(Quasi-)Hamiltonian manifolds of cohomogeneity one
((QUASI-)HAMILTONIAN MANIFOLDS OF COHOMOGENEITY ONE
FRIEDRICH KNOP AND KAY PAULUS
Abstract.
We classify compact, connected Hamiltonian and quasi-Hamiltonian man-ifolds of cohomogeneity one (which is the same as being multiplicity free of rank one).Here 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 mul-tiplicity free manifolds in the special case of rank one. As a result we obtain numerousnew concrete examples of multiplicity free quasi-Hamiltonian manifolds or, equivalently,Hamiltonian loop group actions. Introduction
Let K be a compact connected Lie group. The hierarchy of Hamiltonian K -manifoldsis parameterized by the half-dimension c of their symplectic reduction. Most basic aretherefore manifolds with c = 0 which are called multiplicity free .Multiplicity free manifolds have a very rigid but non-trivial structure. This makes themamenable to classification. In fact, such a classification was achieved by the first namedauthor in [Kno11]. The classification is in terms of pairs ( P , Γ) where P is a convexpolytope and Γ is a lattice. Unfortunately, the compatibility conditions between P and Γ are quite intricate. More precisely, they are expressed in terms of the existence of certainspherical varieties. This makes it a non-trivial task to produce examples of multiplicityfree manifolds using the classification.On the other hand, Alekseev-Malkin-Meinrenken introduced in [AMM98] the concept of quasi-Hamiltonian manifolds . These share many features with Hamiltonian manifolds,e.g., the concept of symplectic reductions. The main difference is that the momentum maphas values in K instead of the coadjoint representation. Loosely speaking, Hamiltonianmanifolds relate to quasi-Hamiltonian ones as finite root systems to affine root systems .The notion of multiplicity free quasi-Hamiltonian manifolds makes sense and in [Kno16]the methods of [Kno11] were adapted to classify compact multiplicity free quasi-Hamiltonianmanifolds (provided K is simply connected). This classification suffers from the same dis-advantage as the one before which is unfortunate since few quasi-Hamiltonian manifoldsare known.This quandary provides the main motivation for the present paper, namely to producea subclass of multiplicity-free (quasi-)Hamiltonian manifolds which can actually be de-scribed in terms of lists and tables. Here, we chose the case where the polyhedron P hasthe lowest reasonable dimension namely one.As it turns out, multiplicity free (quasi-)Hamiltonian manifolds with dim P = 1 areprecisely those of cohomogeneity . This is kind of a coincidence since otherwise thecohomogeneity does not seem to be a very useful invariant of Hamiltonian manifolds. The twisted affine root systems appear through quasi-Hamiltonian manifolds which are twisted byan automorphism of K . In this paper, we work in this more general setting. a r X i v : . [ m a t h . S G ] O c t he classification proceeds in two steps. First, there is a kind of induction procedurewhich produces (quasi-)Hamiltonian K -manifolds from Hamiltonian L -manifolds where L ⊆ K is a (twisted) Levi subgroup. Manifolds which are not induced from smaller onesare called primitive .In a second step, we determine all primitive manifolds. They are listed in Table 8.3. Fora concise description we use a graphic notation which is very close to that of Luna whichhe introduced in [Lun01] to describe spherical varieties.1.1. Theorem.
Let M be a primitive, multiplicity free, (quasi-)Hamiltonian manifold.Then M corresponds to a diagram in Table 8.3. Moreover, to each diagram there corre-sponds a manifold which is either unique (in the quasi-Hamiltonian case) or unique upto rescaling the symplectic structure (in the Hamiltonian case). Do we get new quasi-Hamiltonian manifolds this way? First of all, we recover a couple ofcases which were already known like the spinning -sphere of Hurtubise-Jeffrey [HJ00],Alekseev-Malkin-Woodward [AMW02]) and its generalization, the spinning n -sphere, byHurtubise-Jeffrey-Sjamaar [HJS06]. We also find the Sp(2 n ) -action on P n H by Eshmatov[Esh09].The product of two symmetric spaces K/H × K/H with diagonal action was calleddisymmetric in [Kno16]. It follows easily from the techniques in [AMM98] that all disym-metric manifolds are (possibly twisted) quasi-Hamiltonian. It turns out that most “biho-mogeneous” diagrams for affine root systems are of this type.The Hamiltonian items from the list all seem originate from smooth projective sphericalvarieties of rank one. These are all known, in principle. See also work of Lê [Le98] forstructural results in this case.Taking these cases into account, we are left with affine diagrams which have apparentlynot appeared in the literature. One of them is a generalization of Eshmatov’s examplenamely a quasi-Hamiltonian action of K = Sp(2 n ) on the quaternionic Grassmannians Gr k ( H n +1 ) .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 andWolfgang Ruppert for many explanations, discussions and remarks.2.
Hamiltonian and quasi-Hamiltonian manifolds
We first recall the most important properties of Hamiltonian and quasi-Hamiltonian man-ifolds.In the entire paper, K will be a compact connected Lie group with Lie algebra k . A Hamiltonian K -manifold is a triple ( M, ω, m ) where M is a K -manifold, ω is a K -invariant symplectic form on M and m : M → k ∗ is a smooth K -equivariant map (the momentum map ) such that(2.1) ω ( ξx, η ) = (cid:104) ξ, m ∗ ( η ) (cid:105) , for all ξ ∈ k , x ∈ M, η ∈ T x M. In [AMM98], Alekseev, Malkin and Meinrenken studied this concept in the context of loopgroups. Even though they are infinite dimensional, they managed to reduce Hamiltonian oop 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, ω, m ) where M is a K -manifold, ω is a K -invariant -form and m is a K -equivariant map. But there aredifferences.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 (which may be the identity). Themomentum 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 ) − . Finally, the closedness andnon-degeneracy of ω as well as formula (2.1) have to be adapted. For the details one canconsult the papers [AMM98], [Kno16], or [Mei17]. They are not relevant for the presentpaper.In the following, we want to treat the Hamiltonian and the quasi-Hamiltonian on the samefooting. So we talk about U -Hamiltonian manifolds where U = k ∗ in the Hamiltonianand U = K in the quasi-Hamiltonian case.This momentum map m : M → U gives rise to a map between orbits spaces:(2.2) m/K : M/K → U/K.
