Classification of Reductive Monoid Spaces Over an Arbitrary Field
aa r X i v : . [ m a t h . AG ] A ug Classification of Reductive Monoid Spaces Over anArbitrary Field
Mahir Bilen CanAugust 16, 2018
Abstract
In this semi-expository paper, we review the notion of a spherical space. In particu-lar, we present some recent results of Wedhorn on the classification of spherical spacesover arbitrary fields. As an application, we introduce and classify reductive monoidspaces over an arbitrary field.
The classification of spherical homogenous varieties, more generally of spherical actions,is an important, lively, and very interesting chapter in modern algebraic geometry. Itnaturally encompasses the classification theories of toric varieties, horospherical va-rieties, symmetric varieties, wonderful compactifications, as well as that of reductivemonoids. Our goal in this paper is to give an expository account of some recent workin this field. As far as we are aware, the broadest context in which a classification ofsuch objects is achieved, is in the theory of algebraic spaces. This accomplishment isdue to Wedhorn [34]. Here, we will follow Wedhorn’s footsteps closely to derive someconclusions. At the same time, our intention is to provide enough detail to make basicdefinitions accessible to beginners. We will explain a straightforward application ofWedhorn’s progress to monoid schemes.It is not completely wrong to claim that the origins of our story go back to Leg-endre’s work [18], where he analyzed the gravitational potential of a point surroundedby a spherical surface in 3-space. To describe the representative functions of his en-terprise, he found a clever change of coordinates argument and introduced the set oforthogonal polynomials { P n ( x ) } n ≥ via √ − hx + h = P n ≥ P n ( x ) h n , which are nowknown as Lagrange polynomials. It has eventually been understood that the P n ( x )’sare the eigenfunctions of the operator ∆ = ∂ ∂x + ∂ ∂y + ∂ ∂z restricted to the space of C ∞ functions on the unit 2-sphere. Nowadays, bits and pieces of these elementary factscan be found in every standard calculus textbook but their generalization to higherdimensions can be explained in a conceptual way by using transformation groups. et n be a positive integer, and let Q n denote the standard quadratic form on R n , Q n ( x ) := x + · · · + x n , x = ( x , . . . , x n ) ∈ R n . The orthogonal group, denoted by O ( Q n ), consists of linear transformations L : R n → R n such that Q n ( L ( x )) = Q n ( x ) for all x ∈ R n . It acts transitively on the n − S n − = { x ∈ R n : Q n ( x ) = 1 } , and the isotropy subgroup in O ( Q n ) of apoint x ∈ S n − is isomorphic to O ( Q n − ). It is not difficult to write down a Lie groupautomorphism σ : O ( Q n ) → O ( Q n ) of order two such that the fixed point subgroup O ( Q n ) σ is isomorphic to O ( Q n − ). In other words, S n − has the structure of a sym-metric space , that is, a quotient manifold of the form G/K , where G is a Lie groupand K = { g ∈ G : σ ( g ) = g } is the fixed subgroup of an automorphism σ : G → G with σ = id . It is known that the Laplace-Beltrami operator ∆ n := P ni =1 ∂ ∂x i gen-erates the algebra of O ( Q n )-invariant differential operators on S n − . Moreover, foreach k ∈ N , there is one eigenspace E k of ∆ n with eigenvalue − k ( k + n − E k defines a finite dimensional and irreducible representation of O ( Q n ). In addition, theHilbert space of square integrable functions on S n − has an orthogonal space decom-position, L ( S n − ) = P ∞ k =0 E k . The last point of this example is the most importantfor our purposes; the representation of O ( Q n ) on the polynomial functions on S n − ismultiplicity-free! All these facts are well known and can be found in classical textbookssuch as [14] or [7].We will give another example to indicate how often we run into such multiplicity-freephenomena in the theory of Lie groups. This time we start with an arbitrary compactLie group, denoted by K , and consider C ( K, C ), the algebra of continuous functionson K with complex values. The doubled group K × K acts on K by translations:( g, h ) · x = gxh − for all g, h, x ∈ K. In particular, we have a representation of K × K on C ( K, C ), which is infinite dimen-sional unless K is a finite group. Let F ( K, C ) denote the subalgebra of representativefunctions , which, by definition, are the functions f ∈ C ( K, C ) such that K × K · f is contained in a finite dimensional submodule of C ( K, C ). The theorem of Peterand Weyl states that F ( K, C ) is dense in C ( K, C ). Moreover as a representation of K × K , F ( K, C ) has an orthogonal space decomposition into finite dimensional irre-ducible K × K -representations, each irreducible occurring with multiplicity at mostone.We have a quite related, analogous statement on the multiplicity-freeness of the G × G -module structure of the coordinate ring C [ G ] of a reductive complex algebraicgroup G . Indeed, it is a well known fact that on every compact Lie group K there existsa unique real algebraic group structure, and furthermore, its complexification K ( C )is a complex algebraic group which is reductive. Conversely, any reductive complexalgebraic group has an algebraic compact real form and this real form has the structureof a compact Lie group. Two compact Lie groups are isomorphic as Lie groups if andonly if the corresponding reductive complex algebraic groups are isomorphic (see [8,22]).The unifying theme of these examples, as we alluded to before, is the multiplicity-freeness of the action on the function space of the underlying variety or manifold. It