors are in the coroot lattice of Spin(5), but there is no triangle with special vertices. This example is a variation on the previous one.

Example 7.9. Let X be a thick Euclidean building with associated discrete Euclidean Coxeter complex (E, W_{aff}) of type B_{2}. Then the map (11) is not surjective for n = 3 and L = Q(R^{∨}).

We recall that the simple roots for the root system B_{2} are (1, −1) and (0, 1) and the simple coroots are (1, −1) and (0, 2). The folded triangle T is represented in Figure 4. It has three vertices: s, p and q and two geodesic sides: \overline{sp} and \overline{sq}. The third side is the broken geodesic segment which consists of three pieces: \overline{pv}, \overline{vu} and \overline{uq}. The vectors in the Weyl chamber Δ = \{(x, y) ∈ ℝ^{2} : 0 ≤ y ≤ x\} which represent the corresponding Δ-side lengths are (2, 2), (3, 1) and (2, 2), where (2, 2) represents the broken side. To see that T can be unfolded into a geodesic triangle in the corresponding building X one can either use Lemma 7.6 or simply verify that the vectors (2, 2), (3, 1) and (2, 2) satisfy the inequalities of Section 6.3. Note that the vectors (3, 1) and (2, 2) belong to the coroot lattice.

We now prove that there is no triangle Δ(s′′, p′′, q′′) in the building X whose Δ-side lengths are the vectors (2, 2), (3, 1) and (2, 2) and whose vertices are special vertices of X. Suppose that such triangle exists. As described in section 7.1, we subdivide the side \overline{p′′q′′} and fold Δ(s′′, p′′, q′′) to a billiard triangle T′ in the model apartment A. Without loss of generality we can assume that the side \overline{s′′p′′} folds onto the geodesic segment \overline{oc}, where o is the origin in A and c = (2, 2). Set a = (3, 1), a′ := (1, 3). If \overline{s′′q′′} folds onto any geodesic \overline{or} in the W_{sph}-orbit of \overline{oa} other than \overline{oa} and \overline{oa′}, the distance d(r, c) will be strictly larger than 2\sqrt{2}, which is a contradiction. By symmetry in the line x = y, it suffices to consider the case when \overline{s′′q′′} folds (isometrically) onto the geodesic segment \overline{oa}. Note that d(a, c) is strictly less than 2\sqrt{2}, which is the magnitude of the vector (2, 2). Thus the broken geodesic segment f(\overline{p′′q′′}) (which is the image of \overline{p′′q′′} under folding) has to have at least one break point. Next, observe that the only break points in f(\overline{p′′q′′}) can occur at the vertices of the affine Coxeter complex. Any vertex other than one in the set

S := {(2, 2), (3, 1), (2.5, 0.5), (1.5, 1.5), (1.5, 2.5),
(2.5, 1.5), (2.5, 2.5), (3.5, 0.5), (3.5, 1.5)}

Figure 4: A folded triangle.

would be too far from a, c for a billiard triangle to exist. We note that all points in S \ {a, c} are nonspecial vertices of the Coxeter complex. Thus the only break in f(\overline{p′′q′′}) which can occur at such a point is “backtracking”.

Below we exclude breaks at various points of S and will leave the rest of the possibilities to the reader, since the arguments are similar. Suppose that the broken geodesic path f(\overline{p′′q′′}) is the concatenation of the geodesic segments

\overline{ca}, \overline{ab}, \overline{ba},

where b = (3.5, 1.5), see Figure 5. The point b is one of two break points of f(\overline{p′′q′′}). At the link Σ_{b}(A) consider the directions:

ξ′, η′, ζ′

towards the points a, a and o. Then

∠(ζ′, ξ′) + ∠(ζ′, η′) = 2∠(ζ′, ξ′) < π.

Hence, according to Remark 7.5, the billiard triangle T′ cannot be unfolded into a geodesic triangle Δ(s′′, p′′, q′′) ⊂ X. Contradiction.

Suppose now that the broken geodesic path f(\overline{p′′q′′}) is the concatenation of the geodesic segments \overline{cb}, \overline{ba}, where b = (1.5, 2.5), see Figure 6. The point b is the only break point of f(\overline{p′′q′′}). At the link Σ_{b}(A) consider the directions:

ξ′, η′, ζ′

towards the points c, a and o. Then

∠(ζ′, ξ′) + ∠(ζ′, η′) = 2∠(ζ′, ξ′) < π.

We again reach a contradiction using the Remark 7.5.

Lastly, consider a break point at the vertex b = (2.5, 2.5); the broken geodesic path f(\overline{p′′q′′}) is the concatenation of the geodesic segments

\overline{ca}, \overline{ab}, \overline{ba},

see Figure 6. At the link Σ_{a}(A) of the break point a consider the directions:

ξ′, η′, ζ′

towards the points c, b and o. Again,

∠(ζ′, ξ′) + ∠(ζ′, η′) = 2∠(ζ′, ξ′) < π,

contradiction. (Note that in this example we do not get a contradiction by considering the link of the break point b.)

Remark 7.10. Consider the usual embedding R_{2} → R_{ℓ} of the root system R_{2} = B_{2} to R_{ℓ} = B_{ℓ}. The Δ-side length vectors (2, 2), (3, 1) and (2, 2) correspond to the vectors

α = β = (2, 2, 0, ..., 0), γ = (3, 1, 0, ..., 0) ∈ L = Q(R_{ℓ}^{∨}).

One can show that for each ℓ ≥ 3 and the appropriate discrete building X, the vectors α, β, γ belong to the image of ι : D_{3}^{ref,L}(X) → D_{3}(X).

To get examples of triples of vectors which do not belong to the image of D_{3}^{ref,L}(X) ↪^{ι} D_{3}(X) for buildings X modelled on discrete Euclidean Coxeter complexes of type C_{ℓ}, ℓ ≥ 3, we do the following. We note that under the isomorpism between the C_{2} and B_{2} root systems given by the matrix

[ \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ]

the vectors α, β, γ (from the previous example) correspond to the vectors α′ = β′ = (2, 0) and γ′ = (2, 1) in the coroot lattice of C_{2}. Consider the natural embedding of root systems C_{2} ↪ C_{ℓ} = R_{ℓ}, ℓ ≥ 2. One can verify (similarly to the arguments presented above) that the vectors α′ = β′, γ′ ∈ L = Q(R_{ℓ}^{∨}) satisfy the property that

(α′, β′, γ′) ∈ D_{3}^{L}(X) \ ι(D_{3}^{ref,L}(X)),

where X is a discrete Euclidean building modelled on the Euclidean Coxeter complex associated with the root system C_{ℓ}.

8 Existence of fixed vertices in buildings and computation of the saturation factors for reductive groups

As we have seen in the previous chapter, the map

ι : D_{n}^{ref,L}(X) → D_{n}(X) ∩ L^{n}.  (12)

in general is not surjective. The goal of this chapter is to find conditions on the root systems, etc., which would guarantee existence of polygons P in Euclidean buildings X with the prescribed L-integral Δ-side lengths and vertices at the vertices of X. Moreover, we will find conditions under which the vertices of P are in G · o, where o ∈ X is a certain special vertex. We will also see that the image of ι is always contained in the set

D_{n}^{L,0}(X) = {(τ_{1}, ..., τ_{n}) ∈ D_{n}(X) ∩ L^{n} : ∑_{i=1}^{n} τ_{i} ∈ Q(R^{∨}) = L_{trans}}.

We will show that for each data (W_{aff}, L) (where L_{trans} ⊂ L ⊂ N_{aff}), there exists a natural number k such that for each τ ∈ D_{n}(X) ∩ L^{n}, the vector kτ belongs to the image of the map (11). We will compute the saturation factors k for various classes of (W_{aff}, L). In few cases we are fortunate and k = 1, i.e. the map (12) is onto. We then apply these results to the algebra Problem Q3.

8.1 Existence of fixed vertices

We begin this section with few simple remarks about existence of polygons in X with the vertices of a given type. As before, G ⊂ Aut(X) is such that the G-stabilizer of the model apartment A acts on A through the group W̃, whose translational part is L = L\underline{G}. Thus, if \overline{xy} ⊂ X is a geodesic segment with x ∈ G · o, then y ∈ G · o iff σ(x, y) ∈ L. Therefore the following are equivalent for a polygon P ⊂ X, whose Δ-lengths are in L = L · o:

1. All vertices of P are in G · o.

2. One (say, the first) of the vertices of P is in G · o.

We also have

Lemma 8.1. For a vector τ^{→}∈ D_{n}(X) ∩ L^{n} the following are equivalent:

1. There exists a polygon P ⊂ X with the Δ-lengths τ^{→}, such that all (equivalently, one of) the vertices of P are in G · o.

2. There exists a polygon P ⊂ X with the Δ-lengths τ^{→}, such that all (equivalently, one of) the vertices of P are special.

Proof: It is clear that (1)⇒(2), let’s prove the converse. Let P = x_{1} ⋯ x_{n} be a polygon in X with the Δ-lengths τ^{→}, such that the first vertex x_{1} of P is special. Let y_{1}, ..., y_{n+1} ∈ E be such that \overline{y_{i}y_{i+1}} ⊂ E have the same refined lengths as the segments \overline{x_{i}x_{i+1}}, i = 1, ..., n. By assumption, the vertex y_{1} is special. Recall that the normalizer N_{aff} of W_{aff} in the group of translations of E, acts transitively on the set of special vertices. We identify the model apartment E with an apartment in X containing the origin o. Let T ∈ N_{aff} be such that T (y_{1}) = o. Set z_{i} := T (y_{i}), i = 1, ..., n. Then z_{i} are in L-orbit of o for each i and according to the Transfer Theorem 1.3, there exists a geodesic polygon Q = u_{1} ⋯ u_{n} ⊂ X with the refined side-lengths same as for \overline{z_{i}z_{i+1}}, i = 1, ..., n. In particular, σ(u_{i}, u_{i+1}) = τ_{i} for each i and all vertices of Q are in G · o.

Thus a vector τ^{→} belongs to the image of the map (12) iff there exists a polygon P ⊂ X with special vertices and the Δ-side lengths τ^{→}.

As we have seeing in chapter 6, the existence problem for polygons with the given Δ-side lengths in X is equivalent to the existence of a fixed point for a certain map Φ_{ψ} : X → X. In this section we will try to find conditions under which Φ_{ψ} fixes a special vertex in the building X. We first analyze the case when the Euclidean Coxeter complex is irreducible.

We therefore assume that the affine Coxeter group W_{aff} corresponds to a reduced irreducible root system R of rank ℓ in an ℓ-dimensional vector space V, with the underying affine space E, see chapter 3. Recall that L is a lattice such that

L_{trans} = Q(R^{∨}) ⊂ L ⊂ N_{aff} = P (R^{∨}).  (13)

As before, W̃ denotes the subgroup of Isom(E) generated by elements of W_{aff} and L. Then W̃ is a Euclidean Coxeter group with the linear part W_{sph} and translation part L. Since W̃ normalizes W_{aff}, we get the induced action of the finite abelian group W̃ /W_{aff} ≅ L/L_{trans} = F on the Weyl alcove α of W_{aff}. The notion of L-admissible faces of Weyl alcoves of (E, W_{aff}) introduced in chapter 3, carries over to the building X. Note that in the case L = L_{trans}, the only L-admissible faces are the vertices of X.

Suppose that τ = (τ_{1}, ..., τ_{n}), where τ_{i} ∈ L for each i. Let

ψ = ((m_{1}, ξ_{1}), ..., (m_{n}, ξ_{n}))

be a weighted configuration on ∂_{T its}X of the type τ, where m_{i} = τ_{i}/|τ_{i}|. Consider the map Φ = Φ_{ψ} : X → X. Then Φ = Φ_{n} ∘ ... ∘ Φ_{1} : X → X where each Φ_{i} acts on every apartment a_{i} asymptotic to ξ_{i} by the translation T_{i} (by the vector which has the same Δ-length as τ_{i}). Thus the assumption τ_{i} ∈ L ⊂ N_{aff} implies that each T_{i} is an automorphism of the Euclidean Coxeter complex (which we identify with a_{i}). In particular, the map Φ preserves the simplicial structure of X.

Theorem 8.2. Assume in addition that Φ has a fixed point in X. Then: 1. Φ fixes the barycenter of an L-admissible face in X.

2. If ∑_{i=1}^{n} τ_{i} ∈ L_{trans} then Φ fixes a vertex of X.

3. More generally, if L̂ is a lattice normalized by W_{sph}, such that L_{trans} ⊂ L̂ ⊂ L, and ∑_{i=1}^{n} τ_{i} ∈ L̂, then Φ fixes the barycenter of an L̂-admissible face in X.

4. If Φ fixes a special vertex of X then ∑_{i=1}^{n} τ_{i} ∈ L_{trans}. In particular, if the Coxeter complex associated with X is the irreducible Coxeter complex (E, W_{aff}) of type A_{ℓ}, we have:

∑_{i=1}^{n} τ_{i} ∈ L_{trans} ⟺ Φ fixes a vertex of X.

