Geometric Invariant Theory for principal three-dimensional subgroups acting on flag varieties
aa r X i v : . [ m a t h . R T ] N ov GEOMETRIC INVARIANT THEORYFOR PRINCIPAL THREE-DIMENSIONAL SUBGROUPSACTING ON FLAG VARIETIES
HENRIK SEPP ¨ANEN AND VALDEMAR V. TSANOV
Abstract.
Let G be a semisimple complex Lie group. In this article, we study GeometricInvariant Theory on a flag variety G/B with respect to the action of a principal 3-dimensionalsimple subgroup S ⊂ G . We determine explicitly the GIT-equivalence classes of S -ample linebundles on G/B . We show that, under mild assumptions, among the GIT-classes there arechambers, in the sense of Dolgachev-Hu. The GIT-quotients with respect to various chambersform a family of Mori dream spaces, canonically associated with G . We are able to determinethe three important cones in the Picard group of any of these quotients: the pseudo-effective-,the movable-, and the nef cones. Contents
Introduction 11. Setting 41.1. The flag variety
G/B and GIT for subgroups 41.2. The principal subgroup S ⊂ G S -action on G/B and GIT 92.1. The orbit structure 92.2. The Hilbert-Mumford criterion 102.3. A lemma on Weyl group elements of small length 102.4. The Kirwan stratification and the S -ample cone 122.5. GIT-classes of ample line bundles 142.6. S -Movable line bundles 153. Quotients and their Picard groups 154. The exceptional cases of types A and C G = SL G = Sp Introduction
Let G be a semisimple complex Lie group. We explore the interaction of two remarkable ob-jects from the theory of semisimple groups - flag varieties X = G/B and principal 3-dimensionalsimple subgroups S . One context in which these two objects interact is Geometric InvariantTheory (GIT), quotients, and their variations (VGIT). The theme of this paper is to relateproperties of S -invariant sections of line bundles on X to geometric properties of suitable GIT-quotients of X under S . It is known that, for a given line bundle L , the study of its invariant Both authors are supported by the DFG Priority Programme 1388 “Representation Theory”. V.V.T. wasalso partially supported by project SFB/TR12 at Ruhr-Universit¨at Bochum. sections amounts to a study of a GIT-quotient of X which is adapted to the line bundle L (cf.e.g. [Sj95]). What we have in mind here is rather to have a quotient that is “universal” inthe sense that it reflects the invariant theory of sections of all line bundles. In this sense, thispaper gives an example of pairs ( S, G ) of a semisimple group G and a semisimple subgroup S where the theory of suitable quotients developed in [S14] works particularly well. The quotientvarieties obtained are Mori dream spaces (cf. [HK00]), canonically associated to G . Whatwe present in this article are some initial results, and it becomes evident from these resultsthat this example could be developed further to illustrate further elements of the theoreticalframework of GIT in the sense of [Kir84], [T96], [DH98], [KKV89], [S14]. Our main source forthe properties of principal subgroups is the classical work of Kostant, [Kos59].We summarize some of our results in the following two theorems. The first one given asTheorem 2.6 in the text, and the second one is a compilation of Proposition 2.2, Theorem2.10, Theorem 3.1, Theorem 3.2. To give the statements, we briefly introduce some standardnotation. We recall some definitions and basic properties in the next section. Assume that G is connected and simply connected. Let H ⊂ B ⊂ G be nested Cartan and Borel subgroups.Let X = G/B be the flag variety. Let W = N G ( H ) /H be the Weyl group, with the length l ( w )defined by B , and let w be the longest element. Recall that the H -fixed points in X are givenby Weyl group elements X H = { x w = wB : w ∈ W } . Let Λ be the weight lattice of H , let Λ + and Λ ++ denote the monoids of dominant and strictly dominant weight with respect to B . Wehave Pic( X ) ∼ = Λ, with L λ = G × B C − λ ; all line bundles are G -equivariant. The sets Λ + andΛ ++ represent respectively the sets of effective and ample line bundles. Furthermore, all amplebundles are very ample. By the Cartan-Weyl classification of irreducible modules and the Borel-Weil theorem, the space of global section of any effective line bundle is an irreducible G -moduleand all irreducible G -modules are obtained this way; H ( X, L λ ) = V ∗ λ , for λ ∈ Λ + . Let S ⊂ G be a principal 3-dimensional simple subgroup with Cartan and Borel subgroups H S ⊂ B S ⊂ S .The principal property of S ensures that there are unique H ⊂ B ⊂ G satisfying H S = S ∩ H and B S = S ∩ B . The line bundle L λ on X , being G -equivariant, is also S -equivariant; hencethere are well-defined notions of stability, instability and semistability on X with respect to S and L λ . The set of ample line bundles on X is partitioned into GIT-classes, two line bundlesbeing equivalent if their semistable (or equivalently unstable) loci coincide. In the followingtheorem, we determine the unstable loci of ample line bundles, in terms of the Kirwan-Nessstratification with respect to the squared norm || µ || of a momentum map µ = µ K S : X → i k ∗ S .Here K S ⊂ S is a maximal compact subgroup and µ is defined with respect to the K -invariantK¨ahler structure on X defined by λ , where K ⊂ G is a maximal compact subgroup containing K S . Let h ⊂ h S be a dominant integral element. Then X h = X H and these are exactly thecritical points of the momentum component µ h . Theorem A:
Let λ ∈ Λ ++ . The Kirwan strata of the S -unstable locus in X with respect to L λ are the S -saturations of Schubert cells SBx w , for w ∈ W such that wλ ( h ) >
0, with theSchubert cell Bx w being the prestratum and K S x w being the critical set of || µ || . Thus X us ( λ ) = [ w ∈ W : wλ ( h ) > S wλ , S wλ = SBx w . (1)The dimension of the strata is given by dim S wλ = l ( w ) + 1, and consequentlydim X us ( λ ) = 1+max { l ( w ) : wλ ( h ) > } , codim X X us ( λ ) = − { l ( w ) : wλ ( h ) < } . This explicit description of the unstable loci allows us to determine the GIT-classes of S -ample line bundles on X , and some properties of the GIT quotients, as follows. Theorem B:
Assume that every simple factor of G has at least 5 positive roots. The followinghold:(i) The S -orbits in X of dimension less than 3 are exactly the orbits through H -fixed points, X H = { x w = wB : w ∈ W } . There is a unique 1-dimensional orbit Sx = Sx w ∼ = S/B S ∼ = P . There are | W | − Sx w = Sx w w ∼ = S/H S ∼ =( P × P ) \ diag( P ). The rest of the orbits are three dimensional with trivial or finiteabelian isotropy groups.(ii) All ample line bundles on X are S -ample, i.e., some power admits S -invariant sections.The S -unstable locus of any ample line bundle on X has codimension at least 2. TheGIT-equivalence classes of S -ample line bundles on X are defined by the subdivision ofthe dominant Weyl chamber Λ + R by the system of hyperplanes H w , w ∈ W given by H w = { λ ∈ Λ R : λ ( wh ) = 0 } , where h is an arbitrary fixed nonzero element in the Lie algebra of H S .(iii) The GIT-equivalence classes given by the connected components of Λ ++ R \ ( ∪ w H w ) arechambers, in the sense of Dolgachev-Hu, which in our case means that the semistablelocus consists only of 3-dimensional orbits. The hyperplanes H w are walls, in the senseof Dolgachev-Hu.(iv) The GIT-quotient Y ( C ) = X ss ( C ) //S with respect to a chamber C ⊂ Λ ++ R \ ( ∪ w H w ) is ageometric quotient. The variety Y is a Mori dream space, whose Picard group is a latticeof the same rank as Λ. There is an isomorphism of Q -Picard groups Pic( X ) Q ∼ = Pic( Y ) Q induced by descent in one direction, and pullback followed by extension in the other.There are the following isomorphisms involving the pseudo-effective, movable, and nefcones in the Picard group of Y :Eff( Y ) = Mov( Y ) ∼ = Λ + R , Nef( Y ) ∼ = C . Moreover, every nef line bundle on Y is semiample, i.e., admits a base-point-free power.(v) Fix Y as in (iv). For every λ ∈ Λ + , there exists k ∈ N such that ( V kλ ) S = 0 and a linebundle L on Y such that H ( Y, L j ) ∼ = ( V jkλ ) S for all j ∈ N . Remark:
The assumption made in the above theorem allows us to reduce the technicality ofthe statement. The excluded cases are those G admitting simple factors of type A , A , B .They are taken into account in the main text and in some detail in Section 4. Recall that dim X is equal to the number of positive roots of G . The particularities of the excluded cases are dueto the low dimension of the respective factors of X , which results in low codimension of theunstable locus for some line bundles. Let us note here the following, in order to give an idea ofthe occurring phenomena. The presence of simple factors of G of rank 1 implies the existenceof ample line bundles on X , which are not S -ample, and with respect to which the whole X isunstable. The presence of simple factors of type A or B implies the existence of S -ample linebundles whose unstable loci contain divisors; this interferes in the relations between the Picardgroups of X and its GIT-quotients. An application:
Geometric Invariant Theory finds one of its applications in the theory ofbranching laws for reductive groups, a.k.a. eigenvalue problem. This relates to part (v) of the above theorem. Let us outline the general ideas or order to see how our example fits in, whatphenomena it exhibits, and what questions it presents. If ˆ G ⊂ G is an embedding of reductivecomplex algebraic groups, the branching law consist of the descriptions of the decompositionsof irreducible G -modules over ˆ G . This amounts to descriptions of the so called eigenmonoid(or Littlewood-Richardson monoid) and multiplicities E ( ˆ G ⊂ G ) = { (ˆ λ, λ ) ∈ ˆΛ + × Λ + : Hom ˆ G ( ˆ V ˆ λ , V λ ) = 0 } , m ˆ λλ = dim Hom ˆ G ( ˆ V ˆ λ , V λ ) . The relation to geometric invariant theory comes via the isomorphismsHom ˆ G ( ˆ V ˆ λ , V λ ) ∼ = ( ˆ V ∗ ˆ λ ⊗ V λ ) ˆ G ∼ = H ( ˆ G/ ˆ B × G/B, ˆ L − ˆ w ˆ λ ⊠ L λ ) ˆ G . Thus, the branching laws for reductive groups are contained in the more general laws describinginvariants, where one considers the nulleigenmonoid and respective multiplicities E ( ˆ G ⊂ G ) = { λ ∈ Λ + : V ˆ Gλ = 0 } , m λ = dim V ˆ Gλ . It is known by a theorem of Brion and Knop, that E is a finitely generated submonoid ofΛ + , spanning a rational polyhedral cone Cone( E ) ⊂ Λ + R , the nulleigencone, cf. [E92]. Theequalities of the nulleigencone have been determined, the final result providing a minimallist of inequalities was obtained by Ressayre, building on works of Heckmann, Berenstein-Sjamaar, Belkale-Kumar, cf. [R10]. In the particular case of a principal subgroup S ⊂ G , thenulleigencone was computed as an example by Berenstein and Sjamaar, [BS00]. The globaldescription of the multiplicities still presents an open problem in the general situation. Thereare results concerning specific weights, e.g. Kostant’s multiplicity formula, cf. [V78]. There arealso methods for specific types of subgroups, e.g. Littlemann’s path method, [L95], the methodof Berenstein-Zelevinski, [BZ01]. Recently, the first author has constructed a global Okounkovbody, ∆ Y of a suitable quotient of X = G/B (we do not use this notation elsewhere in thetext), a strongly convex cone with a surjective map p : ∆ Y → Cone( E ), such that the fibres∆ Y (ˆ λ, λ ) = p − (ˆ λ, λ ) are in turn Okounkov bodies, whose volume varies along the ray R + (ˆ λ, λ )asymptotically as the dimension of ( ˆ V ∗ ˆ λ ⊗ V λ ) ˆ G , cf. [S14]. In fact, this is proven more generallyfor ˆ G -invariants, for a pair ( ˆ G, G ) of semisimple groups where ˆ G is a subgroup of G underthe assumption on the existence of chambers among the GIT-classes, an assumption which israther mild in the branching-case, which corresponds to the pair ( ˆ G, ˆ G × G ). The existenceof chambers is not guaranteed for a general action of a reductive subgroup ˆ G ⊂ G on G/B ;for instance, so-called thick walls (cf. [R98]) could appear. In the case of a principal subgroup S ⊂ G , however, our results show that there are no thick walls, and hence, by the results of[S14], the dimensions of the spaces V Sλ of S -invariants could be measured, in an asymptoticsense, by volumes of slices of a convex cone, namely of a a global Okounkov body of a fixedquotient Y = Y ( C ) = X ss ( C ) //S . 1. Setting
The flag variety
G/B and GIT for subgroups.
Let G be a connected, simply con-nected semisimple complex Lie group. Let B ⊂ G be a Borel subgroup and X = G/B be theflag variety of G . Let H ⊂ B be Cartan subgroup and ∆ = ∆ + ⊔ ∆ − be the root systems of G with respect to H , split into positive and negative part with respect to B . Let Π be the set ofsimple roots. Let g = n ⊕ h ⊕ ¯ n be the associated triangular decomposition of the Lie algebraof G , where n = [ b , b ] is the nilradical of b and ¯ n is the nilradical of the opposite Borel ¯ b . TheWeyl group W = N G ( H ) /H acts simply transitively on the Borel subgroups of G containing H , and thus on the set of H -fixed points X H = { x w = wB, w ∈ W } . The B -orbits in X define the Schubert cell decomposition X = [ w ∈ W Bx w . The torus acts on the tangent space T x w X and the weights for this action are w ∆ − . Theunipotent radical N ⊂ B acts transitively on each Schubert cell and, if N x w denotes the stabilizerof x w , the set of positive roots is partitioned roots as ∆ + = ∆( N x w ) ⊔ ∆( N/N x w ), with ∆( N/N x w )being the set of weights for the T -action on T x w Bx w . We have ∆( N/N x w ) = ∆ + ∩ w ∆ − , this setis called the inversion set of w − , if we adhere to the popular notation Φ w = ∆ + ∩ w − ∆ − . Thetwo sets Φ w and Φ w − have the same number of elements, the length l ( w ) of w , also equal to thenumber of simple reflections in a reduced expression for w . Thus dim Bx w = l ( w ). Inversionsets are closed and co-closed under addition in ∆ + , we have ∆ + \ Φ w = Φ w w , where w is thelongest element of w . Thus ∆( N x w ) = Φ w w − and ∆( N/N x w ) = ∆( N x w w ) = Φ w − .Let Λ ∈ h ∗ denote the weight lattice of H , Λ + the set of dominant weights with respect to B and Λ ++ the set of strictly dominant weights, i.e., those belonging to the interior of the Weylchamber. For λ ∈ Λ + , let V λ denote an irreducible G -module with highest weight λ and let v λ be the highest weight vector of G , the unique B -eigenvector. We have an equivariant orbit-map X → G [ v λ ] ⊂ P ( V λ ). and this is the unique closed orbit of G in P ( V λ ). We have ϕ ( x w ) = [ v wλ ].The map ϕ is an embedding if and only if λ ∈ Λ ++ . For λ ∈ Λ + \ Λ + +, the orbit G [ v λ ] is apartial flag variety G/P , where P ⊃ B is a parabolic subgroup of G . We shall focus mostly onthe complete flag variety G/B and the interior of the Weyl chamber.Every line bundle on X admits a unique G -linearization. The Picard group of X is identifiedwith the weight lattice. For λ ∈ Λ, we denote by L λ = G × B C − λ the associated homogeneousline bundle on X . For dominant λ , L λ is the pullback of O P ( V λ ) (1). The Borel-Weil theoremasserts that H ( X. L λ ) ∼ = V λ for λ ∈ Λ + and H ( X, L λ ) = 0 for λ / ∈ Λ \ Λ + . In invariant theory one considers a subgroup ˆ G ⊂ G and the space of ˆ G -invariant vectors V ˆ Gλ .We restrict ourselves to the case when ˆ G is semisimple, and our results concern the very specialcase of a principal simple subgroup of rank 