(Quasi-)Hamiltonian manifolds of cohomogeneity one

We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). The group acting is a compact connected Lie group (simply connected in the quasi-Hamiltonian case). This work is a concretization of a more general classification of multiplicity free manifolds in the special case of rank one. As a result we obtain numerous new concrete examples of multiplicity free quasi-Hamiltonian manifolds or, equivalently, Hamiltonian loop group actions.


Introduction
Let K be a simply connected, compact Lie group.In order to study Hamiltonian actions of the (infinite dimensional) loop group LK on (infinite dimensional) manifolds M, Alekseev-Malkin-Meinrenken, [AMM98], introduced the notion of quasi-Hamiltonian K-manifolds.These are finite dimensional K-manifolds, equipped with a 2-form and a momentum map.
To a certain extent, quasi-Hamiltonian manifolds are very similar to classical Hamiltonian manifolds.This means, in particular, that a quasi-Hamiltonian manifold can be locally described by Hamiltonian manifolds.The most striking difference is that the target of the momentum map is the group K itself (instead of its coadjoint representation k * ).For that reason, quasi-Hamiltonian manifolds lack functoriality properties, like restriction to a subgroup and are therefore more difficult to construct than Hamiltonian manifolds.
The most basic quasi-Hamiltonian manifolds are the conjugacy classes of K.These are precisely the ones on which K acts transitively.The main purpose of this paper is to classify the next case in difficulty, namely compact manifolds of cohomogeneity one, i.e., where K acts with orbits of codimension one.
Our approach is based on the papers [Kno11,Kno23] where the more general class of multiplicity free (quasi-)Hamiltonian manifolds was considered.These are manifolds M for which the momentum map induces an injective map on orbit spaces (M/K → k * /K or M/K → K/ /K, respectively).In op.cit.these have been classified in terms of pairs (P, Γ) where P is a convex polytope (the momentum polytope) and Γ is a lattice (characterizing the principal isotropy group).The compatibility condition between P and Γ is expressed in terms of a root system (affine or finite, respectively) and the existence of certain smooth affine spherical varieties.Even though the latter have been determined in [KVS06] (up to coverings, central tori and C × -fibrations), the condition is hard to handle in practice.So the present paper is also meant to be a (successful) test case for the feasibility of using [Kno11,Kno23] to obtain explicit classification results.More precisely, the case considered here is precisely that of the multiplicity free manifolds which are of rank one, i.e., where rk M := dim P = rk Γ = 1.Our classification proceeds in two steps.First, there is an induction procedure from smaller Hamiltonian manifolds.Manifolds which are not induced are called primitive.So, in a second step, we determine all primitive manifolds.These can be quasi-Hamiltonian or Hamiltonian.So, as a by-product, but of independent interest our classification also encompasses (classical) Hamiltonian manifolds of cohomogeneity one.
1.1.Theorem.Let M be a primitive, multiplicity free, (quasi-)Hamiltonian manifold of rank one.Then M corresponds to a diagram in Table 8.3.Moreover, to each diagram there corresponds a manifold which is either unique (in the quasi-Hamiltonian case) or unique up to rescaling the symplectic structure (in the Hamiltonian case).
In Table 8.3 we presented each case by a diagram which is very close to Luna's [Lun01] for classifying spherical varieties.The classification yields many previously known quasi-Hamiltonian manifolds.For example, the spinning 4-sphere of Hurtubise-Jeffrey [HJ00], Alekseev-Malkin-Woodward [AMW02]) and its generalization, the spinning 2n-sphere, by Hurtubise-Jeffrey-Sjamaar [HJS06] are on our list.We also recover the Sp(2n)-action on P n H discovered by Eshmatov [Esh09] as part of a larger series which seems to be new, namely a quasi-Hamiltonian action of K = Sp(2n) on the quaternionic Grassmannians Gr k (H n+1 ) (item (cc) for the root system C (1) n in Table 8.3).It should be mentioned that there is related work by Lê [Le98] on more qualitative aspects of Hamiltonian manifolds of cohomogeneity one.1.2.Remark.Part of this paper is based on part of the second author's doctoral thesis [Pau18] which was written under the supervision of the first named author.
Acknowledgment: We would like to thank Guido Pezzini, Bart Van Steirteghem and Wolfgang Ruppert for many explanations, discussions and remarks.Last not least, we would like to thank the referee for many suggestions to improve the paper.