Proof: 1. For a fixed point x = x_{1} of Φ, let δ denote the smallest face (a simplex) in X containing x. Consider the apartments a_{1} ⊂ X (containing x_{1}, x_{2}, ξ_{1}), a_{2} ⊂ X (containing x_{2}, x_{3}, ξ_{2}),..., a_{n} ⊂ X (containing x_{n}, x_{1}, ξ_{n}). Then Φ_{1}(δ) ⊂ a_{1} ∩ a_{2}, Φ_{2} ∘ Φ_{1}(δ) ⊂ a_{2} ∩ a_{3},..., Φ_{n} ∘ ... ∘ Φ_{1}(δ) ⊂ a_{n} ∩ a_{1}. Since Φ preserves the simplicial structure of X and δ is the smallest simplex containing x, we get: Φ(δ) = δ.

The map Φ|_{δ} : δ → δ is the composition of maps

(T_{n} ∘ ... ∘ T_{2} ∘ T_{1})|_{δ}.

Let φ_{1} : E → a_{1} be an (isometric) parameterization as in the definition of a space modelled on a Coxeter complex. We can assume that δ′ := φ_{1}^{−1}(δ) is a face of the Weyl alcove α ⊂ E. There exists a parameterization φ_{2} : E → a_{2} so that φ_{2}^{−1}∘φ_{1} = id on the domain of this composition. Similarly (like in the definition of a developing map of a geometric structure) we choose parameterizations φ_{i}, i = 1, ..., n, so that for each i, φ_{i+1}^{−1} ∘ φ_{i} = id on the domain of this composition. Then we get:

(T_{n} ∘ ... ∘ T_{2} ∘ T_{1})|_{δ} = (T_{n} ∘ φ_{n} ∘ φ_{n−1}^{−1} ∘ T_{n−1} ∘ ... ∘ φ_{2} ∘ T_{2} ∘ φ_{2} ∘ φ_{1}^{−1} ∘ T_{1})|_{δ}.

Let T′_{i} := φ_{i}^{−1} ∘ T_{i} ∘ φ_{i} : E → E; these maps are translations by the vectors τ′_{i}, which are in the W_{aff}-orbits of the vectors τ_{i}, i = 1, ..., n. Set T′ := T′_{n} ∘ ... ∘ T′_{1}. Then the “holonomy map”

T′ = φ_{n}^{−1} ∘ T_{n} ∘ φ_{n} ∘ φ_{n−1}^{−1} ∘ ... ∘ φ_{2}^{−1} ∘ T_{2} ∘ φ_{2} ∘ φ_{1}^{−1} ∘ T_{1} ∘ φ_{1} : E → E

sends the simplex δ′ to itself. Moreover,

T′|_{δ′} = φ_{n}^{−1} ∘ Φ ∘ φ_{1}|_{δ′} = φ_{n}^{−1} ∘ φ_{1} ∘ φ_{1}^{−1} ∘ Φ ∘ φ_{1}|_{δ′} = w ∘ φ_{1}^{−1} ∘ Φ ∘ φ_{1}|_{δ′},

where w ∈ W_{aff}. Thus, by letting Φ′|_{δ′} := φ_{1}^{−1} ∘ Φ ∘ φ_{1}|_{δ′}, we conclude that Φ′|_{δ′} admits the extension by w^{−1}T′ to the entire E. Since w^{−1}T′ ∈ W̃, Φ induces an automorphism of the simplex δ by an element of the group W̃. We note that Φ does not necessarily permute all the vertices of δ. Consider however an orbit of ⟨Φ⟩ on the vertex set of δ. It is clear that this orbit spans an L-admissible simplex c in δ ⊂ X; since Φ|_{δ} is a 1-Lipschitz automorphism, it preserves the barycenter of the simplex c and hence the first assertion of the theorem follows.

2. Let us prove the second assertion. Set t = ∑_{i=1}^{n} τ_{i}. Note that the map T′_{i} is the translation in E by the vector τ′_{i} where τ′_{i} = w_{i}(τ_{i}) for some w_{i} ∈ W_{sph}. The composition T′ = T′_{n} ∘ ... ∘ T′_{1} is the translation by the vector

v = ∑_{i=1}^{n} τ′_{i} = ∑_{i=1}^{n} w_{i}(τ_{i}).

We claim that v ∈ L_{trans} iff t ∈ L_{trans}. Equivalently, (in the multiplicative form, where we identify each τ_{i} with the corresponding translation in E) we claim that

v = w_{n}τ_{n}w_{n}^{−1}...w_{1}τ_{1}w_{1}^{−1} ∈ tW_{aff}.

Since the τ_{i}’s normalize the group W_{aff} we have:

v = w_{n}τ_{n}w_{n}^{−1}w_{n−1}τ_{n−1}w_{n−1}^{−1}...w_{1}τ_{1}w_{1}^{−1} = τ_{n}w′_{n}τ_{n−1}w_{n−1}^{−1}...w_{1}τ_{1}w_{1}^{−1} =
= τ_{n}τ_{n−1}w′_{n−1}τ_{n−2}w_{n−1}^{−1}...w_{1}τ_{1}w_{1}^{−1} = ... = τ_{n}...τ_{1}w′_{1} = tw′_{1},

where w′_{i} ∈ W_{aff} for each i. This proves the claim.

Recall that Φ′|_{δ′} admits the extension by w^{−1}T′ (where w ∈ W_{aff}) to the entire model apartment E. Thus, if t ∈ L_{trans} then Φ′|_{δ′} is the restriction of an element g ∈ W_{aff}. Since the alcove α is a fundamental domain for the action of W_{aff} on E and δ′ is a face of a, we conclude that Φ′|_{δ′} = id. Hence Φ fixes a vertex of X.

3. The proof of 3 is analogous to the proof of 2 and we leave it to the reader.

4. Lastly, suppose that Φ fixes a special vertex x of X, then δ = {x}, δ′ = {x′} is a special vertex in E, and Φ′(x′) = x′ implies that T′(x′) = w(x′), where w ∈ W_{aff}. Since x′ is a special vertex, w = T ∘ w′, where w′ ∈ W_{aff} fixes x′ and T ∈ L_{trans}. Thus T′(x′) = T (x′), which implies that T′ ∈ L_{trans}, and hence t ∈ L_{trans}. As an immediate corollary of the above theorem we get:

Corollary 8.3. 1. For each τ^{→}= (τ_{1}, ..., τ_{n}) ∈ D_{n}(X) ∩ L^{n} such that ∑_{i} τ_{i} ∈ L_{trans}, there exists a polygon P ⊂ X such that σ(P) =τ^{→} and the vertices of P are at the vertices of X.

2. If there exists a polygon P ⊂ X with special vertices and σ(P) =τ^{→}, then ∑_{i} τ_{i} ∈ L_{trans}.

Example 8.4. For the irreducible Coxeter complex (E, W_{aff}) of the type C_{2}, there are vectors τ_{1}, τ_{2}, τ_{3} ∈ L = P (R^{∨}) such that Φ fixes a vertex of X but ∑_{i=1}^{3} τ_{i} ∉ Q(R^{∨}) = L_{trans}.

Proof: Consider the billiard triangle T and the geodesic triangle Δ(x, y′, z) ⊂ X as in Example 7.8. Then the vectors τ_{1}, τ_{2} representing the sides \overline{xz} and \overline{xy′} of Δ(x, y′, z) are in the coroot lattice Q(R^{∨}), but the vector τ_{3} representing the side \overline{zy} is not in the coroot lattice. Hence ∑_{i=1}^{3} τ_{i} ∉ Q(R^{∨}).

Thus the fixed point of Φ may not be special. However, if N_{aff} acts transitively on the vertices of E, every vertex is special and we get:

Corollary 8.5. Suppose that the normalizer N_{aff} of W_{aff} in V acts transitively on the vertices of E. Then the image of D_{n}^{ref,L}(X) in D_{n}(X) equals

D_{n}^{L,0}(X) := {(τ_{1}, ..., τ_{n}) ∈ D_{n}(X) ∩ L^{n}, ∑_{i=1}^{n} τ_{i} ∈ L_{trans}}.

The example of a Coxeter complex for which the hypothesis of Corollary 8.5 holds is given by (E, W_{aff}) for which the finite Weyl group W_{sph} is of type A_{ℓ}. Then each vertex of the Coxeter complex (E, W_{aff}) is special.

Next we consider the general case when the Coxeter complex (E, W_{aff}) is reducible. As in section 5.2, we have the corresponding deRham decomposition of the building X:

X = X_{0} × X_{1} × ... × X_{s},

where X_{0} is the flat deRham factor, the group W_{aff}^{i} (i > 0) acts as an irreducible affine Coxeter group on the Euclidean space E_{i}. The Weyl chamber Δ is the product ∏_{i=0}^{s} Δ_{i} of Weyl chambers, where of course Δ_{0} = E_{0}. Let p_{i} denote the orthogonal projection X → X_{i}, i = 0, 1, ..., s; by abusing notation we will also use p_{i} to denote the orthogonal projections E → E_{i}. Given a lattice L′ in E we get the inclusions

L_{trans}^{i} ⊂ p_{i}(L) = L_{i} ⊂ N_{aff}^{i}, i = 1, ..., s.

Note that

D_{n}(X_{0}) = {(τ_{1}, ..., τ_{n}) : ∑_{i=1}^{n} τ_{i} = 0}.

We have

Proposition 8.6. 1. D_{n}(X) = ∏_{i=0}^{s} D_{n}(X_{i}).

2. For each τ^{→}∈ Δ^{n}, there is a polygon P ⊂ X with the Δ-side lengths τ^{→} and the vertices at the (special) vertices of X iff for each i there exists a polygon P_{i} ⊂ X_{i} with the Δ_{i}-side lengths p_{i}(τ^{→}) and the vertices at the (special) vertices of X_{i}.

3. D_{n}^{ref,L′}(X) = ∏_{i=0}^{s} D_{n}^{ref,L′_{i}}(X_{i}).

Proof: We will prove the third assertion since the proofs of (1) and (2) are similar. First of all, if P ⊂ X is a polygon with L′-integral refined side-lengths then its projections P_{i} := p_{i}(P) ⊂ X_{i} are also polygons with L_{i}-integral refined side-lengths. Conversely, if P_{i} ⊂ X_{i} are polygons with L′_{i}-integral refined side-lengths and x_{ij} is the j-th vertex of the polygon P_{i}, we set

x_{j} := (x_{0j}, ..., x_{sj}) ∈ X,

this point is a vertex of X since all x_{ij}’s are vertices of the corresponding buildings X_{i}. This defines the polygon P = x_{1} ⋯ x_{n} in X. It is clear that the refined side-lengths of P are L′-integral.

The above proposition implies that Corollaries 8.3 and 8.5 remain valid for Euclidean buildings with reducible Coxeter complexes.

As in chapter 3, given a lattice L′ in E we define the lattice L as the intersection of L′ and the translation group of E_{1} ⊕ ... ⊕ E_{s}. We now can compute the image of the map (12) “up to saturation”:

Theorem 8.7. 1. Let k = k(W_{aff}, L′) = k(W_{aff}, L) be the saturation factor defined in chapter 3. Then for each τ^{→}∈ D_{n}(X) ∩ (kL′)^{n}, there exists an n-gon P ⊂ X with the special vertices and the Δ-side lengths τ_{1}, ..., τ_{n}.

2. Let R be the root system corresponding to the Coxeter complex (E, W_{aff}) and let k := k_{R} = k(W_{aff}, L_{trans}) be the saturation factor defined in chapter 3. Then for each τ^{→}∈ D_{n}(X) ∩ (kL′)^{n} satisfying

∑_{i} τ_{i} ∈ L_{trans} = Q(R^{∨})

there exists an n-gon P ⊂ X with the special vertices and the Δ-side lengths τ_{1}, ..., τ_{n}.

Proof: 1. We first consider the case when the root system R is reduced, irreducible and has rank equal the dimension of the space V; then L = L′. Let P′ ⊂ X be a polygon with the Δ-lengths \frac{1}{k}τ_{1}, ..., \frac{1}{k}τ_{n}. Since \frac{1}{k}τ_{i} ∈ L for each i, Theorem 8.2 implies that the polygon P′ can be chosen so that its first vertex is the barycenter of an L-admissible simplex in X. Let \overline{x_{i}x_{i+1}}, i = 1, 2, ..., n, be oriented segments in the model apartment E, which represent the refined side-lengths of the polygon P′; the point x_{1} (and hence x_{i} for each i = 2, ..., n+1) is the barycenter of an L-admissible simplex in E. We regard E as the vector space V by identifying the origin with the special vertex o. Then, according to the definition of k = k(W_{aff}, L), the segments \overline{(kx_{i})(kx_{i+1})} = \overline{y_{i}y_{i+1}} have end-points at the special vertices of E. Consider the affine transformation ι : E → E which is the homothety x ↦ kx. Then ι : (E, W_{aff}) → (E, W_{aff}) is an injective endomorphism of the affine Coxeter systems and the Transfer Theorem 1.3 implies that there exists a polygon P in X whose refined side-lengths are represented by \overline{y_{i}y_{i+1}}, i = 1, ..., n. The vertices of P are special and its Δ-side-lengths are τ_{1}, ..., τ_{n}.