By definition, the fibers of this map are the symplectic reductions of M . The smoothones 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 . These manifolds are called multiplicity free .An important invariant of M is its momentum image ( m/K )( M ) ⊆ U/K . Its dimensionis called the rank of M . Multiplicity free manifolds of rank are simply the K -orbits in U . In this paper we study the next more difficult case namely multiplicity free manifoldsof 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 .(2) M is multiplicity free of rank .Proof. Let c := dim (( m/K ) − ( 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 multiplicityfreeness. Let r be the rank of M . Then we have(2.3) dim M/K = 2 c + r. Hence, dim
M/K = 1 if and only if c = 0 and r = 1 . (cid:3) The original paper [AMM98] deals only with the untwisted case τ = id K . The straightforwardadaption to the twisted case has been carried out independently in [BoYa15], [Kno16], and [Mei17]. . Affine root systems
Before we go on with explaining the general structure of U -Hamiltonian manifolds weneed to set up notation for finite and affine root systems. Here, we largely follow theexposition in [Kno16] which is in turn based on [Mac72] and [Mac03].Let a be a Euclidean vector space, i.e., a finite dimensional real vector space equippedwith a scalar product (cid:104)· , ·(cid:105) . Let a be an affine space for a , i.e., a is equipped with a freeand transitive a -action. We denote the set of affine linear functions on a by A ( a ) . Thegradient of a function α ∈ A ( a ) is the element α ∈ a with(3.1) α ( X + t ) = α ( X ) + (cid:104) α, t (cid:105) , X ∈ a , t ∈ a . A reflection s is an isometry of a whose fixed point set is an affine hyperplane. If thathyperplane 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 ∨ = α || α || .3.1. Definition. An affine root system on a is a subset Φ ⊂ A ( a ) such that: (1) R ∩ Φ = ∅ (in particular (cid:54)∈ Φ ), (2) s α (Φ) = Φ for all α ∈ Φ , (3) (cid:104) β, α ∨ (cid:105) ∈ Z for all α, β ∈ Φ , (4) the Weyl Group W = (cid:104) s α , α ∈ Φ (cid:105) acts properly discontinuously on a , (5) R α ∩ Φ = { + α, − α } for all α ∈ Φ .Observe that, with our definition, Φ might be finite or even empty. In that case, theroots 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 ) = (cid:104) α, X (cid:105) for all roots α .If ( a , Φ ) , . . . , ( a s , Φ s ) are affine root systems then(3.2) ( a , Φ ) × . . . × ( a s , Φ s ) := ( a × . . . × a s , p ∗ Φ ∪ . . . ∪ p ∗ s Φ s ) is also one (where the p i are the projections). Conversely, every affine root system admitssuch a decomposition such that the Weyl group W i of Φ i is either trivial or acts irreduciblyon a i . We say that Φ is properly affine if each irreducible factor Φ i is infinite.A chamber of Φ is a connected component of a \ (cid:83) α ∈ Φ { α = 0 } . The closure A of achamber is called an alcove . If Φ is finite then A is called a Weyl chamber . If Φ isirreducible then A is either a simplicial cone if Φ is finite or a simplex if Φ is properlyaffine.A root α ∈ Φ is called simple with respect to an alcove A if A ∩ { α = 0 } is a wall of A .The set of simple roots (for a fixed alcove) will be denoted by S .Put Φ := { α | α ∈ Φ } and Φ ∨ := { α ∨ | α ∈ Φ } . These are possibly non-reduced finiteroot 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 affineroot system and Ξ ⊆ a is a lattice with Φ ⊆ Ξ and (cid:104) Ξ , Φ ∨ (cid:105) ⊆ Z . The integral root systemis simply connected if Ξ = { ω ∈ a | (cid:104) ω, Φ ∨ (cid:105) ⊆ Z } .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 s given. These labels are uniquely characterized by being integral, coprime, and havingthe property that(3.3) δ := (cid:88) α ∈ S k ( α ) α is a positive constant function.4. Classification of multiplicity free Hamiltonian andquasi-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 finiteroot 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 isdetermined 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) ψ : k ∗ = k exp → K = U In the Hamiltonian case, we put for compatibility reasons ψ = id k ∗ . Likewise, we assumethat a scalar product has been selected on k even though the results will not dependenton it.4.1. Theorem.
Let
K, U be as above. Then there is a subspace a ⊆ k ∗ and an integralroot 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 (4.2) K a = { k ∈ K | k · a = a } is connected, a ⊆ k a is a Cartan subalgebra, the weight lattice of K a is Ξ , and (4.3) S ( X ) := { α ∈ S | α ( X ) = 0 } , is a set of simple roots of K a . Here S ⊂ Φ is the set of simple roots with respectto 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 invariantmomentum map is the composition(4.4) m + : M m → U → U/K ∼ → A ⊆ a . Its image P M := m + ( M ) ⊆ A can be shown to be a convex polytope, 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 the isotropy group K a actson the momentum fiber m − ( 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 . .2. Theorem ([Kno11],[Kno16]) . Let M and M be two compact, connected multiplicityfree U -Hamiltonian manifolds with P M = P M and Γ M = Γ M . Then M and M areisomorphic as U -Hamiltonian manifolds. This begs the question which pairs ( P , Γ) arise this way. The key to the answer lies inthe paper [Bri87] of Brion which connects the theory of multiplicity free Hamiltonianmanifolds with the theory of complex spherical varieties. In the following we summarizeonly a simplified version which suffices for our purposes.We start with a connected, reductive, complex group G . An irreducible algebraic G -variety Z is called spherical if a Borel subgroup of G has an open orbit. Now assume alsothat Z is affine and let C [ Z ] be its ring of regular functions. Then the Vinberg-Kimelfeldcriterion [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) Λ Z of dominant integralweights of G such that(4.5) C [ Z ] ∼ = (cid:77) χ ∈ Λ Z V χ where V χ is the simple G -module of highest weight χ . A theorem of Losev [Los06] assertsthat 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 canbe equipped with the structure of a Hamiltonian K -manifold by embedding Z into afinite 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. (2) P Z = R ≥ Λ Z (the convex cone generated by Λ Z ). (3) Γ Z = Z Λ Z (the group generated by Λ Z ).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. 8.6(3)].4.3. Remark.
It follows from the normality of Z that conversely(4.6) Λ Z = P Z ∩ Γ Z . 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 ≥ Λ Z and Γ = Z Λ Z . The (unique) 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) T X P := R ≥ ( P − X ) . 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 for a := ψ ( X ) ∈ U . The model variety for ( T X P , Γ) will be called the local model of ( P , Γ) in X . 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 dimensioninside the affine subspace X + (cid:104) Γ (cid:105) R ⊆ a for any X ∈ P .The classification theorem can now be stated as follows:4.7. Theorem ([Kno11],[Kno16]) . Let K be a connected compact Lie group which isassumed to be simply connected in the quasi-Hamiltonian case. Then the map M (cid:55)→ ( P M , Γ M ) induces a bijection between(1) isomorphism classes of compact, connected multiplicity free U -Hamiltonian man-ifolds and(2) locally spherical pairs ( P , Γ) where P ⊆ A is a compact convex polyhedron and Γ ⊆ Ξ is a subgroup. Remark.