Hamiltonian and quasi-Hamiltonian manifolds
We first recall the most important properties of Hamiltonian and quasi-Hamiltonian manifolds.In the entire paper, K will be a compact connected Lie group with Lie algebra k.A Hamiltonian K-manifold is a triple (M, w, m) where M is a K-manifold, w is a K-invariant symplectic form on M and m : M → k * is a smooth K-equivariant map (the momentum map) such that In [AMM98], Alekseev, Malkin and Meinrenken studied this concept in the context of loop groups.Even though these Loop groups are infinite dimensional, the authors managed to reduce Hamiltonian loop group action to a finite dimensional concept namely quasi-Hamiltonian manifolds.These are very similar to Hamiltonian manifolds.
More precisely, quasi-Hamiltonian manifolds are also triples (M, w, m) where M is a K-manifold, w is a K-invariant 2-form and m is a K-equivariant map.But there are differences.
First of all, the Lie algebra k has to be equipped with a K-invariant scalar product.Moreover, a twist τ ∈ Aut K has to be chosen 1 (which may be the identity).The momentum map m has values in K instead of k * and is equivariant with respect to the τ -twisted conjugation action on K, i.e., g * h := ghτ (g) −1 .Finally, the closedness and non-degeneracy of w as well as formula (2.1) have to be adapted.For the details one can consult the papers [AMM98], [Kno23], or [Mei17].They are not relevant for the present paper.
In the following, we want to treat the Hamiltonian case and the quasi-Hamiltonian on the same footing.So we talk about U -Hamiltonian manifolds where U = k * in the Hamiltonian and U = K in the quasi-Hamiltonian case.
This momentum map m : M → U gives rise to a map between orbits spaces: (2.2) By definition, the fibers of this map are the symplectic reductions of M .The smooth ones are symplectic manifolds in a natural way.In particular, they are even dimensional.
Most important for us are those manifolds for which this dimension is as low as possible, namely 0. These manifolds are called multiplicity free.
An important invariant of M is its momentum image (m/K)(M/K) ⊆ U/K.Its dimension is called the rank of M .Multiplicity free manifolds of rank zero are simply the K-orbits in U .In this paper we study the next more difficult case namely multiplicity free manifolds of rank one.These two conditions can be combined into one.For this recall that the dimension of M/K is the cohomogeneity of M .Then we have: 2.1.Lemma.For a U -Hamiltonian manifold M the following are equivalent: (1) The cohomogeneity of M is 1.
(2) M is multiplicity free of rank one.
Proof.Let c := 1 2 dim ((m/K) −1 (a)) where a is a generic point of the momentum image.As mentioned above, it is an integer.By definition, c = 0 is equivalent to multiplicity freeness.Let r be the rank of M .Then we have Hence, dim M/K = 1 if and only if c = 0 and r = 1.

Affine root systems
Before we go on with explaining the general structure of U -Hamiltonian manifolds we need to set up notation for finite and affine root systems.Here, we largely follow the exposition in [Kno23] which is in turn based on [Mac72] and [Mac03].
Let a be a Euclidean vector space, i.e., a finite dimensional real vector space equipped with a scalar product •, • .Let a be an affine space for a, i.e., a is equipped with a free and transitive a-action.We denote the set of affine linear functions on a by A(a).The gradient of a function α ∈ A(a) is the element α ∈ a with A reflection s is an isometry of a whose fixed point set is an affine hyperplane.If that hyperplane is the zero-set of α ∈ A(a) then one can express s = s α as s α (X) = X − α(X)α ∨ with the usual convention α ∨ = 2α ||α|| 2 .3.1.Definition.An affine root system on a is a subset Φ ⊂ A(a) such that: Observe that, with our definition, Φ might be finite or even empty.In that case, the roots have a common zero which we can use as a base point.This way, we can identify a with a and we have α(X) = α, X for all roots α.If (a 1 , Φ 1 ), . . ., (a s , Φ s ) are affine root systems then ) is also one (where the p i are the projections).Conversely, every affine root system admits such a decomposition such that the Weyl group W i of Φ i is either trivial or acts irreducibly on a i .We say that Φ is properly affine if each irreducible factor Φ i is infinite.
The set of simple roots (for a fixed alcove) will be denoted by S.
These are possibly non-reduced finite root systems on a.We define: 3.2.Definition.An integral root system on a is a pair (Φ, Ξ) where Φ ⊂ A(a) is an affine root system and Ξ ⊆ a is a lattice with Φ ⊆ Ξ and Ξ, Φ ∨ ⊆ Z.The integral root system The classification of irreducible (infinite) affine root systems as it can be found, e.g., in [Kac90] is recalled in Table 8.1.In that table, also the Dynkin label k(α) of each α ∈ S is given.These labels are uniquely characterized by being integral, coprime, and having the property that is a positive constant function.

Classification of multiplicity free Hamiltonian and quasi-Hamiltonian manifolds
We summarize some known facts about the quotient U/K.
If U = k * then it is classical that U/K is parameterized by a Weyl chamber for the finite root system attached to K.