We now consider the general case when the Coxeter complex (E, W_{aff}) is reducible. Let P′ ⊂ X be a polygon with the Δ-lengths \frac{1}{k}τ_{1}, ..., \frac{1}{k}τ_{n}. Since p_{0}(∑_{i=1}^{n} τ_{i}) = 0, we have

\frac{1}{k} ∑_{i=1}^{n} τ_{i} ∈ L′ ∩ V_{1} ⊕ ... ⊕ V_{s} = L.

Hence for each j = 1, ..., s we get

\frac{1}{k} ∑_{i=1}^{n} p_{j}(τ_{i}) ∈ L_{j} = p_{j}(L).

Therefore, by Theorem 8.2 (part 3, where we take L̂ = L_{j}, j = 1, ..., s), the polygon P′ can be chosen so that for each j = 1, ..., s, the vertices of p_{j}(P′) are barycenters of L_{j}-admissible faces of X_{j}. Recall that k = LCM(k_{1}, ..., k_{s}).

By the irreducible case, for each j = 1, ..., s there is a polygon P′_{j} ⊂ X_{j} with special vertices and side-lengths \frac{k_{j}}{k} p_{j}(τ_{1}), ..., p_{j}(τ_{n}). Let \overline{x_{i,j}x_{i+1,j}}, i = 1, 2, ..., n, j = 1, ..., s, be oriented segments in the model apartment E_{j}, which represent the refined sidelengths of the polygon P′_{j}. It follows from the Transfer Theorem 1.3 that for each j = 1, ..., s there exists a polygon P_{j} ⊂ X_{j} whose refined side-lengths are represented by the segments \overline{(m_{j}x_{i,j})(m_{j}x_{i+1,j})}, where m_{j} = k/k_{j}. Note that m_{j} \frac{k_{j}}{k} p_{j}(τ_{i}) = p_{j}(τ_{i}) for each i and j. By the definition of k_{j}, for each j the vertices of P_{j} are special vertices of X_{j}. Now Proposition 8.6 implies that there exists a polygon P ⊂ X with the special vertices, whose projections to X_{j} are P_{j} for each j and the Δ-side lengths of P are τ_{1}, ..., τ_{n}.

2. The proof of this assertion is analogous to the proof of 1. First, suppose that the Coxeter complex is irreducible. Then, according to Theorem 8.2 (part 2), a polygon P′ ⊂ X with the Δ-lengths \frac{1}{k}τ_{1}, ..., \frac{1}{k}τ_{n} can be chosen so that its vertices are vertices of X. Let \overline{x_{i}x_{i+1}}, i = 1, 2, ..., n, be oriented segments in the model apartment E, which represent the refined side-lengths of the polygon P′. Then, by the definition of k = k_{R}, the end-points of the segments \overline{(kx_{i})(kx_{i+1})} are special vertices of X. The rest of the argument is the same as for (1).

8.2 Saturation factors for reductive groups

We now apply the results from the previous section to the generalized Invariant Factor Problem for nonarchimedean reductive Lie groups G. Suppose that \underline{G} is a connected reductive algebraic Lie group over 𝕂, where 𝕂 is a field with discrete valuation v. Let G := \underline{G}(𝕂). The group \underline{G} determines a Bruhat-Tits building X, Bruhat-Tits root system R, the corresponding affine Coxeter group W_{aff} and the extended cocharacter lattice L := L\underline{G} satisfying the double inclusion (13). Define k_{inv fact}(G) := k(W_{aff}, L).

As an immediate corollary of Theorem 8.7, we obtain:

Corollary 8.8. 1. For k = k_{inv fact}(G) and any τ ∈ L^{n} ∩ D_{n}(X), there exists an n-gon P in X with Δ-side-lengths equal to kτ and vertices in G · o.

2. For k = k_{R} and any

τ^{→}∈ D_{n}^{L,0}(X) = {τ^{→}∈ D_{n}(X) ∩ L^{n} : ∑_{i=1}^{n} τ_{i} ∈ Q(R^{∨})},

there exist an n-gon P with Δ-side lengths equal to kτ and vertices in G · o.

3. If \underline{G} is semisimple and simply-connected then D_{n}^{L,0}(X) = D_{n}^{L}(X) and k_{inv fact}(G) = k_{R}.

Proof: Parts (1) and (2) follow from Theorem 8.7 (parts (1) and (2) respectively) and Lemma 8.1.

To prove (3) note that in this case Q(R^{∨}) = L and thus

k_{inv fact}(G) = k_{R} = k(W_{aff}, Q(R^{∨})).

Remark 8.9. We emphasize that k_{inv fact}(G) and k_{R} depend only on G and not the choice of τ.

By specializing to the case n = 3 and using the equivalence of Problem RGTI for Euclidean buildings with the Problem Q3, we get:

Corollary 8.10. For each k ∈ ℕ divisible by k_{inv fact}(G) and each triple (α, β, γ) ∈ k(L ∩ Δ)^{3} which satisfies the generalized triangle inequalities for the space X, there exist representatives A, B, C of the double coset classes

\underline{G}(O)α(π)\underline{G}(O), \underline{G}(O)β(π)\underline{G}(O), \underline{G}(O)γ(π)\underline{G}(O),

such that ABC = 1.

Example 8.11. 1. Let \underline{G} = P Sp(4). Then the Example 7.8 shows that there is a triple (α, β, γ) ∈ (L ∩ Δ)^{3}, L = P (R^{∨}), such that one cannot find A, B, C in the corresponding double cosets of P Sp(4, 𝕂) such that ABC = 1.

2. Let \underline{G} = Spin(5). Then the Example 7.9 shows that there is a triple (α, β, γ) ∈ (L ∩ Δ)^{3}, L = Q(R^{∨}), such that one cannot find A, B, C in the corresponding double cosets of Spin(5, 𝕂) such that ABC = 1.

Thus the saturation constant k = 2 for the groups P Sp(2) and Spin(5) is optimal in the sense that in Corollary 8.10 for \underline{G} = P Sp(4), Spin(5) one cannot take k = 1.

Example 8.12. k_{inv fact}(GL(m)) = 1. In other words, let X be the Bruhat-Tits building associated with the group GL(m, 𝕂), where 𝕂 is a field with nonarchimedean discrete valuation. Let τ^{→}∈ ℤ^{m} have entries arranged in decreasing order. Then τ^{→} satisfies the unrefined generalized triangle inequalities (i.e. τ^{→}∈ D_{n}(X) = D_{n}(ℙ_{m})) if and only if there exists an n-gon P in X with the first vertex at the origin (the point stabilized by GL(m, O)) and Δ-side lengths τ^{→}.

Although this example is a special case of Theorem 8.7 (part 2), we present a complete proof for the benefit of the reader.

Proof: Recall (see Example 3.6) that the Euclidean Coxeter complex (E, W_{aff}) of X is reducible, E = E_{0} ⊕ E_{1}, where E_{1} is the kernel of the map

tr : (x_{1}, ..., x_{m}) ↦ x_{1} + ... + x_{m},

and E_{0} is the span of (1, ..., 1). Let V_{1} be the vector space underlying E_{1}. We may identify V_{1} with the Lie algebra of traceless real diagonal matrices, the real Cartan subalgebra of the Lie algebra of SL(m). Moreover (see Example 3.6), L = L_{GL(m)} = ℤ^{m}, is the cocharacter lattice; the orthogonal projection of L to V_{1} is the coweight lattice of SL(m), and the root system is of type A_{m−1}. On the other hand, suppose that τ^{→}= (τ_{1}, ..., τ_{n}). Then, since τ^{→}∈ D_{n}(X), there exists a polygon P̂ in X with the Δ-side lengths τ_{i}. Let X = X_{0} × X_{1} be the deRham decomposition of X, where X_{0} = E_{0} ≅ ℝ is the flat factor. Since the projection of P̂ to the flat deRham factor E_{0} of X is also a polygon, we get:

p_{0}(τ_{1} + ... + τ_{n}) = tr(τ_{1} + ... + τ_{n}) = 0.

Hence τ_{1} + ... + τ_{n} ∈ L ∩ E_{1} = Q(R^{∨}). Therefore we can apply Theorem 8.2 (part 2 or 4) to conclude that there exists a polygon P′ ⊂ X_{1} whose vertices are (special) vertices of X_{1} and whose Δ-side lengths are p_{1}(τ_{1}), ..., p_{1}(τ_{n}). By combining this with the polygon p_{0}(P̂) via Proposition 8.6 we conclude that there exists a polygon P ⊂ X whose vertices are (special) vertices of X and whose Δ-side lengths are τ_{1}, ..., τ_{n}. Thus k_{inv fact}(GL(m)) = 1.

Remark 8.13. The above computation extends to other split reductive algebraic groups \underline{G}. Let \underline{DG} be the derived (i.e. commutator) subgroup of \underline{G}. Then

k_{inv fact}(\underline{G}) = k_{inv fact}(\underline{DG}).  (14)

Moreover, if \underline{G_{1}}, ..., \underline{G_{s}} are the simple factors of \underline{DG} then

k_{inv fact}(\underline{G}) = LCM(k_{inv fact}(\underline{G_{1}}), ..., k_{inv fact}(\underline{G_{s}})).

The equation (14) follows from three observations:

We first claim that the lattice L\underline{G} ∩ V_{1} ⊕ ... ⊕ V_{s} equals L\underline{DG}. This is because we can assume that the maximal split tori \underline{T} \underline{DG} and \underline{T} \underline{G} of \underline{DG} and \underline{G} are related by

T\underline{DG} = T\underline{G} ∩ \underline{DG}.

Hence the cocharacter lattice of \underline{DG} is the sublattice of L\underline{G} consisting of those cocharacters whose image is contained in \underline{DG}. Passing to tangent vectors at the identity yields the claim.

Second, by definition, the root system R\underline{DG} of \underline{DG} is the same as the root system R\underline{G} of \underline{G} regarded as a system in E_{1} ⊕ ... ⊕ E_{s}. Thus \underline{G} and \underline{DG} have the same affine Weyl group W_{aff}.

Lastly, the projection of the lattice L\underline{G} into V_{1} ⊕ ... ⊕ V_{s} is contained in P (R^{∨}) ∩ V_{1} ⊕ ... ⊕ V_{s}.

With these observations, the same argument as in the case of GL(m) goes through:

k(W_{aff}, L\underline{G}) = k(W_{aff}, L\underline{DG})

(the saturation factor k_{inv fact} depends only on the group W_{aff} and the cocharacter lattice), which implies (14).

We can also give a complete solution (in terms of the stability inequalities) for the Invariant Factor Problem Q3 in the case of nonarchimedean reductive Lie groups G with root system of type A_{m−1}. For instance, \underline{G} could be the quotient of SL(m) by a subgroup of its center. As before, let L denote the lattice L\underline{G} and let X := X\underline{G} be the Bruhat-Tits building associated with \underline{G}.

Theorem 8.14. Let \underline{G} be a reductive algebraic group over 𝕂, so that the group \underline{G} has the associated root system R of type A_{m−1}. Then the natural embedding

D_{3}^{ref,L}(X) ↪ D_{3}^{L}(X) = {(α, β, γ) ∈ D_{3}(X) ∩ L^{3} : α + β + γ ∈ Q(R^{∨})}  (15)

is onto.

Proof: The assertion follows from Corollary 8.8 and the fact that k_{R} = 1 for the root system of type A.

Specializing to the case of GL(m) and SL(m) we get the following corollary:

Corollary 8.15. Let X = ℙ_{m} be the symmetric space of symmetric positive-definite m × m-matrices. Then there exists a solution to the Invariant Factor Problem Q3 for the case G = SL(m, 𝕂) (or GL(m, 𝕂)) and K = SL(m, O) (resp. GL(m, O)) if and only if α, β and γ are integer diagonal matrices so that tr(α + β + γ) = 0 and the projections

α − \frac{1}{m}tr(α)I, β − \frac{1}{m}tr(β)I, γ − \frac{1}{m}tr(γ)I

satisfy the stability inequalities 6.13 for the space X. Here I is the identity matrix.

9 The comparison of Problems Q3 and Q4

9.1 The Hecke ring

For this section we will (for the most part) follow the notation of [Gro]. We urge the reader to consult this paper for more details. However we will let 𝕂 and O be as in section 4.3. We will assume that the valuation v is discrete, the field 𝕂 is locally compact and the residue field is finite of order q.