urns out that the multiplicity-freeness is closely related to the size of orbits of certainsubgroups. To explain this more clearly, for the time being, we confine ourselves to thesetting of affine algebraic varieties that are defined over C . But we have a disclaimer:irrespective of the underlying field of definitions, our tacit assumption throughout thispaper is that all reductive groups are connected unless otherwise noted.Now, let G be a reductive complex algebraic group, and let X be an affine variety onwhich G acts algebraically. We fix a Borel subgroup B , that is, a maximal connectedsolvable subgroup of G . It is well known that every other Borel subgroup of G isconjugate to B . The action of G on X gives rise to an action of G , hence of B , onthe coordinate ring C [ X ]. Let us assume that B has finitely many orbits in X , so, inthe Zariski topology, one of the B -orbits is open. Let χ be a character of B and x be a general point from the dense B -orbit. Let f be a regular function that is onlydefined on the open orbit. Hence, we view f as an element of C ( X ), the field of rationalfunctions on X . If f is an χ -eigenfunction for the B -action, that is, b · f = χ ( b ) f for all b ∈ B, then the value of f on the whole orbit is uniquely determined by χ and the base point x . Indeed, any point x from the open orbit has the form x = b · x for some b ∈ B ,therefore, f ( x ) = b − · f ( x ) = χ ( b − ) f ( x ) . This simple argument shows that there exists at most one χ -eigenvector in C [ X ] whoserestriction to the open B -orbit equals f . As irreducible representations of reductivegroups are parametrized by the ‘highest’ B -eigenvectors, now we understand that thenumber of occurrence of the irreducible representation corresponding to the character χ in C [ X ] cannot exceed 1. In other words, C [ X ] is a multiplicity-free G -module. Theconverse of this statement is true as well; if a linear and algebraic action G × C [ X ] → C [ X ] is multiplicity-free, then a Borel subgroup of G has finitely many orbits in X ,and one of these orbits is open and dense in X (see [3], as well as [32, 24]). This bringsus to the (special case of a) fundamental definition that will occupy us in the rest ofour paper. Definition 1.1.
Let k be an algebraically closed field, and let G be a reductive al-gebraic group defined over k . Let X be a normal variety that is defined over k , andfinally, let ψ : G × X → X be an algebraic action of G on X . If the restriction of theaction to a Borel subgroup has finitely many orbits, then the action is called spherical ,and X is called a spherical G -variety .Let H ⊂ G be a closed subgroup in a reductive group G . The homogenous space G/H is called spherical if BH is an open dense subvariety in G for some Borel subgroup B ⊂ G . A G -variety X is called an equivariant embedding of G/H if X has an openorbit that is isomorphic to G/H . In particular, spherical varieties are normal G -equivariant embeddings of spherical homogenous G -varieties. To see this, let x be ageneral point from the open B -orbit in a spherical variety X , and let H denote theisotropy subgroup H = { g ∈ G : g · x = x } in G . Clearly, G/H is isomorphic to theopen G -orbit, and it is a spherical subvariety of X . It follows that X is a G -equivariantembedding of G/H . t goes without saying that this theory, as we know it, owes its existence to thework of Luna and Vust [20] who classified the equivariant embeddings of sphericalhomogenous varieties over algebraically closed fields of characteristic 0. Their resultsare extended to all characteristics by Knop in [16]. What is left is the classificationof spherical subgroups (over arbitrary fields) and this program is well on its way; seethe recent paper [2] and the references therein. For a good and broad introductionof the field of equivariant embeddings, up to 2011, we recommend the encyclopedictreatment [31] of Timashev.We mentioned in the first paragraph of this introduction that the examples ofspherical embeddings include algebraic monoids. By definition, an algebraic monoidover an algebraically closed field is an algebraic variety M endowed with an associa-tive multiplication morphism m : M × M → M and there is a neutral element forthe multiplication. The foundations of these monoid varieties are secured mainly bythe efforts of Renner and Putcha, who chiefly developed the theory for linear (affine)algebraic monoids (see [28, 25]). From another angle, Brion and Rittatore looked atthe general structure of an algebraic monoid. Amplifying the importance of linearalgebraic monoids, Brion and Rittatore showed that any irreducible normal algebraicmonoid is a homogenous fiber bundle over an abelian variety, where the fiber over theidentity element is a normal irreducible linear algebraic monoid (see Brion’s lecturenotes [5]). In this regard, let us recite a result of Mumford about the possible monoidstructure on a complete irreducible variety ([21, Chapter II]): if a complete irreduciblevariety X has a (possibly nonassociative) composition law m : X × X → X with aneutral element, then X is an abelian variety with group law m . In other words, anirreducible and complete algebraic monoid is an abelian variety. This interesting resultof Mumford is extended to families by Brion in [5].A G -equivariant embedding is said to be simple if it has a unique closed G -orbit.A reductive monoid is an irreducible algebraic monoid whose unit-group is a reduc-tive group. The role of such monoids for the theory of equivariant embeddings wasunderstood very early; Renner recognized in [27] and [26] that the normal reductivemonoids