If U = K we need to assume that K is simply connected which we do from now on.
Then U/K is in bijection with the alcove A for a properly affine root system which is determined by K and the action of τ in the Dynkin diagram of K, cf.[MW04] for details.Recall that in this case, k is equipped with a scalar product.We use it to identify k with k * .Thereby, we obtain a map (4.1) In the Hamiltonian case, we put for compatibility reasons ψ = id k * .Likewise, we assume that a scalar product has been selected on k even though the results will not depend on it.
4.1.Theorem.Let K, U be as above.Then there is a subspace a ⊆ k * and an integral root system (Φ, Ξ) on a such that: (1) If A ⊆ a is any alcove of Φ, then the map ψ/K : A → U/K is a homeomorphism.
(2) If X ∈ A and a := ψ(X) ∈ U , then the isotropy group is connected, a ⊆ k a is a Cartan subalgebra, the weight lattice of K a is Ξ, and is a set of simple roots of K a .Here S ⊂ Φ is the set of simple roots with respect to A.
Since K a depends only on S(X) ⊆ S we also write K a = K S(X) .Let M be a compact, connected U -Hamiltonian manifold.Then the invariant momentum map is the composition Its image P M := m + (M ) ⊆ A can be shown to be a convex polytope [Kir84], the so-called momentum polytope of M .It is the first main invariant of M .
A second invariant comes from the facts that for generic a ∈ P M the isotropy group K a acts on the momentum fiber m −1 + (a) via a quotient A M of K a which is a torus independent of a.Its character group Γ M is a subgroup of the weight lattice Ξ.
This begs the question which pairs (P, Γ) arise this way.The key to the answer lies in the paper [Bri87] of Brion which connects the theory of multiplicity free Hamiltonian manifolds with the theory of complex spherical varieties.In the following we summarize only a simplified version which suffices for our purposes.
We start with a connected, reductive, complex group G.An irreducible algebraic Gvariety Z is called spherical if a Borel subgroup of G has an open orbit.Now assume also that Z is affine and let C[Z] be its ring of regular functions.Then the Vinberg-Kimelfeld criterion [VK78] asserts that Z is spherical if and only if C[Z] is multiplicity free as a G-module.This means that there is a set (actually a monoid) where V χ is the simple G-module of highest weight χ.A theorem of Losev [Los06] asserts that in case Z is smooth, the variety Z is in fact uniquely determined by its weight monoid Λ Z .
Let K ⊆ G be a maximal compact subgroup.Then any smooth affine G-variety can be equipped with the structure of a Hamiltonian K-manifold by embedding Z into a finite dimensional L-module V and using a K-invariant Hermitian scalar product on V to define a momentum map.Then (1) Z is spherical as a G-variety if and only if it is multiplicity free as a Hamiltonian K-manifold.
The first two items were proved by Brion [Bri87] in the context of projective varieties.The version which we need, namely for affine varieties, was proved by Sjamaar in [Sja98].For the last item see Losev [Los06,Prop. 8 So Λ Z and the pair (P Z , Γ Z ) carry the same information.
4.4.Definition.A pair (P, Γ) is called G-spherical if there exists a smooth affine spherical G-variety Z with P = R ≥0 Λ Z and Γ = ZΛ Z .The (unique) smooth variety Z will be called a model for (P, Γ).
Now we go back to U -Hamiltonian manifolds.For any subset P ⊆ A and point X ∈ P we define the tangent cone of P at X as (4.7) Here is a local version of sphericality: 4.5.Definition.Let P ⊆ A be a compact convex polytope and Γ ⊆ Ξ a subgroup.
(1) (P, Γ) is spherical in X ∈ P if (T X P, Γ) is L-spherical where L := K C a is the Levi subgroup corresponding to a := ψ(X) ∈ U .The model variety for (T X P, Γ) will be called the local model of (P, Γ) in X.
(2) The pair (P, Γ) is locally spherical if it is spherical in every vertex of P. 4.6.Remark.It follows from the definition of G-sphericality that in a locally spherical pair P and Γ are necessarily parallel in the sense that P is a polytope of maximal dimension inside the affine subspace X + Γ R ⊆ a for any X ∈ P.
The classification theorem can now be stated as follows: 4.7.Theorem ([Kno11], [Kno23]).Let K be a connected compact Lie group which is assumed to be simply connected in the quasi-Hamiltonian case.Then the map M → (P M , Γ M ) induces a bijection between (1) isomorphism classes of compact, connected multiplicity free U -Hamiltonian manifolds and (2) locally spherical pairs (P, Γ) where P ⊆ A is a compact convex polyhedron and Γ ⊆ Ξ is a subgroup.

Remarks.