We let \underline{G} be a connected reductive algebraic group over 𝕂. We will assume that \underline{G} is split over 𝕂. Then \underline{G} is the general fiber of a group scheme (also denoted \underline{G}) over O with reductive special fiber. We fix a maximal split torus \underline{T} ⊂ \underline{G} defined over O. We put G = \underline{G}(𝕂), K = \underline{G}(O) and T = \underline{T} (𝕂). We let \underline{B} ⊂ \underline{G} be a Borel subgroup normalized by \underline{T} and put B = \underline{B}(𝕂). We let U be the unipotent radical of B whence B = T U. Let X = X\underline{G} denote the Bruhat-Tits building associated with the group \underline{G}; o ∈ X is a distinguished special vertex stabilized by the compact subgroup K.

We have already defined in section 4.3 the free abelian groups (of rank l = dim(\underline{T})) X_{∗}(\underline{T}) and X^{∗}(\underline{T}) and a perfect ℤ-valued pairing ⟨ , ⟩ between them. The first contains the coroots R^{∨}, the second contains the roots R of \underline{G}.

The roots are the characters of \underline{T} that occur in the adjoint representation on the Lie algebra of \underline{G}. The subset R^{+} of the roots that occur in representation on the Lie algebra of \underline{B} forms a positive system and the indecomposable elements of that positive system form a system of simple roots Π. We let W denote the corresponding (finite) Weyl group.

The root basis Π determines a positive Weyl chamber P^{+} in X_{∗}(\underline{T}), by

P^{+} = \{λ ∈ X_{∗}(\underline{T}) : ⟨λ, α⟩ ≥ 0, α ∈ Π\}.

We define a partial ordering on P^{+} by λ > μ iff the difference λ − μ is a sum of positive coroots.

We define the element ρ ∈ X^{∗}(\underline{T}) ⊗ ℤ[1/2] by

2ρ = ∑_{α∈R^{+}} α.

We recall that ρ is the sum of the fundamental weights of R. Then, for all λ ∈ P^{+}, the half-integer ⟨λ, ρ⟩ is positive.

Definition 9.1. The Hecke ring H = H(G, K) is the ring of all locally constant, compactly supported functions f : G ⟶ ℤ which are K-biinvariant. The multiplication in H is by the convolution

f · g(z) = ∫_{G} f(x) · g(x^{−1}z)dx

where dx is the Haar measure on G giving K volume 1.

We claim that the function f · g also takes values in ℤ. Indeed, f and g are finite sums of characteristic functions of K-double cosets. Thus it suffices to prove the claim in the case that f and g are both characteristic functions of K-double cosets. In this case it is immediate that their convolution product is the characteristic function of the set of products of the elements in the two double cosets. This product set is itself a finite union of K-double cosets. This implies that the structure constants m_{λ,μ}(ν) of the Hecke ring, defined below, are nonnegative integers. The characteristic function of K is the unit element of H. For the proof of the next lemma see [Gro, §2].

Lemma 9.2. H is commutative and associative.

In fact much more is true. For λ ∈ X_{∗}(\underline{T}) let c_{λ} be the characteristic function of the corresponding K-double coset λ(π) ∈ K\G/K.

Lemma 9.3. 1. The assignment λ ⟶ c_{λ} induces an isomorphism of free abelian groups ℤ[P^{+}] ⟶ H.

2. The Hecke ring is a filtered ring with the filtration levels indexed by the ordered abelian semigroup P^{+}. In particular, we have

c_{λ} · c_{μ} = c_{λ+μ} + ∑_{ν<λ+μ} m_{λ,μ}(ν)c_{ν}.  (16)

Here and in what follows ∑_{ν<μ} denotes the sum over elements ν from P^{+}. We note

that this sum is finite.

We will prove below, Lemma 9.18, that the structure constants m_{λ,μ}(ν) are polynomials in q with integer coefficients. Here and in what follows we will keep track of the dependence of certain quantities on the cardinality q of the residue field. Thus q will play the role of a variable in what follows. One of the main points here is that the structure constants n_{λ,μ}(ν) of the representation ring R(G^{∨}) do not depend on q. They will be encoded in the coefficients of the polynomials m_{λ,μ}(ν) (along with the coefficients of the Kazhdan-Lusztig polynomials b_{λ}(μ)).

Let {λ_{i}, 1 ≤ i ≤ l} be the fundamental coweights, the basis for the free semigroup P^{+} which is dual to the basis Π of simple roots. Then we have an isomorphism of filtered rings:

Lemma 9.4. H ≅ ℤ[c_{λ_{1}}, ⋯, c_{λ_{l}}].

Proof: Clearly the graded ring associated to the above filtration of H is the semigroup ring ℤ[P^{+}]. But the above elements c_{λ_{i}} freely generate the associated graded ring. It is elementary to prove that this implies they freely generate the original filtered ring.

Remark 9.5. The above argument is essentially the classical argument that the representation ring of a reductive group is a polynomial ring, see for example [Sam, Proposition A, pg. 112].

We recall the definition of the structure constants m_{α,β,γ}(δ): Given double cosets of α(π), β(π), γ(π) in K\G/K, consider the characteristic functions c_{α} , c_{β} and c_{γ} of these double cosets; then decompose the triple product

c_{α} · c_{β} · c_{γ} = ∑_{δ} m_{α,β,γ}(δ) c_{δ}

in the Hecke algebra H. This defines m_{α,β,γ}(δ). Our primary interest is m_{α,β,γ}(1) (where 1 is the trivial character); it will be viewed as a function of the variable q.

In our proof of the saturation conjecture for GL(m) we will need the following lemma, where γ^{∗} denotes the weight contragredient to γ. Then γ^{∗}(π) is a representative for the double coset obtained by inverting the elements in the one represented by γ(π).

Lemma 9.6.

m_{α,β,γ}(1) = vol(Kγ(π)K) m_{α,β}(γ^{∗}).

Proof: We define an inner product (( , )) on C_{0}(G), the space of compactly supported complex-valued functions on G, by

((f, g)) := ∫_{G} f(x)\overline{g(x)}dx.

We first claim that

((c_{α}, c_{β})) = \begin{cases}vol Kα(π)K, & if α = β \\ 0, & otherwise. \end{cases}

Indeed

((c_{α}, c_{β})) = ∫_{G} c_{α}(x)c_{β}(x)dx.

The claim follows by noting that the product function is identically one if the cosets coincide and otherwise it is identically zero.

Next we observe that

((c_{α}, c_{β})) = c_{α} · c_{β^{∗}}(1).

Indeed, the right-hand side is equal to

∫_{G} c_{α}(x)c_{β^{∗}}(x^{−1})dx = ∫_{G} c_{α}(x)c_{β}(x)dx.

Hence

c_{α} · c_{β^{∗}}(1) = \begin{cases}vol Kα(π)K, & if α = β \\ 0, & otherwise. \end{cases}

Finally

m_{α,β,γ^{∗}}(1) = ∑_{δ} m_{α,β}(δ)(c_{δ} · c_{γ^{∗}})(1) = m_{α,β}(γ)vol Kγ(π)K.

9.2 A geometric interpretation of m_{α,β,γ}(1)

In this section we will prove the following

Theorem 9.7. m_{α,β,γ}(1) is the number of triangles in the building X with first vertex o and side-lengths α, β, γ.

We recall that given τ ∈ Δ we define S(o, τ) = {x ∈ X : σ(o, x) = τ}, the “Δ-sphere of radius τ and center at o”.

Lemma 9.8. For each τ ∈ Δ the group K acts transitively on S(o, τ).

Proof: Let x ∈ S(o, τ). By the properties of the action of G on the building X, presented in the List 5.12, there exists g ∈ G such that g · o = o and g · x ∈ Δ ⊂ A where A is the model apartment. Since g · o = o we have g = k ∈ K, whence

\overrightarrow{o k · x} = \overrightarrow{o τ}, (equality of vectors).

If follows that k · x = τ and x = k^{−1}τ.

As a consequence we can identify the right K–quotients of the K–double cosets in G to the Δ–spheres S(o, τ) in the building. The proof of the following lemma is then clear.

Lemma 9.9. Let α ∈ X_{∗}(\underline{T}). Then the map k ⟶ k·α(π) induces a bijection between the quotient Kα(π)K/K and S(o, α(π)).

We define

f : Kα(π)K/K × Kγ^{∗}(π)K/K ⟶ K\G/K

by sending (g_{1}, g_{2}) to the double coset represented by f(g_{1}, g_{2}) := g_{1}^{−1} · g_{2}. The reader will observe that f is well-defined. Thus f induces a map (again denoted f) from S(o, α(π)) × S(o, γ^{∗}(π)) to K\G/K.

Now let α, β, γ ∈ X_{∗}(\underline{T}). We define the set

T_{α,β,γ} := {(x, y) ∈ Kα(π)K/K × Kγ^{∗}(π)K/K : f(x, y) ∈ Kβ(π)K}.

Note that this set is finite since q < ∞. Let Δ(o, x_{1}, x_{2}) be a triangle in the building X with the Δ-side lengths α, β, γ. Then f(x_{1}, x_{2}) ∈ Kβ(π)K and we obtain a map F : Δ(o, x_{1}, x_{2}) ↦ (x_{1}, x_{2}) from the space of triangles in X with side-lengths α, β, γ and first vertex at o into the set T_{α,β,γ}. We leave the proof of the following lemma to the reader.

Lemma 9.10. F is a bijection.

Remark 9.11. The appearance of the contragredient coweight γ^{∗} comes about because we require σ(x_{2}, o) = γ. But σ(x_{2}, o) is the contragredient of the σ(o, x_{2}).

Write

Kα(π)K = ∪_{i=1}^{I}x_{i}K and Kγ(π)K = ∪_{j=1}^{J}y_{j}K,

where both I and J are finite (since q < ∞). The theorem will be a consequence of the following lemma.

Lemma 9.12.

m_{α,β,γ}(1) = #(T_{α,β,γ}).

Proof: We have

m_{α,β,γ}(1) = c_{α} · c_{β} · c_{γ}(1) = ∫_{G}(∫_{G} c_{α}(x)c_{β}(x^{−1}y)c_{γ}(y^{−1})dx)dy

= ∫_{G} (∑_{i=1}^{I} ∫_{K} c_{α}(x_{i}k)c_{β}(k^{−1}x_{i}^{−1}y)dk) c_{γ}(y^{−1})dy

= ∑_{i=1}^{I} c_{α}(x_{i}) ∫_{G} c_{β}(x_{i}^{−1}y)c_{γ}(y^{−1})dy

= ∑_{i=1}^{I} ∑_{j=1}^{J} c_{α}(x_{i}) ∫_{K} c_{β}(x_{i}^{−1}y_{j}k)c_{γ}(k^{−1}y_{j}^{−1})dk

= ∑_{i=1}^{I} ∑_{j=1}^{J} c_{α}(x_{i})c_{β}(x_{i}^{−1}y_{j})c_{γ}(y_{j}^{−1})

= ∑_{i=1}^{I} ∑_{j=1}^{J} c_{β}(x_{i}^{−1}y_{j}) = #(T_{α,β,γ})

As a consequence of Theorem 9.7 and Lemma 9.6, we find that the structure constants for the Hecke algebra are determined by the geometry of the building.

Theorem 9.13. Let α, β, γ ∈ X_{∗}(\underline{T}). Then we have

m_{α,β}(γ) = \frac{#(T_{α,β,γ^{∗}})}{#(S(o, γ(π)))}.  (17)

Proof: We have only to apply Lemma 9.6 and observe that since we are assuming that vol(K) = 1 we have vol(Kγ(π)K) = vol(Kγ(π)K)/K = #(S(o, γ(π))).

9.3 The Satake transform

In this section we define an integral transform S, the Satake transform, from compactly supported, K-biinvariant functions on G to left K-invariant, right U-invariant functions on G.

Let δ : B ⟶ ℝ_{+}^{∗} be the modular function of B, [Gro, §3]. We may regard δ as a left K-invariant, right U-invariant function on G. By the Iwasawa decomposition for G, [Tits, pg. 51], any such function is determined by its restriction to T. For a compactly supported K-biinvariant function f on G we define its Satake transform as a function Sf(g) on G given by

Sf(g) = δ(g)^{1/2} · ∫_{U} f(gu)du.

Here the Haar measure du is normalized so that the open subgroup K ∩U has measure 1.

Then Sf is a left K-invariant, right U-invariant function on G with values in ℤ[q^{1/2}, q^{−1/2}]; this function is determined by its restriction to T/T ∩ K ≅ X_{∗}(\underline{T}). The main result of [Sat] (see also [C, pg. 147]), is that the image of S lies in the subring

(ℤ[X_{∗}(\underline{T})])^{W} ⊗ ℤ[q^{1/2}, q^{−1/2}] ≅ R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}],

where ≅ is a ring isomorphism. Here and in what follows G^{∨} = \underline{G}^{∨}(ℂ). Furthermore we have (see [Sat], [C, pg. 147]):

Theorem 9.14. The Satake transform gives a ring isomorphism

S : H ⊗ ℤ[q^{1/2}, q^{−1/2}] ≅ R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}].

For λ ∈ P^{+} let chV_{λ} be the character of the irreducible representation V_{λ} of G^{∨} (see e.g. [FH, pg. 375]. We may identify chV_{λ} with a W -invariant element of ℤ[X^{∗}(\underline{T}^{∨})] = ℤ[X_{∗}(\underline{T})].