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 of X in P such that(4.8) M ∼ = K × K a Z where M = m − ( P ) , a = ψ ( X ) , and Z ⊆ Z is a K a -stable open subset of the localmodel Z in X .Because of this theorem, we are going to work from now on exclusively on the “combina-torial side”, i.e., with locally spherical pairs. Two reduction steps are immediate.Let(4.9) S ( P ) := { α ∈ S | α ( X ) = 0 for some X ∈ P} . Thus, elements of S ( P ) correspond to walls of A which contain a point of P . Let K := 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 . This observation reducesclassifications largely to pairs with S ( P ) = S . Such pairs will be called genuine .After this reduction, another one is possible. Assume S ⊆ S is a component of theDynkin diagram of S . It corresponds to a (locally) direct semisimple factor L S of G = K C . Suppose also that S ⊆ S ( X ) for all X ∈ P . Then it follows from Remark 4.6that (cid:104) Γ , S (cid:105) = 0 . In turn (4.5) implies that L S will act trivially on every local model Z of ( P , Γ) . This means that also the roots in S can be ignored for determining thesphericality of ( P , Γ) .4.9. Definition.
A genuine polyhedron
P ⊆ A is called primitive if S does not containa component S with S ⊆ S ( X ) for all X ∈ P .The following lemma summarizes our findings4.10. Lemma.
Let
P ⊆ A be a compact convex polyhedron and let be Γ ⊆ Ξ a subgroup.Let (4.10) S c := { α ∈ S | α ( X ) (cid:54) = 0 for all X ∈ P} and let S be the union of all components C of S \ S c with C ⊆ S ( X ) for all X ∈ P . Let Ξ := Ξ ∩ S ⊥ . Then P is primitive for S := S \ ( S c ∪ S ) . Moreover, the pair ( P , Γ) islocally spherical for ( S, Ξ) if and only if it is so for ( S, Ξ) . he existence of a spherical model Z turns out to be quite intricate. There is an algorithmdue to Pezzini-Van Steirteghem [PVS19] but it is fairly involved. The purpose of thispaper is to present a complete classification of locally spherical pairs in the special casewhen rk M = dim P = 1 .In this case the following simplifications occur: the polyhedron P is a line segment P = [ X , X ] with X , X ∈ A and Γ = Z ω with ω ∈ Ξ . It follows from Remark 4.6 that(4.11) X = X + cω for some c (cid:54) = 0 . By replacing ω by − ω if necessary, we may assume that c > . Then Theorem 4.7 boilsdown to:4.11. Corollary.
The map M (cid:55)→ ( P M , Γ M ) = ([ X , X ] , Z ω ) induces a bijection between(1) isomorphism classes of compact, connected multiplicity free U -Hamiltonian man-ifolds of rank one and(2) triples ( X , X , ω ) satisfying (4.11) such that N ω is the weight monoid of a smoothaffine spherical K C S ( X ) -variety Z and N ( − ω ) is the weight monoid of a smoothaffine spherical K C S ( X ) -variety Z . The triples ( X , X , ω ) and ( X , X , − ω ) areconsidered equal. Triples as above will be called spherical . A triple is genuine or primitive if P = [ X , X ] has this property. The varieties Z i are called the local models of the triple.5. 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) C [ Z ] = (cid:77) n ∈ Λ V nω , where ω is a non-zero integral dominant weight, V nω is the simple L -module of highestweight nω , and Λ equals either N or Z .The case Λ = Z is actually irrelevant for our purposes since this case only occurs as localmodel 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 ofthe following cases holds:(1) Z = C ∗ and L acts via a non-trivial character.(2) Z = L /H where ( L , H ) appears in in the first part of Table 8.2 and L acts viaa surjective homomorphism ϕ : L → L .(3) Z = V where ( L , V ) appears in the second part of Table 8.2 and L acts via ahomomorphism ϕ : L → L which is surjective modulo scalars (except for case a 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 argumentgoes 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 → L/H . The rank of the homogeneous space
L/H is at most the rank of Z , so either or .If it is , then L/H is projective (see, e.g., [Tim11, prop. 10.1]), but, being also affine, itis a single point, i.e., L = H . We deduce Z = V , i.e., Z is a spherical module of rank .The classification of spherical modules (Kac [Kac80], see also [Kno98]), yields the casesin (3) .Assume now that L/H has rank . 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 ofhomogeneous spherical varieties of rank one (Akhiezer [Akh83], see also [Bri89], andWasserman [Was96]), yields the cases in (1) and (2) . Observe, that the non-affine casesof Akhiezer’s list have been left out. (cid:3)
Remark.
Some remarks concerning Table 8.2: (1)
Observe that items [ ] d and [ ] d could be made part of the series [ ] d n . Becauseof their singular behavior we chose not to do so. For example both can be em-bedded into A n -diagrams. Moreover, [ ] d are the only cases with a disconnectedDynkin 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 inhomo-geneous ones are specific to our situation. (3)
For a homogeneous model the weight ω is a fixed linear combination of simpleroots (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 uniqueup to a character. This is indicated by the notation ω ∼ π which means that (cid:104) w, α ∨ (cid:105) = 1 for α = α and = 0 otherwise.Let S be the set of simple roots of L , i.e., the set of vertices of a diagram. Then inspectionof Table 8.2 shows that the elements of(5.2) S (cid:48) := { α ∈ S | (cid:104) ω, α ∨ (cid:105) > } are exactly those which are decorated. All other simple roots α satisfy (cid:104) ω, α ∨ (cid:105) = 0 .Another inspection shows that the diagram of a local model is almost uniquely encodedby the pair ( S, S (cid:48) ) . What is getting lost is a factor c of / , or , and the cases a and a become indistinguishable. So we assign the formal symbol c = i to the inhomogeneouscases. This way, the local model is uniquely encoded by the triple ( S, S (cid:48) , c ) with c ∈{ / , , , i } which triggers the following5.3. Definition. A local diagram is a triple D = ( S, S (cid:48) , c ) associated to a local model inTable 8.2. In the homogeneous case, let ω D be the weight given in column 4. If D isinhomogeneous and S is non-empty then α D denotes the unique element of S (cid:48) . Moreover,we put α ∨D := α D∨ . 6. The classification
Let ( X , X , ω ) be a primitive spherical triple. Then we obtain two local models Z , Z which determine two local diagrams D = ( S , S (cid:48) , c ) , D = ( S , S (cid:48) , c ) where S , S ⊆ S .Put(6.1) S p ( ω ) := { α ∈ S | (cid:104) ω, α ∨ (cid:105) = 0 } . .1. Lemma.
Let ( X , X , ω ) be a primitive spherical triple. Then (6.2) S ( X ) ∪ S ( X ) = S and S ( X ) ∩ S ( X ) = S p ( ω ) . Proof.
The first equality holds because the triple is genuine. The inclusion S ( X ) ∩ S ( X ) ⊆ S p ( ω ) follows directly from (4.11). Assume conversely that α ∈ S p ( ω ) . Withoutloss of generality we may assume that also α ∈ S ( X ) . But then also α ∈ S ( X ) by(4.11). (cid:3) From now let i ∈ { , } and j := 3 − i , so that if Z i is a local model then Z j is the other.6.2. Lemma.