(1) The relation between pairs and manifolds can be made more precise.Let M be a U -Hamiltonian manifold and X ∈ P M .Then there exists a neighborhood P 0 of X in P such that (4.8) (2) The construction of locally spherical pairs is quite difficult.Already deciding whether a given pair is locally spherical is intricate.There is an algorithm due to Pezzini-Van Steirteghem [PVS19] for this but we are not going to use it since in our setting it is not necessary.
Because of this theorem, we are going to work from now on exclusively on the "combinatorial side", i.e., with locally spherical pairs.We start with two reduction steps.
4.9.Definition.Let (4.9) S(X) := {α ∈ S | α(X) = 0 for X ∈ P} be the set of simple roots which are zero for a fixed X ∈ P and (4.10) Thus, elements of S(P) correspond to walls of A which contain a point of P. Let K 0 := K S(P) be the corresponding (twisted) Levi subgroup of K. Then it is immediate that (P, Γ) is locally spherical for K if and only if it is so for K 0 .This observation reduces classifications largely to pairs with S(P) = S.

Definition. A polyhedron
There is another reduction.Assume S 0 ⊆ S is a component of the Dynkin diagram of S.
It corresponds to a (locally) direct semisimple factor L S 0 of G = K C .Suppose also that S 0 ⊆ S(X) for all X ∈ P. Then it follows from Remark 4.6 that Γ, S 0 = 0.In turn (4.5) implies that L S 0 will act trivially on every local model Z of (P, Γ).This means that also the roots in S 0 can be ignored for determining the sphericality of (P, Γ).
4.11.Definition.A genuine polyhedron P ⊆ A is called primitive if S does not contain a component S 0 with S 0 ⊆ S(X) for all X ∈ P.
The following lemma summarizes our findings 4.12.Lemma.Let P ⊆ A be a compact convex polyhedron and let be Γ ⊆ Ξ a subgroup.Let and let S 1 be the union of all components C of S \ S c with C ⊆ S(X) for all X ∈ P. Let Ξ := Ξ ∩ S ⊥ 1 .Then P is primitive for S := S \ (S c ∪ S 1 ).Moreover, the pair (P, Γ) is locally spherical for (S, Ξ) if and only if it is so for (S, Ξ).Now the purpose of this paper is to present a complete classification of primitive locally spherical pairs in the special case when rk M = dim P = 1.
In this case the following simplifications occur: the polyhedron P is a line segment P = [X 1 , X 2 ] with X 1 , X 2 ∈ A and Γ = Zω with ω ∈ Ξ.It follows from Remark 4.6 that (4.12) X 2 = X 1 + cω for some c = 0.
By replacing ω by −ω if necessary, we may assume that c > 0. Then Theorem 4.7 boils down to: (1) isomorphism classes of compact, connected multiplicity free U -Hamiltonian manifolds of rank one and (2) triples (X 1 , X 2 , ω) satisfying (4.12) such that Nω is the weight monoid of a smooth affine spherical K C S(X 1 ) -variety Z 1 and N(−ω) is the weight monoid of a smooth affine spherical K C S(X 2 ) -variety Z 2 .The triples (X 1 , X 2 , ω) and (X 2 , X 1 , −ω) are considered equal.
Triples as above will be called spherical.A triple is genuine or primitive if P = [X 1 , X 2 ] has this property.The varieties Z i are called the local models of the triple.

The local models
We proceed by recalling all possible local models, i.e., smooth, affine, spherical L-varieties Z of rank one where L is a connected,reductive, complex, algebraic group.Then (5.1) where ω is a non-zero integral dominant weight, V nω is the simple L-module of highest weight nω, and Λ equals either N orZ.The case Λ = Z is actually irrelevant for our purposes since this case only occurs as local model of an interior point of P (by (4.6)).
In case Λ = N, the weight ω is unique.
5.1.Theorem.Let Z be a smooth, affine, spherical L-variety of rank one.Then one of the following cases holds: (1) Z = C * and L acts via a non-trivial character.
(2) Z = L 0 /H 0 where (L 0 , H 0 ) appears in the first part of Table 8.2 and L acts via a surjective homomorphism ϕ : L → L 0 .(3) Z = V 0 where (L 0 , V 0 ) appears in the second part of Table 8.2 and L acts via a homomorphism ϕ : L → L 0 which is surjective modulo scalars (except for case a 0 when ϕ should be surjective).
Proof.Smooth affine spherical varieties have been classified by Knop-Van Steirteghem in [KVS06] and the assertion could be extracted from that paper.A much simpler argument goes as follows.First, a simple application of Luna's slice theorem (see [KVS06, Cor.2.2]) yields Z ∼ = L × H V where H ⊆ L is a reductive subgroup and V is a representation of H.As the homogeneous space L/H is the image of Z = L × H V under the projection Z → L/H.The rank of the homogeneous space L/H is at most the rank of Z, so either 0 or 1.