In what follows we will need two bases for the free ℤ[q^{1/2}, q^{−1/2}]-module R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}]. The first basis is

S := {S(c_{λ}) : λ ∈ P^{+}},

the second basis is

R := {chV_{λ} : λ ∈ P^{+}}.

The change of basis matrices relating these two bases are both upper triangular (see Lemma 9.15 below) with entries in the ring ℤ[q^{1/2}, q^{−1/2}]. We define the order (at ∞) ord(f) of the element f = ∑_{k=−M}^{k=N} a_{k}q^{k/2} ∈ ℤ[q^{1/2}, q^{−1/2}] by ord(f) = N, provided that a_{N} ≠ 0. Note that if f is a polynomial in q then ord(f) = 2deg(f) where deg(f) is the degree of f in q. We will accordingly extend the degree to ℤ[q^{1/2}, q^{−1/2}] by defining

deg(f) = 1/2 ord(f).

Thus the extended degree takes values in the half integers. We define deg_{S}(F) for F ∈ R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}] to be the maximum of the degrees of the components of F when F is expressed in the basis S. We will use a similar convention (i.e., expanding it terms of the basis S) when we speak of the “leading term” of F. We retain the notation deg (without subscript) for the ordinary notion of degree ∈ \frac{1}{2}ℤ for the Laurent polynomials in ℤ[q^{1/2}, q^{−1/2}]. Analogously, we define deg_{R}(F) when F is expanded in terms of the basis R.

For the following lemma we refer the reader to [Gro, (3.11) and (3.12)], [Lu83, (6.10)]. See [Ha, §2], for the statement of the lemma in the generality we require here.

Lemma 9.15. 1. There exist polynomials a_{λ}(μ) in q such that

S(c_{λ}) = q^{⟨λ,ρ⟩}chV_{λ} + ∑_{μ<λ} a_{λ}(μ)q^{⟨μ,ρ⟩}chV_{μ}.

2. Conversely, there exist polynomials b_{λ}(μ) in q such that

q^{⟨λ,ρ⟩}chV_{λ} = S(c_{λ}) + ∑_{μ<λ} b_{λ}(μ)S(c_{μ}).

As an immediate corollary of this lemma we get

deg_{R}(S(c_{λ})) ≥ 0 = deg_{S}(S(c_{λ})).

and thus

deg_{R}(F) ≥ deg_{S}(F)  (18)

for F ∈ R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}].

The following degree estimate is a consequence of a conjecture of Lusztig, [Lu83, (9.4), pg. 226], who proved it for the root systems of type A_{ℓ}, [Lu81]; it was proved in general by Kato in [Kato] (see also [Gro, pg. 230]). This inequality will play a critical role in what follows. In fact the degree estimate follows because the polynomials b_{λ}(μ) coincide with Kazhdan-Lusztig polynomials associated to certain Weyl group elements w_{μ}, w_{λ} as was realized by Lusztig, [Gro, §4]. These polynomials have many remarkable properties.

Lemma 9.16. For all μ < λ, μ ∈ P^{+} we have:

deg(b_{λ}(μ)) < ⟨λ − μ, ρ⟩ ≤ ⟨λ, ρ⟩.

This lemma when combined with Lemma 9.15 has the following consequences.

Lemma 9.17. For each cocharacter λ ∈ P^{+} \ {0} we have:

deg_{S}(q^{⟨λ,ρ⟩}chV_{λ}) < ⟨λ, ρ⟩.

Proof: By Lemma 9.15 we get

q^{⟨λ,ρ⟩}chV_{λ} = S(c_{λ}) + ∑_{μ<λ} b_{λ}(μ)S(c_{μ}).

Hence Lemma 9.16 implies

deg_{S}(q^{⟨λ,ρ⟩}chV_{λ}) < ⟨λ, ρ⟩.

Lemma 9.18. There exists a polynomial M_{α,β,γ}(q) in the variable q, such that we have the equality of functions in q:

m_{α,β,γ}(1) = M_{α,β,γ}(q).

Furthermore, all structure constants m_{α,β}(γ) are polynomials in q.

Proof: We will prove that all the structure constants m_{α,β}(γ) are polynomials in q. The first statement will follow from this.

Following [Gro] we define ϕ_{λ} ∈ R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}] by

ϕ_{λ} = q^{⟨λ,ρ⟩}chV_{λ}.

The elements \{ϕ_{λ} : λ ∈ X^{∗}(\underline{T}^{∨})\} give a new basis for the ℤ[q^{1/2}, q^{−1/2}]-module R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}].

We claim that the structure constants for the ring relative to this basis are polynomials in q. Indeed, we have

ϕ_{α} · ϕ_{β} =q^{⟨α+β,ρ⟩}chV_{α} · chV_{β} = q^{⟨α+β,ρ⟩} ∑_{γ≤α+β} n_{α+β}(γ) chV_{γ}
= ∑_{γ≤α+β} n_{α+β}(γ) q^{⟨α+β−γ,ρ⟩}ϕ_{γ}.

Now, since γ ≤ α + β, the coweight α + β − γ is a sum of positive coroots and consequently ⟨α + β − γ, ρ⟩ is an nonnegative integer.

We can now prove the second statement. Let α and β be given. We expand S(c_{α}) and S(c_{β}) in terms of the basis of ϕ_{λ}’s thereby introducing the polynomials a_{λ}(μ). We then multiply the resulting expressions. According to the paragraph above, the result is an expression in the ϕ_{λ}’s with polynomial coefficients in q. We then substitute for the ϕ_{λ}’s using Lemma 9.15 introducing the polynomials b_{λ}(μ).

9.4 A solution of Problem Q4 is a solution of Problem Q3

We recall the definition of the structure constants n_{α,β,γ}(δ):

ch(V_{α}) · ch(V_{β}) · ch(V_{γ}) = ∑_{δ∈P^{+}} n_{α,β,γ}(δ) ch(V_{δ}).

We are interesting in comparing the coefficient n_{α,β,γ}(1) (corresponding to the trivial representation 1 of G^{∨}) with m_{α,β,γ}(1).

The main goal of this section is to prove the following

Theorem 9.19. If the triple α, β, γ is a solution of Problem Q4 then it is also a solution of Problem Q3. More precisely,

n_{α,β,γ}(1) ≠ 0 ⟹ m_{α,β,γ}(1) ≠ 0.

The above theorem follows from

Theorem 9.20. Suppose that \underline{G} is of a reductive algebraic group over 𝕂 which is split over 𝕂. Then:

(a) The degree of the polynomial M_{α,β,γ}(q) is at most ⟨α + β + γ, ρ⟩.

(b) The leading coefficient of M_{α,β,γ}(q), i.e., the coefficient at q^{⟨α+β+γ,ρ⟩}, is equal to n_{α,β,γ}(1).

The above theorem immediately implies the following

Corollary 9.21. If n_{α,β,γ}(1) ≠ 0 then ⟨α + β + γ, ρ⟩ ∈ ℤ.

Proof of Theorem 9.20. For the benefit of the reader, below we explain the idea behind the proof of Theorem 9.20. We have to compute the coefficient of 1 in the Hecke triple product c_{α} · c_{β} · c_{γ}. Instead, we will compute the coefficient of 1 in the triple product S(c_{α}) · S(c_{β}) · S(c_{γ}) in the representation ring R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}] where the triple product is expanded relative to the basis S. In order to do this we will use the formula from Lemma 9.15 and compute (initially) in the basis R. However, to prove the theorem we must again apply Lemma 9.15 to rewrite the result in terms of the basis S.

The theorem will follow from the next three lemmas. We owe the first of these to Jiu-Kang Yu.

Lemma 9.22. For all μ ∈ P^{+} satisfying μ < λ we have:

deg(a_{λ}(μ)q^{⟨μ,ρ⟩}) < ⟨λ, ρ⟩.

Proof: The proof is by induction on λ with respect to the partial order <. We remind the reader that each set {μ ∈ P^{+} : μ < λ} is finite. If the set {μ ∈ P^{+} : μ < λ} is empty then there is nothing to prove.

Now assume that we are given λ ∈ P^{+} and that we have proved the above estimate for all predecessors of λ in the partial order <. We have, according to Lemma 9.15,

S(c_{λ}) = q^{⟨λ,ρ⟩}chV_{λ} − ∑_{μ<λ} b_{λ}(μ)S(c_{μ})

and

S(c_{μ}) = q^{⟨μ,ρ⟩}chV_{μ} + ∑_{η<μ} a_{μ}(η)q^{⟨η,ρ⟩}chV_{η}.

Thus

S(c_{λ}) = q^{⟨λ,ρ⟩}chV_{λ} − ∑_{μ<λ} b_{λ}(μ)q^{⟨μ,ρ⟩}chV_{μ} − ∑_{μ<λ} ∑_{η<μ} b_{λ}(μ)a_{μ}(η)q^{⟨η,ρ⟩}chV_{η}.

We make the change of variable μ → τ, η → μ in the last sum to obtain

S(c_{λ}) = q^{⟨λ,ρ⟩}chV_{λ} − ∑_{μ<λ} b_{λ}(μ)q^{⟨μ,ρ⟩}chV_{μ} − ∑_{μ<λ} ∑_{{τ:μ<τ<λ}} b_{λ}(τ)a_{τ}(μ)q^{⟨μ,ρ⟩}chV_{μ}.

Since chV_{μ}, μ ∈ P^{+}, is a basis of the ℤ[q^{−1/2}, q^{1/2}]-module R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}], by combining the previous formula with the part 1 of Lemma 9.15, we obtain:

a_{λ}(μ)q^{⟨μ,ρ⟩} = −b_{λ}(μ)q^{⟨μ,ρ⟩} − ∑_{{τ:μ<τ<λ}} b_{λ}(τ)a_{τ}(μ)q^{⟨μ,ρ⟩}.

The degree of the first term satisfies the required estimate by Lemma 9.16. We estimate the degrees of the terms in the second sum. By the induction hypothesis, deg(a_{τ}(μ)q^{⟨μ,ρ⟩}) < ⟨τ, ρ⟩ and by Lemma 9.16 we have

deg(b_{λ}(τ)) < ⟨λ − τ, ρ⟩.

Consequently, for all μ, we get

deg(b_{λ}(τ)a_{τ}(μ)q^{⟨μ,ρ⟩}) < ⟨λ − τ, ρ⟩ + ⟨τ, ρ⟩ = ⟨λ, ρ⟩.

Corollary 9.23.

deg(a_{λ}(μ)) < ⟨λ − μ, ρ⟩.

Lemma 9.24. For α, β, γ ∈ P^{+} such that α + β + γ is a nontrivial character, we have:

deg_{S}(S(c_{α}) · S(c_{β}) · S(c_{γ}) − q^{⟨α+β+γ,ρ⟩}chV_{α} · chV_{β} · chV_{γ}) < ⟨α + β + γ, ρ⟩.

Proof: By expanding the triple product

S(c_{α}) · S(c_{β}) · S(c_{γ}) =

(q^{⟨γ,ρ⟩}chV_{γ} + ∑_{λ<γ} a_{γ}(λ)q^{⟨λ,ρ⟩}chV_{λ})

·(q^{⟨β,ρ⟩}chV_{β} + ∑_{μ<β} a_{β}(μ)q^{⟨μ,ρ⟩}chV_{μ})

·(q^{⟨α,ρ⟩}chV_{α} + ∑_{ν<α} a_{α}(ν)q^{⟨ν,ρ⟩}chV_{ν})

in terms of the basis R := {chV_{τ}} we get the following types of summands:

q^{⟨α+β+γ,ρ⟩}chV_{α} · chV_{β} · chV_{γ},

q^{⟨α+β,ρ⟩}chV_{α} · chV_{β} ∑_{λ<γ} a_{γ}(λ)q^{⟨λ,ρ⟩}chV_{λ},

q^{⟨γ,ρ⟩}chV_{γ} · (∑_{μ<β} a_{β}(μ)q^{⟨μ,ρ⟩}chV_{μ}) · (∑_{ν<α} a_{α}(ν)q^{⟨ν,ρ⟩}chV_{ν}),

and similar ones obtained by permuting α, β, γ, and finally:

a_{α}(ν)a_{β}(μ)a_{γ}(λ)q^{⟨ν+μ+λ,ρ⟩}chV_{α} · chV_{β} · chV_{γ},

where in the latter case ν < α, μ < β, λ < γ, ν, μ, λ ∈ P^{+}. The first term (q^{⟨α+β+γ,ρ⟩}chV_{α} · chV_{β} · chV_{γ}) cancels out, we estimate the degree of each of the remaining terms separately. We will do it in the case of the 2-nd and 4-th term and leave the 3-rd term to the reader:

First, since λ < γ,

deg_{S}[q^{⟨α+β,ρ⟩}chV_{α} · chV_{β} · a_{γ}(λ)q^{⟨λ,ρ⟩}chV_{λ}] ≤
deg_{R}[q^{⟨α+β,ρ⟩}chV_{α} · chV_{β} · a_{γ}(λ)q^{⟨λ,ρ⟩}chV_{λ}] ≤
⟨α + β, ρ⟩ + deg(a_{γ}(λ)q^{⟨λ,ρ⟩}) <

(by Lemma 9.22)

< ⟨α + β, ρ⟩ + ⟨γ, ρ⟩ = ⟨α + β + γ, ρ⟩.