1, but now we outline the general scheme. The firstnatural questions one may ask are: What is the dimension of the space of invariants and when isit nonempty? How does it vary with λ ? Two central objects associated with these questions arethe nulleigenmonoid, or null-Littlewood-Richardson monoid (which is indeed finitely generatedsubmonoid of Λ + by a theorem of Brion and Knop), and the multiplicity E ( ˆ G ⊂ G ) = { λ ∈ Λ : V ˆ Gλ = 0 } , m λ = dim V ˆ Gλ . Note that V ˆ Gλ has a canonical nondegenerate pairing with ( V ∗ λ ) ˆ G and recall that V ∗ λ ∼ = V − w λ ,whence E ∗ = − w E = E and m λ = m − w λ .The Borel-Weil theorem allows to rephrase these questions in terms of invariant sections ofline bundles, and E ++0 = E ∩ Λ ++ corresponds to the cone C ˆ G ( X ) of ˆ G -ample line bundles inthe Picard group of X . Of particular interest is the variation of the multiplicity as the weightvaries along a ray, m kλ , k ∈ N . Since L kλ = L kλ , this relates to the ring of ˆ G -invariants in thehomogeneous coordinate ring of X ⊂ P ( V λ ), which is given by R ( λ ) = M k ∈ N H ( X, L λ ) = C [ V λ ] /I ( X ) , R k ( λ ) = V ∗ kλ . In this setting, there are the notions of instability, semistability and stability on X , with respectto L λ , defined by X us ( λ ) = X us, ˆ G ( L λ ) = { x ∈ X : f ( x ) = 0 , ∀ f ∈ R ( λ ) ˆ G \ C } X ss ( λ ) = X ss, ˆ G ( L λ ) = { x ∈ X : ∃ f ∈ R ( λ ) ˆ G \ C , f ( x ) = 0 } X s ( λ ) = X s, ˆ G ( L λ ) = { x ∈ X : ˆ G x is finite and ˆ Gx ⊂ X ss ( λ ) is closed } . With these definitions, we have, for λ ∈ Λ + . λ ∈ Cone( E ) ⇐⇒ X ss ( λ ) = ∅ . This relation has been the basis for the descriptions of eigencone initiated with Heckman’sthesis and culminating with Ressayre’s minimal list of inequalities defining Cone( E ). We aregoing to consider subgroups for which Cone( E ) has a fairly simple structure. In fact, in manycases we will have Cone( E ) = Λ + R . We are rather interested in the structure of the unstable,semistable and stable loci. Let us note, however, that the cases when Cone( E ) = Λ + R are indeedof specific interest for the study of branching laws, and we shall see a manifestation of this lateron. Remark 1.1.
The group ˆ G necessarily has closed orbits in X . These closed orbits are flagvarieties of ˆ G . In fact, they are all complete flag varieties, i.e., have the form ˆ G/ ˆ B , because theall isotropy groups in X are solvable. Thus the closed ˆ G -orbits are parametrized by the Borelsubgroups of G containing a fixed Borel subgroup ˆ B ⊂ ˆ G . Note that ˆ Gx ⊂ X closed = ⇒ ˆ Gx ⊂ X us ( λ ) for all λ ∈ Λ ++ . We end this section by recalling the notions of GIT-equivalence classes and chambers, fol-lowing Dolgachev and Hu, [DH98]. We have Λ R ∼ = Pic( X ) R . The dominant Weyl chamber Λ + R is identified with the pseudo-effective cone Eff( X ). Then Cone( E ) ∼ = C ˆ G ( X ) is the ˆ G -amplecone on X . Definition 1.1.
Two ˆ G -ample line bundles L λ and L λ on X are called GIT-equivalent, if theyhave the same semistable loci, i.e., X ss ( λ ) = X ss ( λ ) . The equivalence classes are called GIT-classes. If C is a GIT-class of line bundles, we denote by X ss ( C ) the corresponding semistablelocus. Recall that the equivalence relation on line bundles in C ˆ G ( X ) is extended to an equivalencerelation on C ˆ G ( X ), by a natural extension of the notion of stability and semistability to R -divisors, cf. [DH98]. The following definition we singles out a specific type of equivalenceclasses, chambers, the existence of which has remarkable consequences, as shown by Dolgachevand Hu. The definition we adopt here differs from the original one in [DH98], but is showntherein to be an equivalent characterization. Definition 1.2.
A GIT-class
C ⊂ C ˆ G ( X ) is called a chamber, if X ss ( C ) = X s ( C ) . We also distinguish another type of GIT-class, these without unstable divisors.
Definition 1.3.
A GIT-class
C ⊂ C ˆ G ( X ) is called ˆ G -movable, if codim X ( X us ( C )) ≥ . If L λ is a ˆ G -ample line bundle on X , the GIT-quotient, or Mumford quotient, of X withrespect to L λ is given by Y λ = Proj( R ( λ ) ˆ G ) = X ss ( λ ) / ∼ , where x ∼ y ⇐⇒ ˆ Gx ∩ ˆ Gy = ∅ . The quotient depends only on the GIT-class of λ , say C , so we write Y C = Y λ . Chambersand ˆ G -movable GIT-classes are important, because they give rise to, respectively, geometricquotients and quotients whose R -Picard group is isomorphic to the one of X . In what follows, unless otherwise specified, we apply the notation E , X us etc. for the caseˆ G = S , where S is the principal three dimensional simple subgroup of G defined in the nextsection.1.2. The principal subgroup S ⊂ G . Among the conjugacy classes of three dimensionalsimple subgroups of a semisimple complex Lie group G , there is a distinguished one - the classof principal subgroups, cf. [Kos59]. They admit several characterizations. Since we focus onthe flag variety X = G/B , we define a principal subgroup S ⊂ G to be a three dimensionalsimple subgroup with a unique closed orbit in X . Such an orbit is a rational curve, which wecall the principal curve C ⊂ X . In the next proposition we recall other characterizations ofprincipal subgroups. Proposition 1.1.
Let s ⊂ g be a three dimensional simple subalgebra, i.e., s ∼ = sl C , and let S ⊂ G be the corresponding subgroup. Let { e + , h , e − } ⊂ s be a standard sl -triple. Then thefollowing are equivalent: (i) S has a unique closed orbit in X . (ii) Every Borel subgroup of S is contained in a unique Borel subgroup of G . (iii) These exists a unique triangular decomposition g = n ⊕ h ⊕ ¯ n compatible with s = C e + ⊕ C h ⊕ C e − . (iv) e + is contained in a unique maximal subalgebra n ⊂ g of nilpotent elements. (Theelements with this property are called principal nilpotent elements. They form a singleconjugacy class.) (v) If n ⊂ g is any maximal subalgebra of nilpotent elements containing e + , h ⊂ g is aCartan subalgebra normalizing n and e α ∈ n , α ∈ ∆ + are the root vectors, upon writing e + = X α ∈ ∆ + c α e α , we have c α = 0 for all simple roots α .Furthermore, all subalgebras (respectively subgroups) of g (respectively G ) with the above prop-erties form a single conjugacy class. From now on we fix a principal subgroup S ⊂ G and a triple { e + , h , e − } ⊂ s and theassociated triangular decomposition of g . We may further take the nilpotent element to be thesum of the simple root vectors: e + = X α ∈ Π e α . Then we necessarily have α ( h ) = 2 for all α ∈ Π , (2)which determines h uniquely in h . We denote by h S = C h and b S = C h ⊕ C e + the Cartanand Borel subalgebras of s associated with the given triple, respectively, and by H S and B S thecorresponding subgroups of S . For the attributes of G we use the notation introduced earlierin the text, with reference to the given triangular decomposition. Remark 1.2.
For any finite dimensional G -module V , we have V G = V H ∩ V S . (3)In what follows, we shall make extensive use of restrictions of weights from H to H S . Wedenote the inclusion map by ι : S ⊂ G , and keep the same notation for the restriction of ι tosubgroups; we denote by ι ∗ the resulting restrictions and pullbacks. In particular, for weightswe have ι ∗ : Λ → Λ S ∼ = Z , ν ν ( h ) . This map is determined by its values on the simple roots (2). However, weights, especiallydominant weights, are often given in terms of the fundamental weights ω α . The values offundamental weights on the principal element h can be computed using the classification andstructure of root systems. We record in the next proposition some inequalities, which we needfor our estimates on codimension of unstable loci. Proposition 1.2.