Assume now that L/H has rank one.This means that Z and L/H have the same rank.
Let F ⊆ L be the stabilizer of a point in the open L-orbit of Z such that F ⊆ H.By [Gan10, lem.2.4], the quotient H/F is finite.This implies that the projection Z → H/F has finite fibers.Hence V = 0 and Z = L/H is homogeneous.The classification of homogeneous spherical varieties of rank one (Akhiezer [Akh83], see also [Bri89], and Wasserman [Was96]), yields the cases in (1) and (2).Observe, that the non-affine cases of Akhiezer's list have been left out.
5.2.Remarks.Some remarks concerning Table 8.2: (1) Observe that items [ 1 2 ]d 2 and [ 1 2 ]d 3 could be made part of the series [ 1 2 ]d n .Because of their singular behavior we chose not to do so.For example both can be embedded into A n -diagrams.Moreover, [ 1 2 ]d 2 are the only cases with a disconnected Dynkin diagram.
(2) We encode the local models by the diagram given in the last column of Table 8.2.
For homogeneous models these diagrams are due to Luna [Lun01].The inhomogeneous ones are specific to our situation.
(3) For a homogeneous model the weight ω is a fixed linear combination of simple roots (recorded in the fourth column).Hence it lifts uniquely to a weight of L.
On the other hand, for inhomogeneous models the weight of L is only unique up to a character.This is indicated by the notation ω ∼ π 1 which means that ω, α ∨ = 1 for α = α 1 and = 0 otherwise.
Let S be the set of simple roots of L 0 , i.e., the set of vertices of a diagram.Then inspection of Table 8.2 shows that the elements of
From now let i ∈ {1, 2} and j := 3 − i, so that if Z i is a local model then Z j is the other.
6.3.Definition.Let S be a subset of a graph S. The connected closure C(S , S) of S in S is the union of all connected components of S which meet S .In other words, C(S , S) is the set of vertices of S for which there exists a path to S .
The following lemma shows in particular how to recover S i from the triple (S, S 1 , S 2 ).
6.4.Lemma.Let (X 1 , X 2 , ω) be primitive.Then (1) S i is the connected closure of S i in S \ S j .
(2) S is the connected closure of S 1 ∪ S 2 .
(1) Recall that S \ S j = S(X i ) is the disconnected union of S i and the Dynkin diagram C i of ker ϕ.Inspection of Table 8.2 shows that S i is the connected closure of S i .
6.5.Definition.Let D be the quintuple D = (S, S 1 , c 1 , S 2 , c 2 ) where S is a (possibly empty) Dynkin diagram, S 1 , S 2 are disjoint (possibly empty) subsets of S and c 1 , c 2 ∈ { 1 2 , 1, 2}.Let S i be the connected closure of S i in S \ S j .Then D is a primitive spherical diagram if it has following properties: where k(α ∨ ) is the colabel of α, i.e., the label of α ∨ in the dual diagram of S.
A primitive spherical diagram can be represented by the Dynkin diagram of S with decorations as in Table 8.2 which indicate the subsets S i and the numbers c i .
Proof.All conditions have been verified except for (3) c).If S is affine then the coroots satisfy the linear dependence relation We pair this with ω and observe that ω, α ∨ = 1, −1, 0 according to α = α D 1 , α = α D 2 or otherwise.This implies the claim.
The following is our main result.It will be proved in the next section.

Theorem. Every primitive spherical diagram is isomorphic to an entry of Table 8.3.
Explanation of Table 8.3:The first column gives the type of S. The second lists for identification purposes the type of the local models.The diagram is given in the fifth column.If a parameter is involved, its scope is given in the last column.Observe the boundary cases where we used the conventions b 1 = a 1 , 2b 1 = 2a 1 , c 2 = b 2 , and c 1 = a 1 .In some cases, besides (S, S 1 , c 1 , S 2 , c 2 ) also (S, S 1 , c c 1 , S 2 , c c 2 ) is primitive spherical where c is the factor in the column "factor".An entry of the form [c] n=a indicates that the factor applies only to the case n = a.
Finally, the weight ω can be read off from the third column.More precisely, if D i is homogeneous then ω i indicates the unique lift of ω or −ω to an affine linear function with ω i (X i ) = 0.If both local models D 1 , D 2 are inhomogeneous then ω is only unique up to a character of G. Thus, the notation ω ∼ ω 0 means ω, α ∨ = ω 0 , α ∨ for all α ∈ S. Here, π i ∈ Ξ ⊗ Q denotes the i-th fundamental weight.6.9.Example.The primitive diagrams for A In these cases, we have P = A and ω = 1 2 α 1 , α 1 , 2α 1 , 1 2 α 1 , α 1 , respectively (where S = {α 0 , α 1 }).