Similarly,

deg_{S}[a_{α}(ν)a_{β}(μ)a_{γ}(λ)q^{⟨ν+μ+λ,ρ⟩}chV_{α} · chV_{β} · chV_{γ}] ≤
deg_{R}[a_{α}(ν)a_{β}(μ)a_{γ}(λ)q^{⟨ν+μ+λ,ρ⟩}chV_{α} · chV_{β} · chV_{γ}] =
deg[a_{α}(ν)q^{⟨ν,ρ⟩}] + deg[a_{β}(μ)q^{⟨μ,ρ⟩}] + deg[a_{γ}(λ)q^{⟨λ,ρ⟩}] <

(by Lemma 9.22) < ⟨α + β + γ, ρ⟩.

The lemma follows.

Now we can prove the degree estimate in Theorem 9.20. It suffices to prove the estimate for the image of the polynomial M_{α,β,γ}(q) in the ring H. According to Lemma 9.15,

S(c_{λ}) = q^{⟨λ,ρ⟩}chV_{λ} + ∑_{μ<λ} a_{λ}(μ)q^{⟨μ,ρ⟩}chV_{μ}.

Hence, by Lemma 9.22,

deg_{R}(S(c_{λ})) = ⟨λ, ρ⟩,

and consequently

deg_{S}(S(c_{α}) · S(c_{β}) · S(c_{γ})) ≤ deg_{R}((S(c_{α}) · S(c_{β}) · S(c_{γ})) = ⟨α + β + γ, ρ⟩.

Thus deg(M_{α,β,γ}) ≤ ⟨α + β + γ, ρ⟩ which proves the first assertion of Theorem.

Let N_{α,β,γ}(q^{1/2}) be the polynomial in q^{1/2} with coefficients in the ring R(G^{∨}) ⊗ ℤ[q^{1/2}, q^{−1/2}] given by

N_{α,β,γ}(q^{1/2}) := q^{⟨α+β+γ,ρ⟩}chV_{α} · chV_{β} · chV_{γ}.

Here as usual we assume that the irreducible characters have been expanded in terms of the basis S using Lemma 9.15. We see as a consequence of the previous lemma that the Theorem 9.20 will follow from

Lemma 9.25.

deg_{S}(N_{α,β,γ}(q^{1/2}) − n_{α,β,γ}(1)q^{⟨α+β+γ,ρ⟩}) < ⟨α + β + γ, ρ⟩.

Proof: By definition of the structure constants n_{α,β,γ}(δ) we have

q^{⟨α+β+γ,ρ⟩}chV_{α} · chV_{β} · chV_{γ} = ∑_{δ≤α+β+γ} n_{α,β,γ}(δ) q^{⟨α+β+γ−δ,ρ⟩}q^{⟨δ,ρ⟩}chV_{δ}.

But by Lemma 9.17, δ ≠ 1 ⟹ deg_{S}(q^{⟨δ,ρ⟩}chV_{δ}) < ⟨δ, ρ⟩.

This proves Theorem 9.20 and hence Theorem 9.19.

We now give another explanation why

n_{α,β,γ}(1) ≠ 0 ⟹ ⟨α + β + γ, ρ⟩ ∈ ℤ.

Theorem 9.26. 1. m_{α,β,γ}(1) ≠ 0 ⟹ α + β + γ ∈ Q(R^{∨}) ⟹ ⟨α + β + γ, ρ⟩ ∈ ℤ.

2. n_{α,β,γ}(1) ≠ 0 ⟹ α + β + γ ∈ Q(R^{∨}).

Proof: 1 follows from Theorem 8.2, part 4. Hence 2 follows from 1 via Theorem 9.20, however we give a direct proof below.

The center of G^{∨} acts on each of the factors of the triple tensor product by a scalar, hence it acts by a scalar on the triple tensor product itself. But if n_{α,β,γ}(1) ≠ 0 this scalar is necessarily 1. Hence the center fixes the (highest) weight vector with weight α + β + γ, whence α + β + γ annihilates the center. Accordingly it is in the character lattice of the adjoint group Ad G^{∨}. Thus α + β + γ ∈ Q(R^{∨}).

9.5 A solution of Problem Q3 is not necessarily a solution of Problem Q4

In this section we consider the case two examples: \underline{G} = SO(5), whence G^{∨} = Sp(4, ℂ) and the group \underline{G} of type G_{2} and G^{∨} = G_{2}(ℂ).

We begin with \underline{G} = SO(5); we assume that we have chosen Witt bases for ℂ^{5} and ℂ^{4}. We let T^{∨} be the split torus consisting of diagonal matrices in Sp(4, ℂ) and \underline{T} be the split torus of SO(5) consisting of diagonal matrices. We use the rectangular coordinates x_{1}, x_{2} in the Cartan subalgebra 𝔞^{∨} ⊂ 𝔰𝔭(4, ℂ) such that the simple roots are x_{1} − x_{2}, 2x_{2}. We set

α = β = γ = (1, 1) ∈ X^{∗}(\underline{T}^{∨}) = X_{∗}(\underline{T}).

Note that α + β + γ belongs to the coroot lattice of SO(5), i.e., the condition stated in Theorem 9.26 is satisfied.

We give two proofs of the next lemma, the first computational, the second conceptual.

Lemma 9.27.

n_{α,β,γ}(1) = 0.
m_{α,β,γ}(1) = q^{5} − q.

Thus for this choice of side-lengths Problem Q3 has a solution but Problem Q4 does not.

Remark 9.28. In fact the second equality implies the first because, in this case, ⟨α + β + γ, ρ⟩ = 6. Hence, (according to Theorem 9.20) M_{α,β,γ}(q) = n_{α,β,γ}(1)q^{6} + lower terms. But the coefficient of q^{6} in the above formula for m_{α,β,γ}(1) is 0.

Proof: The first equality is obvious if we observe that under the isomorphism between Sp(4) and Spin(5) the representation of Sp(4, ℂ) with the highest weight (1, 1) corresponds to the standard (vector) representation of SO(5, ℂ) on ℂ^{5} (considered as a nonfaithful representation of Spin(5, ℂ)). Now it is standard that the tensor square of this representation consists of three irreducible summands, the trivial representation (corresponding to the invariant quadratic form), the exterior square of the standard representation and the harmonic quadratic polynomials on ℂ^{5}. Since none of these are equivalent to the standard representation and all representations of Sp(4, ℂ) are self-dual the first statement follows.

The proof of the second statement requires more work. From [Gro, pg. 231], we have:

S(c_{(1,1)}) = q^{2}chV_{(1,1)} − 1.

Upon taking the cube and calculating in the representation ring of Sp(4, ℂ) one obtains

S(c_{(1,1)})^{3} = q^{6}[chV_{(3,3)} + 2chV_{(3,1)} + chV_{(2,0)} + 3chV_{(1,1)}]
−3q^{4}[chV_{(2,2)} + chV_{(2,0)} + 1] + 3q^{2}chV_{(1,1)} − 1.

Upon substituting for the chV_{λ} using Lemma 9.15 (and computing the appropriate Kazhdan-Lusztig polynomials b_{λ}(μ)) the lemma follows.

As for the conceptual proof of (the second part of) the lemma, note that the side-lengths belong to the root-subgroup SL(2) corresponding to the positive root x_{1} + x_{2}. But it is evident that one can construct an equilateral triangle with sidelengths equal to the positive root and vertices of the correct type in the tree for SL(2, 𝕂). Since this tree is totally-geodesic in the building for Sp(4, 𝕂) and the fixed vertex for SL(2, O) in this embedded tree is o (the fixed vertex for Sp(4, O)), the lemma follows. Equivalently if we can realize the trivial double coset as a product of the three SL(2, O)–double cosets in SL(2, 𝕂) belonging to (1, 1), (1, 1), (1, 1) then we can certainly do it in Sp(4, 𝕂).

Remark 9.29. This last sentence is the essence of the counterexample. The solutions to Problems Q3 and Q4 behave differently with respect to inclusions of subgroups. The solutions of Problem Q3 “push forward”, the solutions of Problem Q4 do not.

We now give another example of a triple of coroots α, β, γ such that the triple is a solution to Problem Q3 but not of Problem Q4. We take \underline{G} = G_{2}. Hence G^{∨} = G_{2} as well. We take α = λ_{1} and β = γ = λ_{2}. Here λ_{1} is the first fundamental weight (the highest weight of the unique irreducible seven dimensional representation) and λ_{2} is the the second fundamental weight (the highest weight of the adjoint representation).

Lemma 9.30. m_{λ_{1},λ_{2},λ_{2}}(1) = q^{5}(q + 1)(q^{6} − 1).

Proof: From [Gro, pg. 231, (5.7)], we have

S(c_{λ_{1}}) =q^{3}chV_{λ_{1}} − 1
S(c_{λ_{2}}) =q^{5}chV_{λ_{2}} − q^{3}chV_{λ_{2}} − q^{4}.

Multiplying (using LiE) one obtains

S(c_{λ_{1}}) · S(c_{λ_{2}}) = q^{8}chV_{λ_{1}+λ_{2}} + (q^{8} − q^{6})(chV_{2λ_{1}} + chV_{λ_{1}}) − q^{6}chV_{λ_{2}} − q^{4}S(c_{λ_{1}}) − S(c_{λ_{2}}).

Hence, by Lemma 9.15 we have

m_{λ_{1},λ_{2}}(λ_{2}) = b_{λ_{2},λ_{1}+λ_{2}}(q) + (q^{2} − 1)b_{λ_{2},2λ_{1}}(q) − q − 1.

Using b_{λ_{2},λ_{1}+λ_{2}}(q) = 1 + q and b_{λ_{2},2λ_{1}}(q) = 1 we obtain

m_{λ_{1},λ_{2}}(λ_{2}) = q^{2} − 1.

Now Lemma 9.6 implies that

m_{λ_{1},λ_{2},λ_{2}}(1) = m_{λ_{1},λ_{2}}(λ_{2}) · vol(Kλ_{2}(π)K).

From [Gro, pg. 735], we get vol(Kλ_{2}(π)K) = deg(c_{λ_{2}}) = q^{10} + q^{9} + q^{8} + q^{7} + q^{6} + q^{5}. The lemma follows.

Corollary 9.31.

n_{λ_{1},λ_{2},λ_{2}}(1) = 0.

Proof: From [Gro, pg. 231, (5.7)], we have ⟨λ_{1}, ρ⟩ = 3 and ⟨λ_{2}, ρ⟩ = 5 whence ⟨λ_{1} + λ_{2} + λ_{2}, ρ⟩ = 13. Hence, the structure constant has degree (one) less than the maximum possible degree.

We now give a geometric proof (using unfolding) that m_{λ_{1},λ_{2},λ_{2}}(1) is nonzero. We show that there exists a triangle T′ (with Δ-side-lengths λ_{1}, λ_{2}, λ_{2}) in a discrete Euclidean building X modelled on the discrete affine Coxeter complex of type G_{2}, so that all vertices of T′ are special. We start with the billiard triangle T in Figure 8, contained in a model apartment A ⊂ X; this triangle has two geodesic sides \overline{zx}, \overline{zy} and one broken side \overline{xuy}. Note that all three vertices of this billiard triangle are special vertices of the Coxeter complex. The break point u on the broken side of the billiard triangle belongs to the wall l of the Coxeter complex, the wall l separates the broken side \overline{xuy} from the vertex z = o. Thus one can unfold the billiard triangle T to a geodesic triangle T′ ⊂ X preserving the refined side-lengths by applying Lemma 7.6.

Figure 8: A billiard triangle in the affine Coxeter complex of type G_{2}.

9.6 The saturation conjecture for GL(ℓ)

In this subsection we explain Green’s idea for proving that for GL(ℓ)

α, β, γ is a solution of Problem Q3 ⟹ α, β, γ is a solution of Problem Q4.

Let 𝕂 be a valued field with a discrete complete valuation v and let O ⊂ 𝕂 be the subring of elements with nonnegative valuation. Pick a uniformizer π ∈ 𝕂 for the valuation v. We will assume that the residue field is finite.

Let \underline{G} be a connected reductive group algebraic group over 𝕂 and set G = \underline{G}(𝕂) and let G^{∨} = \underline{G}^{∨}(ℂ) be its Langlands dual. We will assume that \underline{G} is split over 𝕂. Recall that the structure constants m_{α,β}(γ) and n_{α,β}(γ) for the Hecke ring of G and the representation ring of G^{∨} respectively, are given by

c_{α} · c_{β} = ∑_{γ} m_{α,β}(γ)c_{γ},

ch(V_{α}) · ch(V_{β}) = ∑_{γ} n_{α,β}(γ) ch(V_{γ}).