In the bases of fundamental weights for Λ and Λ S the restriction ι ∗ is givenby ι ∗ = (2 2 . . . − : Z ℓ → Z , where A and ℓ are the Cartan matrix and the rank of G , respectively.Denote m = m ( g ) = min { ω α ( h ) : α ∈ Π } . The value of a fundamental weight on h dependsonly on the simple ideal to which the fundamental weight belongs. For the types of simple Liealgebras we have:1) m (A ℓ ) = ℓ for ℓ ≥ .2) m (B ℓ ) = 2 ℓ for ℓ ≥ .3) m (C ℓ ) = 2 ℓ − for ℓ ≥ .4) m (D ℓ ) = 2 ℓ − for ℓ ≥ .5) m (E ) = 16 .6) m (E ) = 27 .7) m (E ) = 58 .8) m (F ) = 16 .9) m (G ) = 6 .In particular, if g has no simple factors of rank 1, we have ω ( h ) ≥ for all fundamentalweights.Proof. The first statement follows from (2) and the characterization of the Cartan matrix asthe matrix for change of basis between the fundamental weights and the simple roots. Thesecond statement is obtained by direct calculation using for instance the tables at the end of[Bou68]. (cid:3)
Lemma 1.3.
Let W S = { , σ = − } be the Weyl group of S and recall that w denotes thelongest element of W . We have ι ∗ ( w λ ) = σι ∗ ( λ ) for all λ ∈ Λ . Proof.
The statement follows immediately from the definition of h and the fact that w sendsthe set of simple roots Π to − Π. (cid:3) Representations, coadjoint orbits and momentum maps.
Let T S ⊂ H S be themaximal compact subgroup of H S and K S ⊂ S be a maximal compact subgroup containing T S . Let K ⊂ G be a maximal compact subgroup containing K S . Then K necessarily containsthe maximal compact subgroup T ⊂ H . We fix a K -invariant positive definite Hermitian form h , i on V λ , which induces the Fubini-Study form on P = P ( V λ ). We take h , i to be C -linear onthe first argument. We shall consider momentum maps for this action. The classical targetspace of momentum maps is k ∗ . For our calculations it is suitable to replace k ∗ , by i k , which isharmless, since the representations are isomorphic. The Killing form is positive definite on i k ;we denote it by ( . | . ) and use the same notation for the induced forms on subspaces and dualspaces. This allows us to embed the weight lattice as Λ ∈ Λ R = i t ∗ ⊂ i k ∗ . We define µ K : P → i k ∗ , µ [ v ]( ξ ) = h ξv, v ih v, v i , [ v ] ∈ P , ξ ∈ i k . (4) We have µ K ( X ) = Kλ . The momentum map for the K S -action is given by restriction µ = µ K S = ι ∗ ◦ µ K : P → i k ∗ S . (5)For ξ ∈ i k S we denote the ξ -component of µ by µ ξ : P → R , µ ξ [ v ] = µ [ v ]( ξ ) . In the following we shall make particular use of the restrictions of µ , || µ || and µ h to thesmooth subvariety X ⊂ P . 2. The S -action on G/B and GIT
The orbit structure.
Here we discuss orbit structure for the action of the principalsubgroup S ⊂ G on the flag variety X = G/B . Assume dim X ≥ Lemma 2.1.
The set of H S fixed points in X is X H S = X H = { x w = wB ; w ∈ W } . The Weylgroup W S = { , σ } acts on X H by σ : X H → X H , x w → x w w . Proof.
Since H S is a regular one-parameter subgroup in G , i.e., it is contained in a uniqueCartan subgroup, we have X H S = X H . The Weyl group W S acts on X H S . We shall use aprojective embedding to show that σ acts in X H as w . Let λ ++ and let ϕX → P ( V λ ) bethe corresponding embedding. Then we have ϕ ( x w ) = [ v wλ ], which defines a W -equivariantembedding X H → Λ, x w wλ . We may further restrict weights by ι ∗ Λ → Λ S and thecomposition µ h = ι ∗ ◦ ϕ : X H → Λ S ∼ = Z , x w wλ ( h )is W S equivariant. We may further choose λ avoiding the hyperplanes defined by wλ ( h ) = 0and wλ ( h ) = w ′ λ ( h ) for all w, w ′ ∈ W , so that the above map µ h is injective. It follows that σ has no fixed points in X H , so it is uniquely determined and by Lemma 1.3 we must have σ ( x w ) = x w w . (cid:3) Proposition 2.2. (i)
The orbits of S in X have dimensions 1, 2 and 3. (ii) The orbits of dimension 1 and 2 are exactly the orbits through the fixed point set of themaximal torus H ⊂ G , X H = { x w = wB ; w ∈ W } . There is a unique 1-dimensionalorbit C = S [ x ] = S [ x w ] ∼ = S/B S ∼ = P . There are | W | − two-dimensional orbits S [ x w ] = S [ x w w ] ∼ = S/H S ∼ = ( P × P ) \ ( diagonal ) for w ∈ W \ { , w } . (iii) The isotropy subgroup of any 3-dimensional orbit is either trivial or finite abelian. (iv)
Every S -orbit in X has a finite number of orbits in its closure.Proof. The 1-dimensional orbit S -orbit in C ⊂ X is unique by definition. A 2-dimensionalorbit has a 1-dimensional isotropy group. In S ∼ = SL C , the 1-dimensional subgroups areconjugate to the unipotent N S = N ∩ S , the Cartan subgroup H S , or its normalizer N S ( H S ).The unipotent subgroup N S has a unique fixed point in X , x , because N is the only maximalunipotent subgroup of G containing N S (see Prop. 1.1, (iv)). Since S x = B S , there are noorbits of the form S/N S in X . It follows that any 2-dimensional orbit must contain an H S -fixedpoint x w . Since H S is a regular one-parameter subgroup in G , i.e., it is contained in a uniqueCartan subgroup, we have X H S = X H . We know from Lemma 1.3 that σ ∈ W S has no fixedpoints in X H and acts by σ ( x w ) = x w w , Hence N S ( H S ) has no fixed points in X . We can Apologies for the notation! conclude that the 2-dimensional orbits are S [ x w ] = S [ x w w ] ∼ = S/H S for w ∈ W \ { , w } . Inparticular they are finitely many. This implies (ii) and also (i), with the assumption dim X ≥ K ⊂ G . The group K acts transitively on X and X ∼ = K/T , where T is a Cartan subgroup of K . Hence Γ must be abelian. (cid:3) Remark 2.1.
Nonabelian finite subgroups of S can be obtained as isotropy subgroups for actionson partial flag varieties G/P . For instance the symmetry group of a tetrahedron appears as theisotropy subgroup of the unique 3-dimensional orbit in P of the principal subgroup S ⊂ SL C ,given by the 4-dimensional irreducible representation of SL . The Hilbert-Mumford criterion.
We shall use the Hilbert-Mumford criterion to detectinstability. Here we present its specific form in the case of the principal subgroup. The crite-rion, in its general formulation, reduces verification of the instability for a reductive group toverification of instability for dominant C × -subgroups. Since S has rank 1, all its C × -subgroupsare Cartan subgroups and are conjugate. Thus the detection of S -instability is reduced to H S -instability. We formulate this in the following lemma, which is a direct application of theHilbert-Mumford criterion for our case, and which is essential for our calculations. Lemma 2.3.
Assume dim X ≥ . Let λ ∈ Λ ++ . Let x ∈ X S , i.e., dim Sx = 3 . Then H S hastwo fixed points in H S x , say x w and x w . The following are equivalent: (i) x is S -unstable with respect to L λ . (ii) x is H S -unstable with respect to L λ . (iii) w λ ( h ) and w λ ( h ) are both nonzero and have the same sign, i.e., w λ ( h ) w λ ( h ) > .In particular, X S ∩ X us,S ( λ ) = X S ∩ X us,H S ( λ ) . Remark 2.2.