The conditions defining a primitive spherical diagram D have been shown to be necessary but it is not clear whether each of them can be realized by a (quasi-)Hamiltonian manifold M .And if so, how unique is M ?We answer these questions in Theorem 6.12 below.To state it we need more notation.6.10.Definition.
(1) For a finite root system Φ let π α be the fundamental weight corresponding to α ∈ S.
(2) If Φ is affine and irreducible (hence A is a simplex) let P α ∈ A be the vertex of A with α(P α ) > 0.
Let D be a local diagram = a 0 .An inspection of Table 8.2 shows that the pairing ω, α ∨ is in fact independent of α ∈ S (actually only α n≥2 and [ 1 2 ]d n≥2 have to be checked).The common value will be denoted by n D .Here is a list: 6.12.Theorem.Let K be simply connected (also in the Hamiltonian case) and let D = (∅) be a primitive diagram for (a, Φ, Ξ).
(1) If Φ is finite then D can be realized by a multiplicity free Hamiltonian manifold of rank one.This manifold is unique up to a positive factor of the symplectic structure.The momentum polytope is given by (6.9) (see Table 6.11 for n D ) if both S i are non-empty.If S 1 = ∅ and S 2 = ∅ then (6.10) X 1 = 0 and X 2 = cω.
In both cases, c is some arbitrary positive factor.(2) If Φ is infinite and irreducible then D can be realized by a unique multiplicity free quasi-Hamiltonian manifold of rank one.The momentum polytope is given by (6.11) (3) If Φ is infinite and reducible (cases A (1) 2 ) then D can be realized by a multiplicity free quasi-Hamiltonian manifold of rank one if and only if the scalar product is chosen to be the same on both factors of K, i.e., if the alcove A is a metric square.This manifold is then unique.
Proof.Let L i ⊆ K C be the (twisted) Levi subgroup having the simple roots S(i) := S \S j .We have to construct (X 1 , X 2 , ω) such that S(X i ) = S(i), X 2 − X 1 ∈ R >0 ω, and ω or −ω generates the weight monoid of a smooth affine spherical L 1 -variety or L 2 -variety, respectively.
If both D 1 and D 2 are homogeneous then there are exactly two choices for ω namely ω D 1 and ω D 2 which are related by ω D 1 + ω D 2 = 0. We claim that ω D i , α ∨ ∈ Z for all α ∈ S.This follows by inspection for α ∈ S i .From ω D 1 = −ω D 2 we get it also for α ∈ S j .Since K is simply-connected, the weights ω D i are integral, i.e., ω ∈ Ξ.
If both D i are inhomogeneous then we need a weight ω with ω, α ∨ D 1 = 1, ω, α ∨ D 2 = −1, and ω, α ∨ = 0 otherwise.If Φ is finite then ω exists and is unique since S is a basis of Ξ ⊗ Q.If S is affine and irreducible then ω exists and is unique because of condition (3)c) of Definition 6.5.In both cases ω is integral.The case of reducible affine root systems will be discussed at the end.This settles the reconstruction of ω.It remains to construct points X 1 , X 2 ∈ A with S(X 1 ) = S(1), S(X 2 ) = S(2) and X 2 − X 1 ∈ R >0 ω.These boil down to the following set of linear (in-)equalities (where the last column just records the known behavior of ω): The inequalities (6.12) for X 1 define the relative interior of a face of the alcove A (observe that S 2 = ∅ if Φ is affine).The first and the third set of inequalities for X 2 then follow from (6.13).Inserting (6.13) into the second set we get equalities for X 1 and c: Define the affine linear function α ∨ := 2 α α.Then (6.14) is equivalent to (6.15) α ∨ (X 1 ) = cn D 2 > 0 for all α ∈ S 2 .
This already shows assertion (1) of the theorem.Now assume that Φ is affine and irreducible.Then there is the additional relation (6.16) Setting X = X 1 , we get (6.17) This means that c is unique and positive.From (6.15) we get (6.18) Evaluation of (6.16) at X = P α yields α ∨ (P α ) = ε k(α ∨ ) .To obtain (6.11) just observe that S 2 has either one or two elements.
Finally, assume Φ is reducible.The mixed types "finite times infinite" do not appear in our context.For the two other cases, the existence of ([X 1 , X 2 ], ω) is clear from the following graphics.In particular, they show why A must be a metric square.
One can combine the classification of primitive diagrams with the Reduction Lemma 4.12.For the formulation of the lemma, we define in case Φ is an irreducible affine root system.