We will say that a dominant weight μ for GL(ℓ) is a partition if all components of the integer vector μ are nonnegative.

The starting point is that if the dominant weights α, β, γ are partitions then there is a standard formula [Mac, pg. 161] that expresses the structure constant m_{α,β}(γ) as the number of finite O-module extensions

1 → A → B → C → 1,

where A ≅ ⊕_{i=1}^{ℓ}O/(π^{α_{i}}), B ≅ ⊕_{i=1}^{ℓ}O/(π^{γ_{i}}) and C ≅ ⊕_{i=1}^{ℓ}O/(π^{β_{i}}).

We will say that such an extension is of type (α, β, γ). Now for GL(ℓ) there is an explicit formula (the Littlewood- Richardson rule) for the structure constant n_{α,β}(γ) (in case α, β, γ are partitions) as the number of Littlewood–Richardson sequences (of partitions) of type (α, β, γ), see [Mac, pg. 68, 90].

Then the observation of J. Green is the following

Lemma 9.32. Suppose there exists an extension of O-modules of type (α, β, γ). Then there exists a Littlewood-Richardson sequence (of partitions) of type (α, β, γ).

Thus for the group GL(ℓ) we find that if α, β, γ^{∗} are partitions and the structure constant m_{α,β}(γ^{∗}) is nonzero then the structure constant n_{α,β}(γ^{∗}) is nonzero (the superscript ∗ denotes the contragredient dominant weight).

We next remove the assumption that α, β, γ^{∗} are partitions. To this end let λ be the cocharacter of GL(ℓ) that sends z ∈ ℚ \ {0} to the scalar matrix with diagonal entries equal to z. If α is a cocharacter then there exists k such that λ^{k} · α is a partition. Now since the matrix λ(π) is scalar it is clear that for any α

c_{α} · c_{λ} = c_{α·λ}.

It then follows that

m_{λ^{k}·α,λ^{k}·β}(λ^{2k} · γ^{∗}) = m_{α,β}(γ^{∗}).

Now the previous operation of convolving with the function c_{λ_{1}} corresponds, under Langlands’ duality, to multiplying a character by χ_{1}, the character of the determinant representation. Thus

m_{α,β}(γ^{∗}) ≠ 0 ⇔ m_{λ^{k}·α,λ^{k}·β}(λ^{2k} · γ^{∗}) ≠ 0 ⇔ n_{χ^{k}·α,χ^{k}·β}(χ^{2k} · γ^{∗}) ≠ 0 ⇔ n_{α,β}(γ^{∗}) ≠ 0.

Finally, by Lemma 9.6 we have

m_{α,β,γ}(1) ≠ 0 ⇔ m_{α,β}(γ^{∗}) ≠ 0.

Also it is immediate that n_{α,β}(γ^{∗}) = n_{α,β,γ}(1). Thus, if Problem Q3 can be solved for α, β, γ then Problem Q4 can be as well. By combining this with Theorem 9.19, we conclude that for GL(ℓ) Problems Q3 and Q4 are equivalent. Since we have proved that 1 is a saturation factor for the Problem Q3 it is also a saturation factor for Q4.

10 Appendix: Decomposition of tensor products, Mumford quotients and triangles in symmetric spaces

In this section we will assume that G^{∨} is a reductive complex Lie group with Lie algebra 𝔤^{∨} and the (spherical) Weyl group W^{∨}. We let K^{∨} be a maximal compact subgroup of G^{∨} and let X^{∨} := G^{∨}/K^{∨} be the associated symmetric space. We let 𝔤^{∨} = 𝔨^{∨} ⊕ 𝔭^{∨} denote the corresponding Cartan decomposition of the Lie algebra of G^{∨}. Let 𝔞^{∨} be a Cartan subalgebra in 𝔭^{∨}.

There exists an algebraic group \underline{G}^{∨} over ℚ such that G^{∨} = \underline{G}^{∨}(ℂ). We let \underline{G} denote the Langlands’ dual of \underline{G}^{∨} and set G := \underline{G}(ℂ), let K be a maximal compact subgroup of G. We set X := G/K and let 𝔤 = 𝔭 ⊕ 𝔨 be the corresponding Cartan decomposition of the Lie algebra of G. Let 𝔞 be a Cartan subalgebra of 𝔭. Then 𝔞^{∨} and 𝔞 are dually paired by ⟨ , ⟩. Let W ↷ 𝔞 denote the action of the spherical Weyl group of G. We let \underline{T} ⊂ \underline{G} and \underline{T}^{∨} ⊂ \underline{G}^{∨} denote maximal split tori which correspond to each other under the Langlands’ duality; we let T, T^{∨} be their groups of complex points. As usual we assume that 𝔭, 𝔭^{∨} are given invariant Euclidean metrics denoted (·, ·). For each covector λ ∈ (𝔭^{∨})^{∗} we let λ^{#} ∈ 𝔭^{∨} denote the corresponding vector (under the isometry # : (𝔭^{∨})^{∗} → 𝔭^{∨} given by the metric (·, ·)).

Remark 10.1. As we already know, see chapter 2, the character group X^{∗}(\underline{T}^{∨}) is isomorphic to a lattice in (𝔞^{∨})^{∗} = X^{∗}(\underline{T}^{∨}) ⊗ ℝ; similarly, X_{∗}(\underline{T}) is isomorphic to a lattice in 𝔞 = X_{∗}(\underline{T}) ⊗ ℝ. We recall that the first isomorphism can be described as follows. Let λ ∈ X^{∗}(\underline{T}^{∨}). Then the restriction dλ|_{id} to 𝔞^{∨} is an element Φ^{∨}(λ) of (a^{∨})^{∗}. This defines a map

Φ^{∨} : X^{∗}(\underline{T}^{∨}) → (𝔞^{∨})^{∗},

which is an isomorphism from the cocharacter lattice onto a lattice in (𝔞^{∨})^{∗}.

We have the natural isomorphism of the Euclidean Coxeter complexes:

(𝔞, W ⋉ 𝔞) ⟶^{ι} ((𝔞^{∨})^{∗}, W^{∨} ⋉ (𝔞^{∨})^{∗}),

which extends the isomorphism of (co)character lattices

ι : X_{∗}(\underline{T}) → X^{∗}(\underline{T})

(see chapter 2). To simplify our notation we will retain the notation λ for ι(λ), for each λ ∈ 𝔞. We will also need the isomorphism

# : ((𝔞^{∨})^{∗}, W^{∨} ⋉ (𝔞^{∨})^{∗}) → (𝔞^{∨}, W^{∨} ⋉ 𝔞^{∨}).

given by the metric.

We let Δ ⊂ 𝔞 denote the (positive) Euclidean Weyl chamber for W_{sph}; then X_{∗}(\underline{T}) = L_{trans} = L is the translational part of W_{aff}. Hence the points of Δ ∩ X_{∗}(\underline{T}) are the L-integral side-lengths.

The vector space 𝔭 has the natural structure of a Cartan motion space modelled on (𝔞, W ⋉𝔞). We also get two more Cartan motion spaces modelled on the Euclidean Coxeter complex 𝔞^{∨}, namely the dual spaces (𝔭^{∨})^{∗} and (𝔨^{∨})^{∗} (see Example 5.5). These spaces are isomorphic to each other under the map (𝔭^{∨})^{∗} → (𝔨^{∨})^{∗} given by J : v ↦ \sqrt{−1}v. The group K^{∨} acts on (𝔨^{∨})^{∗} via the coadjoint action. For each fundamental weight λ belonging to an apartment (𝔞^{∨})^{∗} ⊂ (𝔭^{∨})^{∗} we define the coadjoint orbit, O_{λ} ⊂ (𝔨^{∨})^{∗}, which is the Ad^{∗}K^{∨}-orbit of \sqrt{−1}λ.

Now, our Theorem 1.2 implies that the isomorphisms J ∘ ι and # ∘ J^{−1} induce bijections

D_{n}(𝔭) ↔ D_{n}((𝔨^{∨})^{∗}) ↔ D_{n}(𝔭^{∨}).  (19)

We will restrict our attention to n = 3.

We will use the following notation. Let λ ∈ (𝔞^{∨})^{∗} be a dominant weight for the torus T^{∨}. Then we let V_{λ} be an irreducible G^{∨}-module with the highest weight λ (so V_{λ} is unique up to isomorphism). We let λ^{∗} be the highest weight of the dual representation, so λ^{∗} = w_{0}(−λ) where w_{0} is the longest element in W.

The key point in what follows is the famous theorem of Borel and Weil (see e.g. [Vo, Chapter 1]):

Theorem 10.2. 1. Let α ∈ (𝔞^{∨})^{∗} be a dominant weight. Then the coadjoint orbit O_{α} as above carries a natural homogeneous complex structure, so that the coadjoint action of K^{∨} on O_{α} extends to a holomorphic action of G^{∨}. The dominant weight α defines a very ample line bundle 𝕃_{α} over O_{α} and the action of G^{∨} on O_{α} extends to a holomorphic action G^{∨} ↷ 𝕃_{α}.

2. The space of holomorphic sections Γ(O_{α}, 𝕃_{α}) of 𝕃_{α} is isomorphic as a G^{∨}- module to the G^{∨}-module V_{α}.

Now let α and β be dominant weights. In this section we will discuss the problem of finding the possible irreducible constituents of tensor products V_{α} ⊗ V_{β}. Of course this is equivalent to finding for which triples α, β, γ, the space of G^{∨}-invariants

(V_{α} ⊗ V_{β} ⊗ V_{γ})^{G^{∨}} ≅ Hom_{G^{∨}}(V_{γ^{∗}}, V_{α} ⊗ V_{β})

is nonzero.

The main goal of this section is to prove Theorem 10.3, that relates the existence of triangles with integral side-lengths in X (equivalently, in 𝔭) and of nonzero invariants in triple tensor products of irreducible representations of G^{∨}. The main part of the theorem is well known to the experts (the equivalence of 1 and 4). We have used the above isomorphism between 𝔞 and (𝔞^{∨})^{∗} in order to identify vectors α, β, γ with elements of Δ in the assertions 5 and 6.

Theorem 10.3. For each triple of dominant weights α, β, γ ∈ (𝔞^{∨})^{∗} the following are equivalent:

1. There exists a positive integer k > 0 such that (V_{kα} ⊗ V_{kβ} ⊗ V_{kγ})^{G^{∨}} ≠ 0.

2. There exists a triangle in 𝔭^{∨} with Δ-side lengths α^{#}, β^{#}, γ^{#}.

3. There exists a triangle in X^{∨} with Δ-side lengths α^{#}, β^{#}, γ^{#}.

4. There exists a triangle in (𝔨^{∨})^{∗} with Δ-side lengths ι(α), ι(β), ι(γ).

5. There exists a triangle in 𝔭 with Δ-side lengths α, β, γ.

6. There exists a triangle in X with Δ-side lengths α, β, γ.

Remark 10.4. We note that the equivalence of 2, 4 and 5 follows immediately from the double bijection (19), the equivalence of 5 and 6 and the equivalence of 2 and 3 is a special case of Theorem 1.2. It remains therefore to prove the equivalence of 1 and 2.

Remark 10.5. Theorem 10.3 implies that if (V_{α} ⊗ V_{β} ⊗ V_{γ})^{G^{∨}} ≠ 0 then there exists a triangle in X with side-lengths α, β, γ. However the converse is false as we have seen in section 9.5.

Theorem 10.3 will follow from the two propositions below.

Consider the line bundles 𝕃_{α}, 𝕃_{β} and 𝕃_{γ} defined in the previous section. The outer tensor product 𝕃_{α} ⊠ 𝕃_{β} ⊠ 𝕃_{γ} is a very ample G^{∨}–line bundle 𝕃_{α,β,γ} over the product of coadjoint orbits

O_{α,β,γ} := O_{α} × O_{β} × O_{γ},

which gives a G^{∨}-equivariant embedding of O_{α,β,γ} into ℙ((V_{α} ⊗ V_{β} ⊗ V_{γ})^{∗}).

Let M_{α,β,γ} := (O_{α,β,γ})//G^{∨} be the associated Mumford quotient of O_{α,β,γ} by the diagonal action of the group G^{∨}. This quotient coincides with the quotient of the space of weighted configurations of flags defined previously, see Remark 6.10.

Proposition 10.6.

M_{α,β,γ} ≠ ∅ ⇔ (2) holds.

The proposition will follow from the next two lemmas. In order to relate M_{α,β,γ} to a moduli space of triangles we need to recall a lemma about symplectic quotients of products of coadjoint orbits. Again we give a proof for the benefit of the reader.

Lemma 10.7. 1. The momentum map f for the diagonal action of K^{∨} on the product O_{α,β,γ} is given by the formula f(λ, μ, ν) = λ + μ + ν.

2. The symplectic quotient O_{α,β,γ}//Ad^{∗}(K^{∨}) is equal to the moduli space of triangles in (𝔨^{∨})^{∗} with side-lengths α, β, γ.