Let x ∈ X . Consider the orbit H S x . The set of H S -fixed points Sx H S hastwo elements. The nontrivial element σ ∈ W S acts on both Sx H S and N S ( H S ) H S . The set N S ( H S ) H S has either 2 or 4 elements; its image under µ h belongs to Z and is stable under σ which acts here by -1. A lemma on Weyl group elements of small length.
In our calculation of dimensionsof unstable loci, we shall need estimates for the length of Weyl group elements w such that wλ ( h ) < λ . We obtain such estimates using Proposition1.2. We record here the following lemma, which will help us detect cases with empty semistablelocus and cases with unstable divisors (see Theorem 2.6). Recall that any dominant weight λ ∈ Λ + can be written in terms of the fundamental weights as λ = X α ∈ Π λ α ω α with λ α = n λ,α = 2 ( λ | α )( α | α ) . Recall also that E = E ( S ⊂ G ) = { λ ∈ Λ + : V Sλ = 0 } denotes the nulleigenmonoid for theprincipal subgroup. Lemma 2.4.
Let λ ∈ Λ ++ . (i) If w, w ′ ∈ W are related by the Bruhat order on the Weyl group as w ′ ≺ w , then w ′ λ ( h ) > wλ ( h ) . (ii) If G has no simple factors of type A , then wλ ( h ) > for all w ∈ W with l ( w ) ≤ .More precisely, if s α λ ( h ) < for some α ∈ Π , then α is orthogonal to all other simpleroots and thus corresponds to a simple factor of G of type A . Furthermore, we have λ / ∈ Cone( E ) . (iii) If G has no simple factors of type A , A , C , then wλ ( h ) > for all w ∈ W with l ( w ) ≤ . If s β s α λ ( h ) < for some α, β ∈ Π , then one of the following occurs:1) α, β are the simple roots of a simple factor of G of type A ;2) α, β are the simple roots of a simple factor of G of type C ;3) At least one of the roots α, β is the simple root of a simple factor of G of type A .Proof. The Bruhat order is defined by w ′ (cid:22) w if x w ′ ∈ Bx w ⊂ X with w ′ ≺ w if w ′ = w . Thelinear span of the Schubert variety in V λ is the Demazure B -module V B,wλ whose weights areexactly the weights of V λ contained in wλ + Q + . Thus w ′ λ = wλ + q for some sum of positiveroots q . Since q ( h ) >
0, we have w ′ λ ( h ) > wλ ( h ). This proves part (i).Let us consider the action of a simple reflection. Since the principal element h is defined by α ( h ) = 2 for α ∈ Π, we have s α ν ( h ) = ν ( h ) − n ν,α for α ∈ Π , ν ∈ Λ . For part (ii), we recall that for α ∈ Π, we have n ω α ,α = 1 and n ω β ,α = 0 for β ∈ Π \ { α } . Hence s α λ ( h ) = λ ( h ) − λ α = λ α ( ω α ( h ) −
2) + X β ∈ Π \ α λ β ω β ( h ) . By Proposition 1.2 we have ω α ( h ) − < α is orthogonal to all other simpleroots. This implies that, if the root system ∆ has no simple subsystems of type A , then s α λ ( h ) > α ∈ Π. If α is orthogonal to all other simple roots, then s α λ ( h ) < ⇐⇒ λ α > X β ∈ Π \ α λ β ω β ( h ) . Suppose s α λ ( h ) < G = S × G be the factorization of G so that α is thesimple root of S . Let S ⊂ G be the projection of S to the second factor, which is a principalsubgroup of G . Then we have S diag ֒ → S × S ⊂ S × G . We can now consider G -modulesas tensor products of S - and G -modules, so that V G,λ = V S ,λ ⊗ V G ,λ , with λ = λ α and λ = P β ∈ Π \ α λ β ω β . We have( V G,λ ) S = ( V S ,λ α ⊗ V G ,λ ) S ∼ = Hom S ( V S ,λ α , V G ,λ ) = 0 if λ α > X β ∈ Π \ α λ β ω β ( h ) . This proves part (ii). Let us now turn to part (iii) and Weyl group elements of length 2. Suchelements have the form w = s β s α with two distinct simple roots α, β ∈ Π. We have s β s α λ = s β ( λ − λ α α ) = λ − λ α α − λ β β + λ α n β,α β . Hence s β s α λ ( h ) = λ ( h ) − λ α + λ β ) + 2 n β,α λ α = λ α ( ω α ( h ) − − n β,α )) + λ β ( ω β ( h ) −
2) + P γ ∈ Π \{ α,β } λ γ ω γ ( h ) . Since α, β are simple roots, we have n β,α ≤
0. We need to estimate the numbers ( ω β ( h ) − ω α ( h ) − − n β,α )). In part (ii) we already observed that ( ω β ( h ) − ≥ β isorthogonal to all other simple roots, in which case ( ω β ( h ) −
2) = −
1. We shall now estimatethe second number:
The inequality ω α ( h ) ≥ − n α,β )(6) holds for all α, β ∈ Π , except when α is orthogonal to all other simple roots, or when α and β are the simple roots of a simple ideal ˆ g of g of type A or C with α being the long simpleroot. Indeed, recall that n β,α ∈ { , − , − , − } , whence 2(1 − n β,α ) ∈ { , , , } . Thus the inequality ( ω α ( h ) − − n β,α )) < g of g to which α belongs. We consider the possible values of n β,α .If n β,α = 0, i.e., α and β are orthogonal, then ( ω α ( h ) − − n β,α )) = ω α ( h ) − < α is orthogonal to all other simple roots.If n β,α = −
3, then α and β are the long and short simple roots of ˆ g = g , respectively. Inthis case we have ω α ( h ) = 10 > ω β ( h ) = 6), so ( ω α ( h ) − − n β,α )) > n β,α = −
2, then ˆ g has two root lengths, α is long and β is short. According to Proposition1.2, if ω α ( h ) <
6, then ˆ g must have type C , C . For the long simple root of C ℓ we have ω α ( h ) = ℓ . Hence C is excluded, and we are left with C , where we have indeed ω ( h ) − − n α ,α ) = 4 − − n β,α = −
1, then we have the inequality ω α ( h ) <
4. Proposition 1.2 implies that thisoccurs exactly when ˆ g has type A or A .With this our claim about the inequality (6) is proved and the proof of the lemma is complete. (cid:3) The Kirwan stratification and the S -ample cone. For this subsection, we fix λ ∈ Λ ++ , the associated ample line bundle L λ on X , the projective embedding ϕ : X ⊂ P = P ( V λ )and the resulting momentum map µ = µ K S defined in (5). We have a K S equivariant function || µ || : X → R , This function defines an S -invariant Morse-type stratification of X , described in the projectivesetting by Ness and in a more general symplectic setting by Kirwan, [Kir84], [N84]. The strataare parametrized by the critical values of || µ || . The unstable locus consists of the strata arisingfrom nonzero critical values. In our case K S ∼ = SU , which acts transitively on the sphere in itscoadjoint representation. Hence the nonzero critical K S -orbits are exactly the orbits throughthe points of vanishing of the vector field induced by our fixed element h ∈ i k S . Since h isregular, X h = X H = { x w ; w ∈ W } . The corresponding critical values are || µ ( x w ) || = wλ ( h ) . According to Kirwan’s theorem the strata are parametrized by the critical points for which theintermediate values µ ( x w ) ∈ i t ∗ S are dominant, which in our case means positive. We are leadto consider the following partition of W , defined for the (any) dominant weight λW = W + ( λ ) ⊔ W − ( λ ) ⊔ W ( λ ) W + ( λ ) = { w ∈ W : wλ ( h ) > } W ( λ ) = { w ∈ W : wλ ( h ) = 0 } W − ( λ ) = { w ∈ W : wλ ( h ) < } . (7)By Lemma 1.3 and Lemma 2.1, we have w wλ ( h ) = − wλ ( h ) and x w w ∈ K S x w , hence w W ± ( λ ) = W ∓ ( λ ) , w W ( λ ) = W ( λ ) . Lemma 2.5. (i)