Let ( X , X , ω ) be a primitive spherical triple and D , D as above. Then (6.3) S = S (cid:48) ˙ ∪ S p ( ω ) ˙ ∪ S (cid:48) Moreover, (6.4) S ( X i ) = S (cid:48) i ˙ ∪ S p ( ω ) = S \ S (cid:48) j . Proof.
It follows from Theorem 5.1 that every α ∈ S ( X i ) (a simple root of L ) is either in S i (a simple root of L ) or a simple root of ker ϕ . In the latter case, we have (cid:104) ω, α ∨ (cid:105) = 0 .Now let α ∈ S and assume first (cid:104) ω, α ∨ (cid:105) > . Since the triple is genuine we have S = S ( X ) ∪ S ( X ) . If α ∈ S ( X ) then actually α ∈ S . This contradicts (cid:104)− ω, α ∨ (cid:105) ≥ forall α ∈ S . Thus α ∈ S ( X ) . By the same reasoning we have α ∈ S . But then α ∈ S (cid:48) by (5.2).Analogously, (cid:104) ω, α ∨ (cid:105) < implies α ∈ S (cid:48) . This proves (6.3). The second equality (6.4)now follows from Lemma 6.1. (cid:3) Definition.
Let S (cid:48) be a subset of a graph S . The connected closure C ( S (cid:48) , S ) of S (cid:48) in S is the union of all connected components of S which meet S (cid:48) . In other words, C ( S (cid:48) , S ) is the set of vertices of S for which there exists a path to S (cid:48) .The following lemma shows in particular how to recover S i from the triple ( S, S (cid:48) , S (cid:48) ) .6.4. Lemma.
Let ( X , X , ω ) be primitive. Then(1) S i is the connected closure of S (cid:48) i in S \ S (cid:48) j .(2) S is the connected closure of S (cid:48) ∪ S (cid:48) .(3) S = S ∪ S .Proof. (1) Recall that S \ S (cid:48) j = S ( X i ) is the disconnected union of S i and the Dynkindiagram C i of ker ϕ . Inspection of Table 8.2 shows that S i is the connected closure of S (cid:48) i . (2) Let C ⊆ S be a component with C ∩ ( S (cid:48) ∪ S (cid:48) ) = ∅ . Then C ⊆ S p ( ω ) = S ( X ) ∩ S ( X ) in contradiction to primitivity. (3) By (1) , the connected closure of S (cid:48) ∪ S (cid:48) in S is S ∪ S . (cid:3) Definition.
Let D be quintuple a D = ( S, S (cid:48) , c , S (cid:48) , c ) where S is a (possibly empty)Dynkin diagram, S (cid:48) , S (cid:48) are disjoint (possibly empty) subsets of S and c , c ∈ { , , , i } .Let S i is the connected closure of S (cid:48) i in S \ S (cid:48) j . Then D is a primitive spherical diagram if it has following properties: (1) S = S ∪ S . (2) The triples D i := ( S i , S (cid:48) i , c i ) are local diagrams. (3) a) If both D i are homogeneous then ω D + ω D = 0 . ) If D i is homogeneous and D j is inhomogeneous with S j (cid:54) = ∅ then (cid:104) ω D i , α ∨D j (cid:105) = − . c) If both D i are inhomogeneous with both S i (cid:54) = ∅ and S is affine and irreduciblethen k ( α ∨D ) = k ( α ∨D ) where k ( α ∨ ) is the colabel of α , i.e., the label of α ∨ inthe dual diagram of S .A primitive spherical diagram can be represented by the Dynkin diagram of S withdecorations as in Table 8.2 which indicate the subsets S (cid:48) i and the numbers c i .6.6. Example.
Consider the following diagram on F (1)4 :(6.5) / It represents the quintuple with S (cid:48) = { α } , c = / , S (cid:48) = { α } , c = 1 . Hence S = { α , α , α } , and S = { α , α , α } . Comparing with Table 8.2 we see that the localdiagrams are of type d and c , respectively. The diagram is bihomogeneous so we needto check condition (3)a ). Indeed(6.6) ω D + ω D = ( 12 α + α + 12 α )+( α +2 α + α ) = 12 ( α +2 α +3 α +4 α +2 α ) = 0 (compare with the labels of F (1)4 in Table 8.1). Thus, the above diagram is primitivespherical.The point of Definition 6.5 is of course:6.7. Corollary.
Let ( X , X , ω ) be a primitive spherical triple with local diagrams S , S (cid:48) , c and S , S (cid:48) , c ) . Then ( S, S (cid:48) , c , S (cid:48) , c ) is a primitive spherical diagram.Proof. All conditions have been verified except for (3) c ). If S is affine then the corootssatisfy the linear dependence relation(6.7) (cid:88) α ∈ S k ( α ∨ ) α ∨ = 0 . We pair this with ω and observe that (cid:104) ω, α ∨ (cid:105) = 1 , − , according to α = α D , α = α D or otherwise. This implies the claim. (cid:3) The following is our main result. It will be proved in the next section.6.8.
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 foridentification purposes the type of the local models. The diagram is given in the fifthcolumn. If a parameter is involved, its scope is given in the last column. Observe theboundary cases where we used the conventions b = a , b = 2 a , c = b , and c = a . Insome cases, besides ( S, S (cid:48) , c , S (cid:48) , c ) also ( S, S (cid:48) , c c , S (cid:48) , c c ) is primitive spherical where c is the factor in the column “factor”. An entry of the form [ c ] n = a indicates that the factorapplies only to the case n = a .Finally, the weight ω can be read off from the third column. More precisely, if D i ishomogeneous 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 , D are inhomogeneous then ω is only unique up to acharacter of G . Thus, the notation ω ∼ ω means (cid:104) ω, α ∨ (cid:105) = (cid:104) ω , α ∨ (cid:105) for all α ∈ S . Here, π i ∈ Ξ ⊗ Q denotes the i -th fundamental weight. .9. Example.
The primitive diagrams for A (1)1 and A (2)2 are(6.8) (cid:4) (cid:4) (cid:4) In these cases, we have P = A and ω = α , α , α , α , α , respectively (where S = { α , α } ).The conditions defining a primitive spherical diagram D have been shown to be necessarybut 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. Tostate it we need more notation.6.10. Definition. (1)
For a finite root system Φ let π α be the fundamental weightcorresponding 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 ) > .Let D be a local diagram (cid:54) = a . An inspection of Table 8.2 shows that the pairing (cid:104) ω, α ∨ (cid:105) is in fact independent of α ∈ S (cid:48) (actually only α n ≥ and [ ] d n ≥ have to be checked). Thecommon value will be denoted by n D . Here is a list:6.11. Table.
The numbers n D D a a a n ≥ b n ≥ b n ≥ c n ≥ d n ≥ d n ≥ f g g b (cid:48) b (cid:48) a n ≥ c n ≥ n D Theorem.