Proof: The first statement will follow by considering the product action of (K^{∨})^{3} on its coadjoint orbit O_{α,β,γ}. First it is standard that the momentum map for the coadjoint action on an orbit is the inclusion of that orbit. Second it is also standard that the momentum map for the restriction of the action of the diagonal subgroup of a product action is the sum of the individual momentum maps.

As a consequence of the first statement we have

O_{α,β,γ}//Ad(K^{∨}) = {(λ, μ, ν) ∈ O_{α,β,γ} : λ + μ + ν = 0}/Ad^{∗}(K^{∨}).

But this quotient is clearly the required moduli space of triangles.

We now can prove the result we need.

Lemma 10.8. The moduli space of triangles in X^{∨} with Δ-side lengths α^{#}, β^{#}, γ^{#} is canonically homeomorphic with the Mumford quotient M_{α,β,γ} defined above.

Proof: By a theorem of Kempf and Ness [KN79], the Mumford quotient M_{α,β,γ} is canonically homeomorphic to the symplectic quotient O_{α,β,γ}//Ad^{∗}(K^{∨}).

But clearly the metric ( , ) induces a homeomorphism between this symplectic quotient and the other symplectic quotient:

O_{α^{#},β^{#},γ^{#}}//Ad(K^{∨}),

where O_{α^{#},β^{#},γ^{#}} is the product of adjoint orbits of K in 𝔨^{∨}. Hence, by transport of structure we obtain

O_{α^{#},β^{#},γ^{#}}//Ad(K^{∨}) = {(λ, μ, ν) ∈ O_{α^{#},β^{#},γ^{#}} : λ + μ + ν = 0}/Ad(K^{∨}),

which is the moduli space of triangles in 𝔨^{∨} with the Δ-side lengths α^{#}, β^{#}, γ^{#} ∈ 𝔞^{∨}. Thus

M_{α,β,γ} ≠ ∅ ⟺ (α^{#}, β^{#}, γ^{#}) ∈ D_{3}(𝔨^{∨}).

We complete the proof of the theorem by proving

Proposition 10.9.

M_{α,β,γ} ≠ ∅ ⇔ (1) holds.

The proposition will follow from the next two lemmas.

Let S be the graded ring associated to the projective variety which is the image F (O_{α,β,γ}) of the product of orbits under the embedding F given by the sections of 𝕃_{α,β,γ}. Then by definition S^{(k)} is the set of restrictions to F (O_{α,β,γ}) of the homogeneous polynomials of degree k in the projective coordinates given by S^{(1)}. Thus

S^{(k)} = Γ(O_{α,β,γ}, 𝕃_{α,β,γ})^{⊗k}/I^{(k)}.

Here I^{(k)} is the ideal of polynomials of degree k in the sections of 𝕃_{α,β,γ} that vanish on O_{α,β,γ}. Equivalently, I^{(k)} is the degree k component of the ideal I of polynomials that vanish on σ(O_{α,β,γ}) in the ring of polynomials on ℙ((V_{α} ⊗ V_{β} ⊗ V_{γ})^{∗}).

Now another definition of the Mumford quotient is P roj(S^{G^{∨}}) where S^{G^{∨}} is the subring of G^{∨}-invariants of the graded ring S. We define another graded ring R by

R = ⨁_{k=0}^{∞} Γ(O_{α,β,γ}, (𝕃_{α} ⊠ 𝕃_{β} ⊠ 𝕃_{γ})^{⊗k}).

Finally we define a graded G^{∨} × G^{∨} × G^{∨} module R′ by

R′ = ⨁_{k=0}^{∞}(V_{kα} ⊠ V_{kβ} ⊠ V_{kγ}).

We will abuse notation and use R′ to denote the restriction of the previous module to the diagonal. Then, we have an isomorphism of G^{∨}-modules

R′ = ⨁_{k=0}^{∞}(V_{kα} ⊗ V_{kβ} ⊗ V_{kγ}).

First we show that R and R′ are canonically isomorphic.

Lemma 10.10. There is a canonical isomorphism of graded (G^{∨} ×G^{∨} ×G^{∨})-modules R ≅ R′.

Proof: The lemma follows from the following two equations. First, let M_{1} and M_{2} be complex manifolds and 𝕃_{1}, resp. 𝕃_{2} be a holomorphic line bundle over M_{1}, resp. M_{2}. Then

Γ(M_{1} × M_{2}, 𝕃_{1} ⊠ 𝕃_{2}) ≅ Γ(M_{1}, 𝕃_{1}) ⊗ Γ(M_{2}, 𝕃_{2}).

Furthermore, for a natural number k, a dominant weight α and the corresponding orbit O_{α} we have

Γ(O_{α}, 𝕃_{α}^{⊗k}) ≅ Γ(O_{α}, 𝕃_{kα}).

Next we show that S and R are isomorphic. This is the exceptional feature of the homogeneous situation. The key point, the surjectivity of the natural map below, was pointed out to us by Lawrence Ein.

Lemma 10.11. There is a natural G^{∨} × G^{∨} × G^{∨} equivariant isomorphism from the graded ring S to the graded ring R.

Proof: To obtain the desired map from the graded ring S to the graded ring R we observe that for any complex manifold M and holomorphic line bundle 𝕃 over M there is a natural map (usually not onto) from Γ(M, 𝕃)^{⊗k} to Γ(M, 𝕃^{⊗k}). Hence there is a natural map from S^{(k)} to R^{(k)}. The exceptional feature here is that this natural map is onto.

To see this we note that by the theorem of Borel and Weil the action of G^{∨} × G^{∨} ×G^{∨} on R^{(k)} is irreducible. But the image of S^{(k)} in R^{(k)} is an invariant subspace. Hence S^{(k)} maps onto R^{(k)}. Clearly the map is injective (because we have divided by the ideal I^{(k)}).

Theorem 10.3 follows by taking G^{∨}-invariants from the G^{∨}-isomorphism S ≅ R′. Indeed, there exists a triangle with Δ-side lengths α,β,γ ⇔ the Mumford quotient M_{α,β,γ} is nonempty ⇔ (⨁_{k=1}^{∞} S^{(k)})^{G^{∨}} ≠ 0 ⇔ ⨁_{k=1}^{∞}(V_{kα} ⊗ V_{kβ} ⊗ V_{kγ})^{G^{∨}} ≠ 0.

Remark 10.12. The reason that the existence of a nonzero G^{∨}-invariant does not follow from the existence of a triangle is that the lowest degree G^{∨}-invariant might not have degree 1, that is, it is possible that

(R^{(1)})^{G^{∨}} = {0} but (R^{(k)})^{G^{∨}} ≠ 0 for some k > 1.

References

[AW] S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Letters, vol. 5 (1998), p. 817–836.

[AMW] A. Alekseev, E. Meinrenken and C. Woodward, Linearization of Poisson actions and singular values of matrix products, Ann. Inst. Fourier (Grenoble), vol. 51 (2001), no. 6, p. 1691–1717.

[Bel] P. Belkale, Local systems on ℙ^{1}-S for S a finite set, Compositio Math., vol. 129 (2001) p. 67–86.

[Ba] W. Ballmann, “Lectures on spaces of nonpositive curvature. With an appendix by Misha Brin.” DMV Seminar, vol. 25. Birkhauser Verlag, Basel, 1995.

[BeSj] A. Berenstein and R. Sjamaar, Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion, Journ. Amer. Math. Soc., vol. 13 (2000), no. 2, p. 433–466.

[Bo] N. Bourbaki, “Groupes et Algébres de Lie”, Chap. 4, 5, 6, Masson, 1981.

[BT] F. Bruhat and J. Tits, Groupes rductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., No. 41, (1972), p. 5–251.

[C] P. Cartier, Representations of p-adic groups, In: “Automorphic forms, representations, and L-functions”, Proc. AMS, 33 (1979), p. 111–155.

[DW] H. Derksen, J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc., vol. 13, (2000), n. 3, p. 467–479.

[Fu] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.), vol. 37 (2000), no. 3, p. 209–249.

[FH] W. Fulton, J. Harris, “Representation theory. A first course.”, Springer Verlag, 1991.

[G] F.Gauss, Letter to Bolyai, “Collected works”, vol. 8, p. 222–223.

[Gro] B. Gross, On the Satake isomorphism, In: “Galois representations in arithmetic algebraic geometry (Durham, 1996)”, London Math. Soc. Lecture Notes, vol. 254, (1998) p. 223–237.

[HL] P. Heinzner and F. Loose, Reduction of complex Hamiltonian G-spaces, Geom. Funct. Anal., vol. 4 (1994), no. 3, 288–297.

[Ha] T. J. Haines, On the matrix coefficients of the Satake isomorphism: complements to the paper of M. Rapoport, Manuscripta Math., vol. 101 (2000), p. 167–174.

[Hum] J. Humphreys, “Reflection groups and Coxeter groups”, Cambridge University Press, 1990.

[KM] M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Diff. Geometry, vol. 44, (1996), p. 479–513.

[KLM] M. Kapovich, B. Leeb, J. J. Millson, Polygons in symmetric spaces and buildings, Preprint, 2002.

[KMT] M. Kapovich, J. J. Millson and T. Treloar, The symplectic geometry of polygons in hyperbolic 3-space, Asian J. Math. (Kodaira’s issue), vol. 4 (2000), no. 1, p. 123–164.

[Kato] S. I. Kato, Spherical functions and a q-analogue of Kostant’s weight multiplicity formula, Invent. Math., vol. 66 (1982), p. 461–468.

[KN79] G. Kempf and L. Ness, The length of vectors in representation spaces, In: “Algebraic Geometry Proceedings. Copenhagen 1978”, Springer Lecture Notes in Mathematics, vol. 732 (1979), p. 233–243.

[K68] T. Klein, The multiplication of Schur functions and extensions of p-modules, London Math. Soc. J., vol 43 (1968), p. 280–284.

[K69] T. Klein, The Hall polynomial, J. Algebra, vol. 12 (1969), p. 61–78.

[KL] B. Kleiner, B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Publ. Math. IHES, vol. 86 (1997), p. 115–197.

[Kly98] A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Mathematica, vol. 4 (1998), p. 419–445.

[Kly99] A. Klyachko, Random walks on symmetric spaces and inequalities for matrix spectra, Linear Algebra Appl. Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem (Coimbra, 1999), vol. 319 (2000), no. 1-3, p. 37–59.

[K] A. Knutson, Weight varieties, PhD thesis, MIT, 1996.

[KT] A. Knutson and T. Tao, The honeycomb model of GL_{n}(ℂ) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., vol. 12 (1999), no. 4, p. 1055–1090.

[Ko] S. Kochen, The model theory of local fields, ISILC Logic conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974), Springer Lecture Notes in Math., vol. 499, p. 384–425.

[Ku] S. Kumar, private communication.

[La] M. C. Laskowski, An application of Kochen’s Theorem, Preprint, 2002.

[LM] B. Leeb, J. J. Millson, Convex functions on symmetric spaces and geometric invariant theory for spaces of weighted configurations on flag manifolds. Preprint.

[Li] P. Littelmann, Generalization of the Littlewood–Richardson Rule, Invent. Math., vol. 116 (1994), p. 329–346.

[Lu81] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math., vol. 42 (1981), p. 169–178.

[Lu83] G. Lusztig, Singularities, character formulas, and a q-analogue of weight multiplicities, Astérisque, vol. 101 (1983), p. 208–227.

[Mac] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford (1995).

[Mi] J. J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math., vol. 104, (1976), p. 235–247.

[MR] J. J. Millson and M. S. Raghunathan, Geometric construction of cohomology of arithmetic groups I., Proc. Indian Acad. Sci. 90, (1981) (V. K. Patodi memorial issue), p. 103–123.

[Sam] H. Samelson, Notes on Lie algebras, Springer-Verlag, 1990.

[Sat] I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Publ. Math. IHES, vol. 18 (1963), p. 1–69.

[Sj] R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math., vol. 141 (1995), no. 1, p. 87–129.

[Sp] T. A. Springer, Reductive groups , In: Automorphic forms, representations and L-functions, Proc. Symp. AMS, vol. 33 (1979), p. 3–27.

[SpSt] T. A. Springer and R. Steinberg, Conjugacy classes, In: “Seminar on Algebraic Groups and Related Finite Groups”, Springer Lecture Notes in Math., vol. 131 (1970), p. 167–266.

[Tits] J. Tits, Reductive groups over local fields, In: “Automorphic forms, representations and L-functions”, Proc. Symp. AMS, vol. 33 (1979), p. 29–69.

[Tr] T. Treloar, The symplectic geometry of polygons in the 3-sphere, Canad. J. Math., vol. 54, (2002), p. 30–54.

[Vo] D. Vogan,“Unitary representations of reductive Lie groups”, Princeton University Press, 1987.

Michael Kapovich: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA kapovich@math.utah.edu

Bernhard Leeb: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany bernhard@riemann.mathematik.uni-tuebingen.de

John J. Millson: Department of Mathematics, University of Maryland, College Park, MD 20742, USA jjm@math.umd.edu