The critical values of || µ || on X are wλ ( h ) , for w ∈ W + ( λ ) , and ,whenever it is attained. The unstable connected critical sets are exactly the K S -orbitsthrough the points x w for w ∈ W + ( λ ) . (ii) For every w ∈ W we have Bx w = { x ∈ X : lim t →−∞ exp( th ) x = x w } . (iii) The Schubert variety Bx w is contained in X us ( λ ) if and only if either wλ ( h ) > or X ss ( λ ) = ∅ .Proof. Part (i) is a summary of the paragraph preceding the lemma. For part (ii), recall fromSection 1.1 that the set of H -weights of the tangent space T x w Bx w is exactly the inversionset Φ w − = ∆ + ∩ w ∆ − . Since the inversion set is closed under root addition, it gives rise to a subgroup N w contained the unipotent radical N of B . The subgroup N w acts simplytransitively on the Schubert cell, Bx w = N w x w . The inversion set is also the set of the positiveeigenvalues of h in T x w X . The H S -action on the Schubert cell N w x w can be linearized usingthe fact that the exponential map of G restricted to n is biholomorphic. Thus for x ∈ Bx w , wehave lim t →−∞ exp( th ) x = x w . Since the Schubert cells constitute a cell decomposition of X and each cell adheres to its own H S -fixed point, we can deduce part (ii).For part (iii), let us notice that the Schubert cell Bx w and the Schubert variety Bx w arepreserved by the H -action. The H -fixed points in the Schubert variety are given by the Bruhatorder Bx wH = Bx wH S = { x w ′ ; w ′ (cid:22) w } . From Lemma 2.4 we get w ′ λ ( h ) ≥ wλ ( h ) for every w ′ such that x w ′ ∈ Bx w with strict inequality whenever w ′ = w . Since λ ( h ) >
0, we have that0 ∈ Conv( µ ( Bx wH )) if and only if wλ ( h ) ≤
0. We may now deduce that Bx w ⊂ X us ( λ ) if wλ ( h ) >
0. We have to show that wλ ( h ) ≤ wλ ( h ) = 0, then the point x w is semistable. Suppose now that wλ ( h ) < w ′ λ ( h ) = 0 for all w ′ ≺ w . Then 0 ∈ Conv( µ ( Bx wH )) and hence Bx w contains H S -semistablepoints. If Bx w ∩ X S = ∅ , then from Lemma 2.3 we see that the H S -semistable points in thatintersection are also S -semistable. Note that Bx w ∩ X S = ∅ whenever l ( w ) ≥
3. Thus itremains to consider the situation Bx w ∩ X S = ∅ . Then we have wλ ( h ) < l ( w ) ≤ Bx w ⊂ S u ∈ W Sx u . These conditions occur only in two cases: case 1) w = s α with α being thesimple root of a factor of G of type A ; however from Lemma 2.4 we know that s α λ ( h ) < X ss ( λ ) = ∅ ; case 2) w = s α s β with α and β being the simple roots of G and G beingof type A × A ; in this case we have X ss ( λ ) = ∅ if and only if s α λ ( h ) = s β λ ( h ) = 0, so theSchubert cell contains semistable points whenever they exist. (cid:3) Theorem 2.6.
The Kirwan strata of the S -unstable locus in X with respect to L λ are the S -saturations of Schubert cells SBx w , for w ∈ W + ( λ ) , with the Schubert cell Bx w being theprestratum and K S x w being the critical set of || µ || . Thus X us ( λ ) = [ w ∈ W + ( λ ) S wλ , S wλ = SBx w . (8) The dimension of the strata is given by dim S wλ = l ( w ) + 1 , and consequently dim X us ( λ ) = 1+max { l ( w ) : w ∈ W + ( λ ) } , codim X X us ( λ ) = − { l ( w ) : w ∈ W − ( λ ) } . Proof.
With the preceding lemma in view, the theorem is deduced by a direct application ofKirwan’s theorem. From part (i) of the lemma we know that the nonzero critical values of || µ || are wλ ( h ) , for w ∈ W + ( λ ), with critical set K S x w . Let S wλ denote the Kirwan stratumcontaining x w . The fact that Bx w is the prestratum follows from parts (ii) and (iii) of Lemma2.5. This implies S wλ = SBx w and proves formula (8).The dimension formula dim S wλ = l ( w ) + 1 follows from dim Bx w = l ( w ), S ∩ B = B S andthe fact that the tangent line C e − · x w ∼ = s / b S is transversal to T x w ( Bx w ). (cid:3) The above theorem allows us, in particular, to detect the cases when the semistable locus isnonempty. Thus we can determine the S -ample cone C S ( X ), which, as we already mentionedis identified with the nulleigencone Cone( E ( S ⊂ G )) ⊂ Λ + R . This description is already knownfrom the work of Berenstein and Sjamaar, [BS00]. We formulate it below. Corollary 2.7.
The S -ample cone on X is defines by the following inequalities: C S ( X ) ∼ = Cone( E ) = { λ ∈ Λ + R : λ ( s α h ) ≥ for all α ∈ Π with α ⊥ Π \ { α }} . In particular, if G has no simple factors of rank 1, then all ample line bundles on X are S -ample, i.e. C S ( X ) ∼ = Cone( E ) = Λ + R . Proof.
Theorem 2.6 implies that X = X us ( λ ) if and only if there exists a simple root α ∈ Πsuch that s α λ ( h ) <
0. In Lemma 2.4 we have shown that this inequality holds if and only if α ⊥ Π \ { α } and upon writing λ = P β ∈ Π λ β ω β in the basis of fundamental weights, we have λ α > X β ∈ Π \{ α } λ β ω β ( h ) . This proves the corollary. (cid:3)
We record the following geometric formulation of the above corollary.
Corollary 2.8. If L λ is an ample and S -ample line bundle on X , i.e., λ ∈ Λ ++ ∩ Cone( E ) ,then all Schubert divisors intersect the S -semistable locus X ss ( λ ) . If furthermore λ belongsto the interior of the S -ample cone, i.e., λ ∈ Λ ++ ∩ Int Cone( E ) , then all Schubert divisorsintersect the S -stable locus. GIT-classes of ample line bundles.
The description of the unstable loci of S -ample linebundles on X given in Theorem 2.6 allows us to determine the GIT-equivalence classes of linebundles according to the definitions given at the end of Section 1.1. In particular, we determinethe chambers, in the sense of Definition 1.2. Recall the partition W = W + ( λ ) ⊔ W − ( λ ) ⊔ W ( λ )of the Weyl group defined in (7). Lemma 2.9.
The following hold: (i)
The partition (7) depends only on the GIT-class of λ , say C . Different GIT-classescorrespond to different partitions. We denote W ± ( λ ) = W ± ( C ) . (ii) For every GIT-class C and every ν ∈ Cone( E ) , the following are equivalent: (a) ν ∈ C ;(b) X ss ( C ) ⊂ X ss ( ν ) ;(c) W + ( ν ) ⊂ W + ( C ) . (iii) The 2-dimensional S orbits in X ss ( C ) are exactly the orbits Sx w with w ∈ W ( C ) . (iv) Suppose dim X ≥ . A GIT-class C is a chamber if and only if W ( C ) = ∅ .Proof. Part (i) follows from the definition of GIT-classes and Theorem 2.6. For part (ii), theequivalence between (a) and (c) follows from the property w W ± ( λ ) = W ∓ ( λ ), w W ( λ ) = W ( λ ); the equivalence between (b) and (c) follows from Theorem 2.6. For part (iii) recallfrom Proposition 2.2, the orbits of dimension 1 and 2 are exactly Sx w for w ∈ W , and we have Sx w = Sx w w . The H S -fixed point x w is semistable if and only if µ ( x w ) = ι ∗ ( wλ ) = 0 for λ ∈ C ,i.e. w ∈ W ( C ). Part (iv) follows from part (ii), since our group S is 3-dimensional and C is achamber if and only if X ss ( C ) consists only of 3-dimensional S -orbits. (cid:3) Summing up the preceding results we obtain the following.
Theorem 2.10.
The partition of the S -ample cone Cone( E ) into GIT-equivalence classes isgiven by the system of hyperplanes H w = { λ ∈ Λ R : λ ( wh ) = 0 } , w ∈ W .