Let K be simply connected (also in the Hamiltonian case) and let D (cid:54) =( ∅ ) be a primitive diagram for ( a , Φ , Ξ) .(1) If Φ is finite then D can be realized by a multiplicity free Hamiltonian manifoldof rank one. This manifold is unique up to a positive factor of the symplecticstructure. The momentum polytope is given by (6.9) X i = c n D (cid:88) α ∈ S (cid:48) j π α (see Table 6.11 for n D ) if both S (cid:48) i are non-empty. If S (cid:48) (cid:54) = ∅ and S (cid:48) = ∅ then (6.10) X = 0 and X = 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 freequasi-Hamiltonian manifold of rank one. The momentum polytope is given by (6.11) X i = (cid:40) P α if S (cid:48) j = { α } k ( α ∨ ) k ( α ∨ )+ k ( β ∨ ) P α + k ( β ∨ ) k ( α ∨ )+ k ( β ∨ ) P β if S (cid:48) j = { α, β } . (3) If Φ is infinite and reducible (cases A (1)1 × A (1)1 and A (2)2 × A (2)2 ) then D can berealized by a multiplicity free quasi-Hamiltonian manifold of rank one if and onlyif the scalar product is chosen to be the same on both factors of K , i.e., if thealcove 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 (cid:48) j .We have to construct ( X , X , ω ) such that S ( X i ) = S ( i ) , X − X ∈ R > ω , and ω or ω generates the weight monoid of a smooth affine spherical L -variety or L -variety,respectively.If both D and D are homogeneous then there are exactly two choices for ω namely ω D and ω D which are related by ω D + ω D = 0 . We claim that (cid:104) ω D i , α ∨ (cid:105) ∈ Z for all α ∈ S .This follows by inspection for α ∈ S i . From ω D = − ω D 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 D is homogeneous and D is inhomogeneous we must put ω = ω D . By condition (2)b )of Definition 6.5 we have (cid:104)− ω, α ∨D (cid:105) = 1 . Let β ∈ S \ { α D } . Then β is not connected toany α ∈ S by Lemma 6.4 (1) . From ω ∈ Q S we get (cid:104) ω, β ∨ (cid:105) = 0 . Hence ω ∈ Ξ and both N ω and N ( − ω ) form the weight monoid of a smooth affine spherical L - or L -variety,respectively.If both D i are inhomogeneous then we need a weight ω with (cid:104) ω, α ∨D (cid:105) = 1 , (cid:104) ω, α ∨D (cid:105) = − ,and (cid:104) ω, α ∨ (cid:105) = 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 systemswill be discussed at the end.This settles the reconstruction of ω . It remains to construct points X , X ∈ A with S ( X ) = S (1) , S ( X ) = S (2) and X − X ∈ R > ω . These boil down to the following setof linear (in-)equalities (where the last column just records the known behavior of ω ):(6.12) α X X ωα ∈ S (cid:48) : α ( X ) = 0 α ( X ) > (cid:104) ω, α (cid:105) > α ∈ S (cid:48) : α ( X ) > α ( X ) = 0 (cid:104) ω, α (cid:105) < α (cid:54)∈ S (cid:48) ∪ S (cid:48) : α ( X ) = 0 α ( X ) = 0 (cid:104) ω, α (cid:105) = 0 (6.13) X = X + cω with c > . The inequalities (6.12) for X define the relative interior of a face of the alcove A (observethat S (cid:48) (cid:54) = ∅ if Φ is affine). The first and the third set of inequalities for X then followfrom (6.13). Inserting (6.13) into the second set we get equalities for X and c :(6.14) α ( X ) = c (cid:104)− ω, α (cid:105) > for all α ∈ S (cid:48) Define the affine linear function α ∨ := (cid:107) α (cid:107) α . Then (6.14) is equivalent to(6.15) α ∨ ( X ) = cn D > for all α ∈ S (cid:48) . This already shows assertion (1) of the theorem.Now assume that Φ is affine and irreducible. Then there is the additional relation(6.16) (cid:88) β ∈ S k ( β ∨ ) β ∨ ( X ) = ε ≡ const. > , X ∈ a . Setting X = X , we get(6.17) cn D (cid:88) β ∈ S (cid:48) k ( β ∨ ) = ε. This means that c is unique and positive. From (6.15) we get(6.18) α ∨ ( X ) = [ (cid:88) β ∈ S (cid:48) k ( β ∨ )] − ε valuation of (6.16) at X = P α yields α ∨ ( P α ) = εk ( α ∨ ) . To obtain (6.11) just observe that S (cid:48) has either one or two elements.Finally, assume Φ is reducible. The mixed types “finite times infinite” do not appearin our context. For the two other cases, the existence of ([ X , X ] , ω ) is clear from thefollowing graphics. In particular, they show why A must be metric square. A (1)1 × A (1)1 A (1)1 × A (1)1 A (2)2 × A (2)2 α = 0 α = 0 α = 0 α = 0 α = 0 α = 0 ω = ( α + α (cid:48) ) ω = α + α (cid:48) ω = α + α (cid:48) (cid:3) Example.
Consider the diagram D (2)4 ( dd ) (6.19)Then k ( α ∨ ) = k ( α ∨ ) = 1 and k ( α ∨ ) = k ( α ∨ ) = 2 , and ω = α + α and ω = α + α and(6.20) X = 23 P α + 13 P α , X = 13 P α + 23 P α Here is a picture of P inside A :6.14. Remark. (1) A disymmetric manifold is the product of two symmetric spaces for thesame group K with K acting diagonally. It is easy to see that all disymmetric manifoldsare twisted quasi-Hamiltonian (see [Kno16, §11]). In fact, one can show that almost allbihomogeneous primitive diagrams are of this type. The exceptions are D (2)4 ( dd ) , D (2)4 ( b (cid:48) b (cid:48) ) D (3)4 ( ag ) , and D (3)4 ( ad ) . For example the case A (1)3 ( dd ) (without the factor / ) correspondsto the disymmetric SU(4) -manifold
SU(4) SO (4) × SU(4)Sp(4) . (2) The three primitive diagrams for A (1)1 (see Example 6.9 correspond to the manifolds S (the so-called “spinning 4-sphere”, [HJ00, AMW02]), S × S and P ( C ) , respectively(see [Kno16]). (3) Generalizing the example above, the diagram A (1) n − ( aa ) with n ≥ corresponds tothe “spinning n -sphere” S n discovered by Hurtubise-Jeffrey-Sjamaar [HJS06]. (4) The cases C (1) n ≥ ( cc ) are realized by Sp(2 n ) acting on the quaternionic Grassmannians M = Gr d ( H n +1 ) (see [Kno16]). This is a generalization of a result by Eshmatov [Esh09]for d = 1 .One can combine the classification of primitive diagrams with the Reduction Lemma 4.10.For its formulation we define(6.21) k ∨ ( S c ) := gcd { k ( α ∨ ) | α ∈ S c } . n case Φ is an irreducible affine root system.6.15. Definition.