6.15.Definition.Assume Φ is finite or irreducible.A spherical diagram is a 6-tuple (S, S c , S 1 , c 1 , S 2 , c 2 ) with: (4) Assume D 1 and D 2 are both inhomogeneous with α i := α D i .Assume also that Φ is affine and irreducible.Then Again, the point of the definition is: 6.17.Lemma.Let (X 1 , X 2 , ω) be a spherical triple.Put S c := S \ (S(X 1 ) ∪ S(X 2 )) and let (S i , S i , c i ) be the local diagram at X i .Then (S, S c , S 1 , c 1 , S 2 , c 2 ) is a spherical diagram.
Proof.The two last conditions (3) and (4) of Definition 6.15 make sure that there exists ω ∈ Ξ which gives rise to the appropriate local model at X i (for (4), see the argument in the proof of Lemma 6.17).We show the existence of a matching polytope first in the finite case.In this case, we may assume that all roots α = α are linear and S is linearly independent.Additionally to the inequalities (6.12) for α ∈ S c we get the inequalities α(X 1 ), α(X 2 ) > 0 for α ∈ S c .Because of linear independence, the values α(X 1 ) with α ∈ S can be prescribed arbitrarily.By Theorem 6.12, we can do that in such a way that all inequalities for α ∈ S c are satisfied.Now we choose α(X 1 ) > > 0 for all α ∈ S c .Since c in (6.13) is not affected by this choice, this yields α(X 2 ) > 0 for all α ∈ S c , as well.Now let Φ be affine, irreducible.If S c = ∅ then the spherical diagram would be in fact primitive.This case has been already dealt with.So let S c = ∅ and fix α 0 ∈ S c .Then S f := S \ {α 0 } generates a finite root system.We may assume that all α ∈ S f are linear.Then the existence of a solution (X 1 , X 2 ) satisfying all inequalities for all α ∈ S f has been shown above.Now observe that the set of these solutions form a cone.If we choose a solution sufficiently close to the origin we get α 0 (X 1 ), α 0 (X 2 ) > 0. Then S c = {α 3 }, S 1 = {α 5 , α 6 , α 7 } and S 2 = {α 4 , α 5 }.

6.20.
Example.In addition to the primitive diagrams of Example 6.9, the root system A (1) 1 supports the following spherical diagrams: (6.23)If one identifies the alcove A with the interval [0, 1] then P = [t, 1] in the first three cases, P = [0, t] in cases 4 through 6 and P = [t 1 , t 2 ] in the last case where 0 < t < 1 and 0 < t 1 < t 2 < 1 are arbitrary.6.21.Example.Up to automorphisms, all spherical diagrams supported on A (1) 2 are listed in the top row of The bottom row lists the corresponding momentum polytopes.Observe that each simple root of a Dynkin diagram in the first row corresponds to the opposite edge of the alcove below it.6.22.Remark.If Φ is a product of more than one affine root system then there are problems with the metric of A as we have already seen for the primitive case where A must be a metric square.We have not explored this case in full generality.7. Verification of Theorem 6.8 Recall i ∈ {1, 2} and j := 3 − i.Let (S, S 1 , c 1 , S 2 , c 2 ) be a primitive diagram.Recall that S i is the connected closure of S i in S \ S j .Put (7.1) S p 0 := S 1 ∩ S 2 .Observe that these are exactly the simple roots that are not decorated in the diagram.Our strategy is to construct S by gluing S 1 and S 2 along S p 0 .For this we have to make sure that S i remains the connected closure of S i .7.1.Lemma.Let S be a graph with subsets S 1 , S 1 , S 2 , S 2 such that S i ⊆ S i ⊆ S \ S j .Assume that S = S 1 ∪ S 2 and S i = C(S i , S i ).Then the following are equivalent: (1) S i is the connected closure of S i in S \ S j .
(2) The following two conditions hold: a) S 1 ∩ S 2 is the union of connected components of Proof.Because of S i = C(S i , S i ), the assertion S i = C(S i , S \ S j ) means that there are no edges between S i and This statement breaks up into two parts: There are no edges between S i ∩ S j and (S j \ S j ) \ (S i ∩ S j ) which is just condition (2)a).And there are no edges between S i \ S j and (S j \ S i ) \ S j which is just condition (2)b).