The chambers are the connected components of
Cone( E ) \ ∪ w H w . S -Movable line bundles. Recall that a line bundle L λ on X is called S -movable, ifcodim X ( X us ( λ )) ≥
2. Assume λ ∈ Λ ++ ∩ Int Cone( E ), so that X us ( λ ) = X . Corollary 2.8tells us that X us ( λ ) cannot contain Schubert divisors. According to Theorem 2.6, we havecodim X X us ( λ ) = − { l ( w ) : wλ ( h ) < } . The dimension formula implies:codim X X us ( λ ) = 1 if and only if there exists w ∈ W with l ( w ) = 2 and wλ ( h ) < Theorem 2.11. If G has no simple factors with of type A , A or C , then for every λ ∈ Λ ++ the S -unstable locus X us ( λ ) has codimension at least 2 in X . In other words, every ample linebundle on X is S -movable.More generally, an S -ample line bundle L λ is S -movable if and only if s α s β λ ( h ) ≥ for allpairs of simple roots satisfying one of the following: (a) α, β are simple roots of a simple factor of G of type A or C ; (b) α is the simple root of a a simple factor of G of type A . The exceptional cases of rank 2 are considered in some detail in Section 4.3.
Quotients and their Picard groups
In this section we use the language of divisors rather than line bundles, as it is more suitablefor the context. This is unproblematic since X is smooth. Thus, C S ( X ) is interpreted as a coneof R -divisors on X (recall from Section 1.1 that C S ( X ) is the S -ample cone on X , identifiedwith the eigencone Cone( E ) ⊂ Λ + R ). We assume from now on that all simple factors of G have at least 5 positive roots. This isequivalent to the hypothesis of Theorem 2.11, whence we getcodim X ( X us ( λ )) ≥ λ ∈ Λ ++ . Let C be a fixed chamber in C S ( X ), let Y := Y ( C ) := X ss ( C ) //S be the corresponding quo-tient, and let π : X ss ( C ) → Y be the quotient morphism. Since the unstable locus of C is ofcodimension at least two, the results of [S14] apply and we obtain the following. Theorem 3.1.
The quotient Y is a Mori dream space. Moreover, (i) there is an isomorphism of Q -Picard groups Pic( Y ) Q ∼ = Pic( X ) Q ;(ii) there is an isomorphism of cones Eff( Y ) ∼ = C S ( X ) . (9) Moreover, for any line bunde L on X , there exists k ∈ N and a line bundle L on Y such that H ( X, L jk ) S ∼ = H ( Y, L j ) , j ∈ N . (10) Remark 3.1.
The fact that the unstable locus X us ( C ) is of codimension at least two yieldsthe identities (9) and (10) which allow us to, essentially, identify the Cox ring of Y with thesubring of S -invariants of the Cox ring of X . However, even without the assumption that achamber have a small unstable locus, the corresponding GIT-quotient would be a Mori dreamspace (cf. [M15, Thm. 5.5] ). In fact, we can say more about the convex geometry of the divisors on Y . We first recallthat the stable base locus , Bs( D ), of a Cartier divisor D on a variety Z is the intersection of the base loci of all positive multiples of D ;Bs( D ) = \ m ≥ Base( mD ) . An effective divisor D is said to be movable if codim(Bs( D ) , Z ) ≥
2. In the case when Z hasa finitely generated Picard group, we define the movable cone Mov( Z ) to be the closed convexcone in Pic( Z ) R generated by all movable divisors.We now have the following result about cones of divisors on the quotient Y . Theorem 3.2.
Let Y = X ss ( C ) //S be the quotient above. Then there is a equality of cones Eff( Y ) = Mov( Y ) , and an isomorphism of cones Nef( Y ) ∼ = C . Moreover, every nef Q -divisor D on Y is semiample, i.e., some positive multiple of D isbasepoint-free.Proof. In order to prove the first identity, let D be an effective divisor in the interior of Eff( Y ).The divisor π ∗ D on X ss ( C ) then extends uniquely to a divisor on X , which we also denote by π ∗ D . By the isomorphism (10) the stable base locus of D is given byBs( D ) = π ( X us ( π ∗ D ) ∩ X ss ( C )) . (11)Since the unstable locus of any divisor in C S ( X ) is of codimension at least two, and since thefibres of π all have the same dimension, (11) shows that D is movable. Hence, Int(Eff( Y )) ⊆ Mov( Y ). Since both cones are rational polyhedral, Y being a Mori dream space, the inclusionEff( Y ) ⊆ Mov( Y ) follows. This proves the first identity.For the second identity, we first note that every ample divisor is identified with a divisor in C by the isomorphism (9). Indeed, if D is a divisor on Y such that π ∗ D * C , then, by Theorem2.10, there exists a w ∈ W such that SBx w ⊆ X ss ( C ) ∩ X us ( π ∗ D ), so that π ( SBx w ) lies in thestable base locus Bs( D ). In particular, D cannot be ample. Hence, Ample( Y ) ⊆ C , so that wealso have Nef( Y ) ⊆ C . On the other hand, if A ∈ C is any R -divisor on the boundary of C , theinclusion of semistable loci X ss ( C ) ⊆ X ss ( A )holds [DH98], so that X us ( A ) ⊆ X us ( C ). If A is a Q -divisor, the identity (11) applied to mA ,for m ∈ N such that mA is an effective integral divisor, then shows that mA = π ∗ D for asemiample divisor D . This proves the second identity as well as the final claim. (cid:3) The exceptional cases of types A and C The results of the last section were derived under the assumption that G has no simplefactors of type A , A , C . Such simple factors create some complications, as seen in Theorem2.11 and, on the structural level, in Lemma 2.4. Here we present some details on the cases A and C . The particularity of these cases is in a sense due to the small dimension of the flagvarieties; dim SL /B = 3 and dim Sp /B = 4. The GIT-quotients by the action of a principal3-dimensional subgroup S with respect to any ample line bundle are, respectively, a point and arational curve. The Picard groups of these quotients are, respectively, trivial and infinite cyclic.The Picard groups of the flag varieties are lattices of rank 2, in both cases, so the isomorphismPic( X ) R ∼ = Pic( Y ) R does not hold. This is due to the fact that, in both cases, there are nomovable chambers among the GIT-classes. Referring to Theorem B stated in the Introduction, we observe a failure in parts (iv) and (v). Part (i) and (iii) hold without alteration. Part (ii)holds with some modifications.In the following two subsections we keep the general notation and relate it to specific modelsusing projective geometry. Let us recall some generalities. We use the embedding of X intothe product of (fundamental) partial flag varieties of G given by maximal parabolic subgroupscontaining B . We are dealing with classical groups of rank 2, so we have X ⊂ X × X definedby an incidence relation on the subspaces in the flags. To gain some understanding on the S -orbits, we use the fact that the principal subgroup has a unique closed orbit in each flagvariety - a rational normal curve. The 2-dimensional orbits are then constructed in terms ofthe tangential varieties to these rational curves. Recall that the tangential variety of a smoothprojective variety Y ⊂ P ( V ) is defined as the union T Y := ∪ [ v ] ∈ Y P ( T v ˆ Y ) ⊂ P ( V ); this union isa closed subvariety in P ( V ). We also denote by T [ v ] Y = P ( T v ˆ Y ) the linear projective subspaceof P ( V ) tangential to Y at a given point [ v ]. The tangential variety if a rational normal curvein C ⊂ P ( V ) is a surface preserved by the projective automorphism group of the curve ( SL or P SL ). If C is a line, then T C = C . If deg C ≥