Assume Φ is finite or irreducible. A spherical diagram is a -tuple ( S, S c , S (cid:48) , c , S (cid:48) , c ) with: (1) S (cid:48) , S (cid:48) , S c ⊆ S are pairwise disjoint (2) ( S , S (cid:48) , c , S (cid:48) , c ) is a primitive diagram where S is the connected closure of S (cid:48) ∪ S (cid:48) in S \ S c . Set D i = ( S i , S (cid:48) i , c i ) where S i is the connected closure of S (cid:48) i in S \ S (cid:48) j . (3) If D i is homogeneous then (cid:104) w D i , α ∨ (cid:105) ∈ Z for all α ∈ S c . (4) Assume D and D are both inhomogeneous with α i := α D i . Assume also that Φ is affine and irreducible. Then k ∨ ( S c ) divides k ( α ∨ ) − k ( α ∨ ) .6.16. Remark.
The condition (3) is only relevant if D i is of type d n ≥ or b (cid:48) .Again, the point of the definition is:6.17. Lemma.
Let ( X , X , ω ) be a spherical triple. Put S c := S \ ( S ( X ) ∪ S ( X )) andlet ( S i , S (cid:48) i , c i ) be the local diagram at X i . Then ( S, S c , S (cid:48) , c , S (cid:48) , c ) is a spherical diagram.Proof. Only (4) needs an argument. The weight ω satisfies (cid:104) ω, α ∨D (cid:105) = 1 , (cid:104) ω, α ∨D (cid:105) = − , (cid:104) ω, α ∨ (cid:105) ∈ Z for α ∈ S c , and (cid:104) ω, α ∨ (cid:105) = 0 for all other α ∈ S . Hence (4) follows from thelinear dependence relation (6.7). (cid:3) Theorem.
Let (Φ , Ξ) be simply connected and D a non-empty spherical diagramon Φ . Let Φ be finite or affine, irreducible. Then D is realized by a spherical triple ( X , X , ω ) .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 argumentin the proof of Lemma 6.17). We show the existence of a matching polytope first in thefinite case. In this case, we may assume that all roots α = α are linear and S is linearlyindependent. Additionally to the inequalities (6.12) for α (cid:54)∈ S c we get the inequalities α ( X ) , α ( X ) > for α ∈ S c . Because of linear independence, the values α ( X ) with α ∈ S can be prescribed arbitrarily. By Theorem 6.12, we can do that in such a way thatall inequalities for α (cid:54)∈ S c are satisfied. Now we choose α ( X ) >> for all α ∈ S c . Since c in (6.13) is not affected by this choice, this yields α ( X ) > for all α ∈ S c , as well.Now let Φ be affine, irreducible. If S c = ∅ then the spherical diagram would be in factprimitive. This case has been already dealt with. So let S c (cid:54) = ∅ and fix α ∈ S c . Then S f := S \ { α } generates a finite root system. We may assume that all α ∈ S f are linear.Then the existence of a solution ( X , X ) satisfying all inequalities for all α ∈ S f hasbeen shown above. Now observe that the set of these solutions form a cone. If we choosea solution sufficiently close to the origin we get α ( X ) , α ( X ) > . (cid:3) A spherical diagram is drawn like a primitive diagram where the roots α ∈ S c are indicatedby circling them.6.19. Example.
Consider the diagram on D :(6.22) (cid:4) Then S c = { α } , S = { α , α , α } and S = { α , α } . .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) (cid:4) (cid:4) If one identifies the alcove A with the interval [0 , then P = [ t, in the first three cases, P = [0 , t ] in cases 4 through 6 and P = [ t , t ] in the last case where < t < and < t < t < are arbitrary.6.21. Example.
Up to automorphisms, all spherical diagrams supported on A (1)2 are listedin the top row ofThe bottom row lists the corresponding momentum polytopes. Observe that each simpleroot of a Dynkin diagram in the first row corresponds to the opposite edge of the alcovebelow it.6.22. Remark. If Φ is a product of more than one affine root system then there areproblems 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 ∈ { , } and j := 3 − i . Let ( S, S (cid:48) , c , S (cid:48) , c ) be a primitive diagram. Recall that S i is the connected closure of S (cid:48) i in S \ S (cid:48) j . Put(7.1) S p := S ∩ S . Our strategy is to construct S by gluing S and S along S p . For this we have to makesure that S i remains the connected closure of S (cid:48) i .7.1. Lemma.
Let S be a graph with subsets S (cid:48) , S , S (cid:48) , S such that S (cid:48) i ⊆ S i ⊆ S \ S (cid:48) j .Assume that S = S ∪ S and S i = C ( S (cid:48) i , S i ) . Then the following are equivalent:(1) S i is the connected closure of S (cid:48) i in S \ S j .(2) The following two conditions hold:a) S ∩ S is the union of connected components of S i \ S (cid:48) i .b) If α ∈ S \ S is connected to α ∈ S \ S in S then α ∈ S (cid:48) and α ∈ S (cid:48) .Proof. Because of S i = C ( S (cid:48) i , S i ) , the assertion S i = C ( S (cid:48) i , S \ S (cid:48) j ) means that there areno edges between S i and(7.2) S \ ( S i ∪ S (cid:48) j ) = ( S j \ S (cid:48) j ) \ ( S i ∩ S j ) = ( S j \ S i ) \ S (cid:48) j . This statement breaks up into two parts: There are no edges between S i ∩ S j and ( S j \ S (cid:48) 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 (cid:48) j which is just condition (2)b ). (cid:3) .1. The case S p (cid:54) = ∅ . We start our classification with:7.2.
Lemma.
Let D be primitive diagram with S p (cid:54) = ∅ such that there is at least oneedge between S (cid:48) and S (cid:48) . Then D ∼ = A (1) n ≥ ( aa ) : (7.3) (cid:4) (cid:4) Proof.
Let α ∈ S p and let there be an edge between α ∈ S (cid:48) and α ∈ S (cid:48) . Inspection ofTable 8.2 implies that there are paths from α and α to α respectively. Together withthe edge they form a cycle in S which implies that S is of type A (1) n ≥ with α , α beingadjacent. Therefore the local diagrams are either of type a m or a m . Since only diagramsof the latter type can be glued such that S p (cid:54) = ∅ and S is a cycle we get A (1) n ≥ ( aa ) as theonly possibility. (cid:3) Thus, we may assume from now on that S is the union of S and S minimally gluedalong S p , i.e., with no further edges added. To classify these diagrams we proceed by thetype of S p . Helpful is the following table which lists for all isomorphy types of S p thepossible candidates for S and S . The factor c is suppressed.7.3. Table.
Gluing data S p Candidates for S and S A a a b c n ≥ c d g c A a a b (cid:48) A a a d A n ≥ a n +2 a n +1 B b c c B b f B n ≥ b n +1 C n ≥ c n +2 c n +1 D d D n ≥ d n +1 A A c d A C n ≥ c n +2 Remark.