7.1.The case S p 0 = ∅.We start our classification with: 7.2.Lemma.Let D be a primitive spherical diagram with S p 0 = ∅ such that there is at least one edge between S 1 and S 2 .Then D ∼ = A (1) n≥2 (aa): Proof.Let α ∈ S p 0 and let there be an edge between α 1 ∈ S 1 and α 2 ∈ S 2 .Inspection of Table 8.2 implies that there are paths from α 1 and α 2 to α respectively.Together with the edge they form a cycle in S which implies that S is of type A (1) n≥2 with α 1 , α 2 being adjacent.Therefore the local diagrams are either of type a m or a m .Since only diagrams of the latter type can be glued such that S p 0 = ∅ and S is a cycle we get A (1) n≥2 (aa) as the only possibility.Thus, we may assume from now on that S is the union of S 1 and S 2 minimally glued along S p 0 , i.e., with no further edges added.To classify these diagrams we proceed by the type of S p 0 .Helpful is the following table which lists for all isomorphy types of S p 0 the possible candidates for S 1 and S 2 .The factor c is suppressed.

Table. Gluing data
7.4.Remark.For c 3 , the graph S \ S consists of two A 1 -components.Therefore c 3 is listed twice in the A 1 -row.Also the case S p 0 = D 4 is listed separately since D 4 has automorphisms which don't extend to D 5 .
Using the table, it easy to find all primitive triples with S p 0 = ∅.Since the case-by-case considerations are lengthy we just give an instructive example namely where S p 0 = A 2 .Here the following nine combinations have to be considered: We omitted the possibility of a factor of 1 /2 for b 3 .Observe that in three cases two different gluings are possible.It turns out that all cases lead to a valid spherical diagram (namely D  8.3) except for a 4 ∪ b 3 and one of the gluings of a 3 ∪ b 3 which do not lead to affine Dynkin diagrams.
7.2.The case S p 0 = ∅.Here, according to Lemma 7.1, S is the disjoint union of S 1 and S 2 stitched together with edges between S 1 and S 2 .
A rather trivial subcase is when S 2 = S 2 = ∅.Then D is just a local diagram all of which appear in Table 8.3.
Therefore, assume from now on that S 1 , S 2 = ∅.Since then 1 ≤ |S i | ≤ 2, the number N of edges between S 1 and S 2 is at most 4.This yields 5 subcases.7.2.1.N = 0.In this case, S is the disconnected union of S 1 and S 2 .If the triple were bihomogeneous we cannot have ω D 1 + ω D 2 = 0.If D 1 were homogeneous and D 2 inhomogeneous then ω D 1 , α ∨ D 2 = 0 = −1.Therefore, the triple is bi-inhomogeneous and we get the three items A m≥1 × A n≥1 , A m≥1 × C n≥2 , and C m≥2 × C n≥2 near the end of the table .7.2.2.N = 1.In this case the diagram D is the disjoint union of two local diagrams connected by one edge between some α 1 ∈ S 1 and α 2 ∈ S 2 .One can now go through all pairs of local diagrams and all possibilities for the connecting edge.This works well if one or both local diagrams are of type (d 2 ).Otherwise, it is easier to go through all possible connected Dynkin diagrams for S and omit one of its edges.The remaining diagram admits very few possibilities for D 1 and D 2 .This way one can check easily that the table is complete with respect to this subcase.Let's look, e.g., at S = F (1) 4 (7.5) Omitting one edge yields (7.6) Each component has to support a local diagram such that the circled vertex is in S .This rules out all cases except the third one where the local diagrams could be of type a n or a n .This yields (7.7) The first diagram violates Definition 6.5(3)c), the second is primitive and is contained in Table 8.3, the third violates (3)b), and the fourth violates (3)a).

(ag)
ω 1 = α 0 ω 2 = 2α 1 + α 2 2 (ad) The finite simple cases (bc) The reducible cases (1) 2 (da) ω 1 = α 1 + α 0 (5.2) S := {α ∈ S | ω, α ∨ > 0} are exactly those which are decorated.All other simple roots α satisfy ω, α ∨ = 0. Another inspection shows that the diagram of a local model is almost uniquely encoded by the pair (S, S ).What is getting lost is a factor c of 1 /2, 1 or 2, and the cases a 1 and a 1 become indistinguishable.So we assign the formal symbol c = i to the inhomogeneous cases.This way, the local model is uniquely encoded by the triple (S, S , c) with c ∈ { 1 /2, 1, 2, i} which triggers the following 5.3.Definition.A local diagram is a triple D = (S, S , c) associated to a local model in Table 8.2.In the homogeneous case, let ω D be the weight given in column 4. If D is inhomogeneous and S is non-empty then α D denotes the unique element of S .Moreover, we put α ∨ D := α D ∨ .

A
spherical diagram is drawn like a primitive diagram where the roots α ∈ S c are indicated by circling them.6.19.Example.Consider the diagram on D 7 : (6.22) ), D 5 (aa), D 4 (aa), A 4 (aa ), B 4 (b a) and D (2) 3 (b b ) in the notation of Table local diagrams are both d 2 .This yields A (1) 3 (dd).This concludes the verification of the table.

Table .
Primitive diagrams