2, there are two orbits of the automorphismgroup in T C - the curve itself and its complement T C \ C . The stabilizer of a point in T C \ C is a torus if deg C > C is a conic. The reader may refer to thebook [Lan12] for more information on tangential varieties. We now proceed with our examples.4.1. The case G = SL . We have here X = SL /B and S = SO ⊂ SL with respect to somenondegenerate quadric κ ∈ S ( C ) ∗ . According to Proposition 2.2, there are 4 orbits of S in X . The orbits can be described geometrically, using the model for X as the space of completeflags on C , or equivalently flags in P . The flag variety is an incidence variety in the productof Graßmannians, which in this case are X = P and its dual X = ( P ) ∗ . In the following wefreely interpret the points in ( P ) ∗ as projective lines in P . Thus X = { ( p, L ) ∈ P × ( P ) ∗ : p ∈ L } ⊂ P × ( P ) ∗ . The principal subgroup S has exactly one closed orbit in each of the projective planes, which wedenote by C ⊂ P and C ⊂ ( P ) ∗ . These are two planar conics. The isomorphism C → ( C ) ∗ induced by κ sends C to C via p T p C . Thus C consists of the lines tangent to C . Noticethat S acts transitively on the complement of either conic, P \ C j . We have1-dimensional: O = C = Sx = { ( p, L ) ∈ X : p ∈ C , L = T p C } ∼ = S/B S ;2-dimensional: O = Sx s = { ( p, L ) ∈ X : p / ∈ C , L tangent to C } ∼ = S/H S ; O = Sx s = { ( p, L ) ∈ X : p ∈ C , L secant to C } ∼ = S/H S ;3-dimensional: O = { ( p, L ) ∈ X : p / ∈ C , L secant to C } , where s , s are the simple reflections generating the Weyl group of G .The unstable loci of effective line bundles on X are the following: X us ( λ ) = C ∪ Sx s ∪ Sx s , λ ∈ Λ ++ ; X us ( kω ) = C ∪ Sx s , k ≥ X us ( kω ) = C ∪ Sx s , k ≥ . Thus there is a single GIT-class of ample line bundles on X , C = Λ ++ R , and the correspondingquotient is a point, Y = X ss ( C ) //S = { pt } . Thus Pic( Y ) = 0. This corresponds to the factthat S us spherical in G and thus H ( X, L λ ) is a multiplicity free S -module for every λ ∈ Λ + .In particular, dim H ( X, L λ ) ≤ The case G = Sp . Let Ω be a nondegenerate skew-symmetric 2-form on C and G = Sp be the corresponding symplectic group. The partial flag varieties of Sp are X = P and theLagrangian Graßmannian Q = Gr Ω (2 , C ). Since sp ∼ = so , the Pl¨ucker embedding presentsthe Lagrangian Graßmannian as a nondegenerate quadric Q ⊂ P . In the following we ofteninterpret the points of Q as projective lines in P without supplementary notation. We considerthe flag variety X = G/B as an incidence variety in the product P × Q given by X = Sp /B = { ( p, L ) ∈ P × Q : p ∈ L } , dim X = 4 . We begin with a description of the orbits of the principal subgroup SL ∼ = S ⊂ Sp . Note that S is mapped to a principal subgroup in SL and SL , under the two fundamental representationsof Sp , respectively, i.e., the representations remain irreducible for S . Thus S has unique closedorbits C ⊂ P and C ⊂ P , with C ⊂ Q ; these are rational normal curves of degrees 3 and4, respectively. Note that the tangent line T p C is Lagrangian for any p ∈ C . As in the caseof SL , we have C = { L ⊂ P : L tangent to C } . Furthermore, S has exactly three orbits in P and the same number in Q . Note that the tangential variety of C ⊂ P is contained in Q .The complements T C j \ C j are the 2-dimensional S -orbits in P and Q , respectively. Also notethat L ∈ T C implies L ∩ C = ∅ , where L is interpreted once as a point in Q and once as aline in P . For L ∈ Q , we denote by S L the stabilizer of L in S ; note that, when L is viewedas a line in P , S L may act nontrivially on L . In particular, for L ∈ T C \ C , the stabilizer S L is a Cartan subgroup of S and has three orbits in L : two fixed points and a copy of C × . Theprincipal curve C ⊂ X and the 2-dimensional S -orbits in X are given by:1-dimensional: O = C = Sx = Sx s s s s { ( p, L ) ∈ X : p ∈ C , L = T p C } ;2-dimensional: O = Sx s = { ( p, L ) ∈ X : p / ∈ C , L ∈ C } ; O = Sx s = { ( p, L ) ∈ X : p ∈ C , L ∈ T C \ C } ; O = Sx s s = { ( p, L ) ∈ X : p ∈ T C \ C , L ∈ T C \ C , p ∈ L S L } . The rest of the S -orbits are 3-dimensional.Let us now consider ample line bundle on X and their unstable loci. The restriction ofweights from H to H S is given by ι ∗ : Λ → Z ∼ = Λ S , ι ∗ ( ω ) = 3 , ι ∗ ( ω ) = 4 , ι ∗ ( a ω + a ω ) = 3 a + 4 a . To prove this formulae one may use Proposition 1.2, or compute here directly: by the principalproperty ι ∗ ( α j ) = 2 for both simple roots α , α (with | α | < | α | ); we have ω = (2 α + α )and ω = α + α , whence the formulae.We denote χ = 3 α + 2 α ; this is the integral generator of the ray in Λ + R corresponding to R + h via the Killing form. Proposition 4.1. (i)
For λ = mχ with m ∈ N , the line bundle L λ has S -unstable locusof dimension 2 (hence codimension 2 in X ), which is given by X us ( λ ) = C ∪ O ∪ O . (ii) For λ ∈ Λ + \ N χ , let ω j be the fundamental weight belonging to the same connectedcomponent of Λ + R \ R + χ as λ . Then the line bundle L λ has D j = SBx s j s i as an S -unstable divisor.Proof. The proposition follows directly from the description of the Kirwan stratification given inTheorem 2.6 applied to the case in hand, together with the remark that
SBx s j = Sx s j ∼ = S/H S ,and the dimension formulae dim SBx s i = 2 = dim X −
2, dim
SBx s j s i = 3 = dim X − (cid:3) Proposition 4.2.
There are 3 GIT-classes of ample bundles on X , besides the class of thetrivial line bundle, given by C = N χ , C = Λ ∩ Relint(Span R + { ω , χ } ) , C = Λ ∩ Relint(Span R + { ω , χ } ) Furthermore, the unstable loci are given by X us ( C ) = D , X us ( C ) = D , X us ( C ) = D ∩ D = C ∪ O ∪ O . The GIT-classes C j are chambers; the GIT-class C is S -movable.For λ ∈ Λ ++ , the quotient X ss ( λ ) //S is isomorphic to P .Proof. The statements on the unstable loci follow from the above proposition. We may nowobserve that all semistable orbits for C j are 3-dimensional, so that C j is a chamber. Also,codim X ( X us ( C )) = 2, hence the class C is S -movable. All three GIT-quotients are one dimen-sional. The quotients X us ( C j ) //S , j = 1 , P is the only 1-dimensional Mori dream space, we have X ss ( C j ) //S ∼ = P . Since C is S -movable, the quotient X ss ( C ) has a discrete Picard group, andhence it is also isomorphic to P . (cid:3) The divisor D j is a P -bundle over the tangential variety T C j . Indeed, we may define insta-bility in X with respect to the effective (but not ample) line bundle L ω j , by vanishing of theinvariant ring R ( ω j ) S . The image of the map X → P ( V ω j ) given by the sections of L ω j is the par-tial flag variety X j . This map factors through the projection π j : X ⊂ X × X → X j , whichis a P -bundle. Both invariant rings are polynomial in one variable, with R ( ω j ) S = C [ f j ],deg( f ) = 4 and deg( f ) = 3. The group S has three orbits in X j , one in each dimension1,2,3, with T C j being the 2-dimensional orbit-closure. Thus ( X j ) us ( ω j ) = T C j and hence X us ( ω j ) = π − j ( T C j ). The stabilizer of any point y ∈ T C j \ C j is a torus C × ⊂ S , and this torusacts on the fibre π − j ( y ) ∼ = P with three orbits - two fixed points and their complement. We canconclude that S has a unique open orbit in X us ( ω j ) with finite isotropy. Take y j = π j ( x s j ) (with i = j ) so that S y = H S . Then ( π − j ( y j )) H S = { x s j , x s j s i } and for x ∈ ( π − j ( y j )) H S \ { x s j , x s j s i } we have Sx = X us ( ω j ). Now note that π − j ( y j ) \ { x s j } ⊂ Bx s j s i , whence Sx ⊂ SBx s j s i . Since SB s j s i is irreducible, we have X us ( ω j ) = Sx = SBx s j s i = D j = X us ( C j ) . Acknowledgement:
We would like to thank Peter Heinzner for useful discussions and support.
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