For c , the graph S \ S (cid:48) consists of two A -components. Therefore c islisted twice in the A -row. Also the case S p = D is listed separately since D hasautomorphisms which don’t extend to D .Using the table, it easy to find all primitive triples with S p (cid:54) = ∅ . Since the case-by-caseconsiderations are lengthy we just give an instructive example namely where S p = A .Here the following nine combinations have to be considered:(7.4) a ∪ a , a ∪ a , a ∪ a (2 × ) , a ∪ b (cid:48) , a ∪ b (cid:48) (2 × ) , b (cid:48) ∪ b (cid:48) (2 × ) We omitted the possibility of a factor of / for b (cid:48) . Observe that in three cases twodifferent gluings are possible. It turns out that all cases lead to a valid spherical diagramexcept for a ∪ b (cid:48) and one of the gluings of a ∪ b (cid:48) which do not lead to affine Dynkindiagrams.7.2. The case S p = ∅ . Here, according to Lemma 7.1, S is the disjoint union of S and S stitched together with edges between S (cid:48) and S (cid:48) . rather trivial subcase is when S = S (cid:48) = ∅ . Then D is just a local diagram all ofwhich appear in Table 8.3.Therefore, assume from now on that S , S (cid:54) = ∅ . Since then ≤ | S (cid:48) i | ≤ , the number N of edges between S (cid:48) and S (cid:48) is at most . This yields subcases.7.2.1. N = 0 . In this case, S is the disconnected union of S and S . If the triplewere bihomogeneous we cannot have ω D + ω D = 0 . If D were homogeneous and D inhomogeneous then (cid:104) ω D , α ∨D (cid:105) = 0 (cid:54) = − . Therefore, the triple is bi-inhomogeneous andwe get the three items A m ≥ × A n ≥ , A m ≥ × C n ≥ , and C m ≥ × C n ≥ near the end of thetable.7.2.2. N = 1 . In this case the diagram D is the disjoint union of two local diagramsconnected by one edge between some α ∈ S (cid:48) and α ∈ S (cid:48) . One can now go through allpairs of local diagrams and all possibilities for the connecting edge. This works well if oneor both local diagrams are of type ( d ) . Otherwise, it is easier to go through all possibleconnected Dynkin diagrams for S and omit one its edges. The remaining diagram admitsvery few possibilities for D and D . This way one can check easily that the table iscomplete 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 (cid:48) . Thisrules out all cases except the third one where the local diagrams could be of type a n or a n . This yields(7.7) (cid:4) (cid:4) , (cid:4) , (cid:4) , The first diagram violates Definition 6.5 (3)c ), the second is primitive and is contained inTable 8.3, the third violates (3)b ), and the fourth violates (3)a ).7.2.3. N = 2 . In this case, at least one of the S (cid:48) i , say S (cid:48) , has two different elements α , α (cid:48) which are connected to elements α , α (cid:48) ∈ S (cid:48) , respectively.Assume first that α (cid:54) = α (cid:48) . Then both local models are either of type a n ≥ or d . Onechecks easily that this yields the cases(7.8) A (1) n ≥ ( aa ) , C (1) n ≥ ( ad ) , D (2) n +1 ( ad ) , A (1)1 × A (1)1 ( dd ) , or A (2)2 × A (2)2 ( dd ) . The second subcase is α = α (cid:48) . If S is of type a n ≥ one ends up with A (1) n ≥ ( aa ) , d = 1 .Otherwise S is of type d . Now one can go through all local diagrams for S and allpossible edges between α , α (cid:48) and α ∈ S (cid:48) .7.2.4. N = 3 . In this case, there are distinct elements α , α (cid:48) ∈ S (cid:48) and α , α (cid:48) ∈ S (cid:48) whichform a string α , α , α (cid:48) , α (cid:48) . Both local diagrams are of type d which leaves only the case D (2)4 ( dd ) .7.2.5. N = 4 . Here α , α , α (cid:48) , α (cid:48) forms a cycle, so S is of type A (1)3 . The local diagramsare both d . This yields A (1)3 ( dd ) .This concludes the verification of the table. . Tables
Table.
Affine Dynkin diagrams and their labels A (1)1 A (1) n , n ≥ B (1) n , n ≥ C (1) n , n ≥ D (1) n , n ≥ E (1)6 E (1)7 E (1)8 F (1)4 G (1)2 A (2)2 A (2)2 n , n ≥ A (2)2 n − , n ≥
11 12 2 2 2 D (2) n +1 , n ≥ E (2)6 D (3)4 .2. Table.
Primitive local models
Homogeneous models L H ω Diagram a PGL(2) GL(1) α a PGL(2) N ( C ∗ ) 2 α a n ≥ PGL( n + 1) GL( n ) /µ n +1 α + . . . + α n b n ≥ SO(2 n + 1) SO(2 n ) α + . . . + α n b n ≥ SO(2 n + 1) O(2 n ) 2 α + . . . + 2 α n c n ≥ PSp(2 n ) Sp(2) × µ Sp(2 n − α +2 α + . . . +2 α n − + α n d n ≥ SO(2 n ) SO(2 n − α + . . . + α n − + α n − + α n / d n ≥ PSO(2 n ) SO(2 n −
1) 2 α + . . . +2 α n − + α n − + α n d SO(4) SO(3) α + α (cid:48) / d SO(3) × SO(3) SO(3) α + α (cid:48) d SO(6) SO(5) α + α + α / d PSO(6) SO(5) α + 2 α + α f F Spin(9) α + 2 α + 3 α + 2 α g G SL (3) 2 α + α g G N ( SL (3)) 4 α + 2 α b (cid:48) Spin(7) G α + α + α / b (cid:48) SO(7) G α + 2 α + 3 α Inhomogeneous models L V ω Diagram a GL(1) C ∼ ∅ a n ≥ GL( n + 1) C n +1 ∼ π (cid:4) c n ≥ GSp(2 n ) C n ∼ π (cid:4) .3. Table.
Primitive diagrams Φ case ω = ω = − ω factor diagram scope The empty case ∅ ( ∅ ) ω (cid:54) = 0 ∅ The affine simple cases A (1) n ≥ ( aa ) ω = α + . . . + α d − ω = α d + . . . + α n [2] n =1 α α d ≤ d ≤ n ( dd ) ω = ω = α + α ω = ω = α + α ( dd ) ω = α + 2 α + α ω = α + 2 α + α ( aa ) ω ∼ π − π n (cid:4) (cid:4) B (1) n ≥ ( bd ) ω = α + α + 2 α + . . . + 2 α d − ω = 2 α d + . . . + 2 α n α d ≤ d ≤ n ( bb ) ω = α + α + . . . + α n ω = α + α + . . . + α n b (cid:48) a ) ω = ( α + 2 α + 3 α ) (cid:4) / ( d a ) ω = ( α + α ) / (cid:4) C (1) n ≥ ( cc ) ω = α + 2 α + . . . + 2 α d − + α d ω = α d + 2 α d +1 + . . . + 2 α n − + α n [2] n =2 α d ≤ d < n ( ad ) ω = ( α + α n ) ω = α + . . . + α n − [2] n =2 / ( cc ) ω ∼ π d − − π d (cid:4) (cid:4) α d ≤ d ≤ n D (1) n ≥ ( dd ) ω = α + α + 2 α + . . . + 2 α d − ω = 2 α d + . . . + 2 α n − + α n − + α n α d ≤ d ≤ n − dd (cid:48) ) ω = 2 α + 2 α + . . . + 2 α n − + α n − + α n ω = 2 α + 2 α + . . . + 2 α n − + α n − + α n ( aa ) ω = α + α + . . . + α n − + α n − ω = α + α + . . . + α n − + α n F (1)4 ( bf ) ω = α + α + α + α ω = α + 2 α + 3 α + 2 α ( cd ) ω = α + α + α ω = α + 2 α + α / ( a a ) ω = α + α (cid:4) G (1)2 ( g a ) ω = α + 2 α (cid:4) ( a a ) ω = α (cid:4) ( d a ) ω = α + α (cid:4) / A (2)2 nn ≥ ( ab ) ω = 2 α + 2 α + . . . + 2 α n − ω = α n ( b c ) ω = α + α + . . . + α d − (cid:4) α d ≤ d ≤ n A (2)2 n − n ≥ ( a c ) ω = α + 2 α + . . . + 2 α n − + α n ω = α ( a c ) ω = α + α + α ω = α + 2 α + . . . + 2 α n − + α n case ω = ω = − ω factor diagram scope ( ad ) ω = α + α + 2 α + . . . + 2 α n − ω = α n ( d c ) ω = α + α + α + . . . + α d − (cid:4) / α d ≤ d ≤ n ( cc ) ω ∼ π − π (cid:4) (cid:4) D (2) n +1 n ≥ ( bb ) ω = α + . . . + α d − ω = α d + . . . + α n α d ≤ d ≤ n ( ad ) ω = α + . . . + α n − ω = α + α n [ ] n =2 ( dd ) ω = α + α ω = α + α ( b (cid:48) b (cid:48) ) ω = 3 α + 2 α + α ω = α + 2 α + 3 α ( d a ) ω = α + α / (cid:4) ( cc ) ω ∼ π − π (cid:4) (cid:4) E (2)6 ( af ) ω = α ω = 2 α + 3 α + 2 α + α ( bc ) ω = α + 2 α + 2 α + α ω = α + α + α ( a a ) ω = α + α + α (cid:4) D (3)4 ( ag ) ω = α ω = 2 α + α ad ) ω = α + α ω = 2 α ( a a ) ω = α + α (cid:4) The finite simple cases A n ≥ ( a a ) ω = α + . . . + α d − (cid:4) α d ≤ d ≤ n ( d a ) ω = α + α / (cid:4) ( d a ) ω = α + 2 α + α (cid:4) ( aa ) ω ∼ π d − − π d (cid:4) (cid:4) α d ≤ d ≤ n ( aa (cid:48) ) ω ∼ π − π n (cid:4) (cid:4) n ≥ a ) ω = α + . . . + α n [2] n =1 ( d ) ω = α + 2 α + α ( a ) ω ∼ π (cid:4) B n ≥ ( b a ) ω = α d + . . . + α n α d (cid:4) ≤ d ≤ n ( d a ) ω = α + α / (cid:4) ( d c ) ω = α + 2 α + α (cid:4) / ( b (cid:48) a ) ω = α + 2 α + 3 α (cid:4) ( aa ) ω ∼ π n − − π n (cid:4) (cid:4) ( ac ) ω ∼ π − π (cid:4)(cid:4) ( b ) ω = α + . . . + α n b (cid:48) ) ω = α + 2 α + 3 α C n ≥ ( a c ) ω = α + . . . + α d − (cid:4) α d ≤ d ≤ n ( c a ) ω = α + 2 α + . . . + 2 α n − + α n (cid:4) ( ac ) ω ∼ π d − − π d (cid:4) α d (cid:4) ≤ d ≤ n ( c ) ω = α + 2 α + . . . + 2 α n − + α n ( c ) ω ∼ π (cid:4) D n ≥ ( d a ) ω = α d + . . . + α n − + α n − + α n / α d (cid:4) ≤ d < n ( a a ) ω = α + . . . + α n − (cid:4) case ω = ω = − ω factor diagram scope ( d a ) ω = α + 2 α + α + 2 α (cid:4) ( aa ) ω ∼ π n − − π n (cid:4)(cid:4) ( d ) ω = 2 α + . . . + 2 α n − + α n − + α n F ( c a ) ω = α + 2 α + α (cid:4) ( a a ) ω = α + α (cid:4) ( b c ) ω = α + α + α (cid:4) ( aa ) ω ∼ π − π (cid:4) (cid:4) ( f ) ω = α + 2 α + 3 α + 2 α G ( a a ) ω = α (cid:4) ( aa ) ω ∼ π − π (cid:4)(cid:4) ( g ) ω = 2 α + α The reducible cases A (1)1 × A (1)1 ( dd ) ω = α + α (cid:48) ω = α + α (cid:48) δ (cid:107) α i (cid:107) = δ (cid:48) (cid:107) α (cid:48) i (cid:107) A (2)2 × A (2)2 ( dd ) ω = α + α (cid:48) ω = α + α (cid:48) / δ (cid:107) α i (cid:107) = δ (cid:48) (cid:107) α (cid:48) i (cid:107) A × A (1)1 ( d a ) ω = α + α (cid:48) (cid:4) / A × A (2)2 nn ≥ ( d c ) ω = α + α (cid:48) (cid:4) A × C (1) n ≥ ( d c ) ω = α + α (cid:48) (cid:4) / A × G (1)2 ( d a ) ω = α + α (cid:48) (cid:4) A × A n ≥ ( d a ) ω = α + α (cid:48) (cid:4) A × C n ≥ ( d c ) ω = α + α (cid:48) (cid:4) A × B n ≥ ( d a ) ω = α + α (cid:48) (cid:4) A × C n ≥ ( d a ) ω = α + α (cid:48) (cid:4) / A × G ( d a ) ω = α + α (cid:48) (cid:4) A m ≥ × A n ≥ ( aa ) ω ∼ π − π (cid:48) (cid:4)(cid:4) A m ≥ × C n ≥ ( ac ) ω ∼ π − π (cid:48) (cid:4)(cid:4) C m ≥ × C n ≥ ( cc ) ω ∼ π − π (cid:48) (cid:4)(cid:4) A × A ( d ) ω = α + α (cid:48) References [Akh83] D. Akhiezer,
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Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen
E-mail address : [email protected] Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen
E-mail address : [email protected]@kaypaulus.de