are simple G × G -equivariant embeddings of reductive groups. Explicatingthis observation, Rittatore showed in [29, Theorem 1] that every irreducible algebraicmonoid M is a simple G ( M ) × G ( M )-equivariant embedding of its unit-group G ( M ).In the same paper, by using Knop’s work on colored fans, which we will describe inthe sequel, Rittatore described a classification of reductive monoids in terms of coloredcones. This classification is a generalization of the earlier classification of the reductivenormal monoids by Renner [27] and Vinberg [33].Let k be an algebraically closed field. We define a toric variety over k as a normalalgebraic variety on which the torus ( k ∗ ) n acts (faithfully) with an open orbit. Toricvarieties are prevalent in the category of spherical varieties in the following sense: if X is a spherical G -variety over k , then the closure in X of the T -orbit of a generalpoint from the open B -orbit will be a toric variety. Here, T stands for a maximaltorus contained in B . Also, let us not forget the fact that the affine toric varieties areprecisely the commutative reductive monoids, [29, Theorem 3].Now let k be an arbitrary field, and let T be a torus that is isomorphic to ( k ∗ ) n ,where both the isomorphism and the torus T are defined over k . The technical term or such a torus is k -split or split torus over k . Toric varieties defined by split tori areparametrized by combinatorial objects, the so-called fans. This fact is due to Demazurein the smooth case [12] and Danilov for all toric varieties [10]. A fan F in Q n is a finitecollection of strictly convex cones such that 1) every face of an element C from F liesin F ; 2) the intersection of two elements of F is a face of both of the cones. Here, bya cone we mean a subset of Q n that is closed under addition and the scaling action of Q ≥ . A face of a cone C is a subset of the form { v ∈ C : α ( v ) = 0 } , where α is a linearfunctional on Q n that takes nonnegative values on C . Let us define two more notionsthat will be used in the sequel. A cone is called strictly convex , if it does not contain aline. The relative interior of C , denoted by C , is what is left after removing all of itsproper faces.The toric varieties defined by nonsplit tori are quite interesting and their classifi-cation is significantly more intricate. Parametrizing combinatorial objects in this case,as shown by Huruguen in [15] are fans that are stable under the Galois group of asplitting field. To explain this, we extend our earlier definition of toric varieties asfollows. Let k be a field, and let k denote an algebraic closure of k . Let T be a torusdefined over k . A normal T -variety Y is called a toric variety over k if Y ( k ) is a toricvariety with respect to T ( k ). (This notation will be made precise in Section 2.) Let uscontinue with the assumption that Y is a toric variety with respect to a k -split torus T , and let k ′ ⊆ k be a field extension with T not necessarily split over k ′ . Let Γ denotethe Galois group of the extension. Since all tori become split over a finite separablefield extension, we will assume also that k ′ ⊂ k is a finite extension. Of course, it mayhappen that Y is not defined over k ′ . If it is defined, then Y ( k ′ ) is called the(!) k ′ -formof Y . In [15, Theorem 1.22], Huruguen gives two necessary and sufficient conditionsfor the existence of a k ′ -form. The first of these two conditions is rather natural inthe sense that the fan of Y ( k ) is stable under the action of Γ . The second conditionis also concrete, however, it is more difficult to check. Actually, it is a criterion aboutthe quasiprojectiveness of an equivariant embedding in terms of the fan of the em-bedding. Its colored version, namely a quasiprojective colored fan, which is also usedby Huruguen, is introduced by Brion in [4]. We postpone the precise definition of aquasiprojective colored fan to Section 4.1, but let us mention that, in [15], Huruguenshows by an example that quasiprojectiveness is an essential requirement for Y to havea k ′ -form.Using the underlying idea that worked for toric varieties, in the same paper, Hu-ruguen proves more. Let Y be a spherical homogenous variety for a reductive group G defined over a perfect field k . (The perfectness assumption is a minor glitch; it will notbe needed once we start to work with the algebraic spaces as Wedhorn does. Indeed,it is required here so that the homogenous varieties have rational points.) Let k be analgebraic closure of k , and let Γ denote the Galois group of k ⊂ k . The introduction ofthe absolute Galois group, which is often too big, is not a serious problem since in thesituations that we are interested in, the absolute Galois group factors through a finitequotient. Let Y be a G -equivariant embedding of a spherical homogenous G -variety Y that is defined over k . In [15, Theorem 2.26], Huruguen shows that a G -equivariantembedding Y of Y has a k -form if and only if the colored fan of Y ( k ) is Γ -stable andit is “quasiprojective with respect to Γ .” Once again, Huruguen shows by examples hat the failure of the second condition implies the nonexistence of k -forms.As Wedhorn shows in [34] via algebraic spaces, as soon as we get over the contrivedemotional barrier set in front of us by algebraic varieties, we happily see that theexistential questions (about k -forms) disappear. Roughly speaking, in some sense,algebraic spaces are to schemes, what schemes are to algebraic varieties. Such is thetransition from spherical varieties to spherical spaces. Definition 1.2.
For an arbitrary base scheme S and a reductive group scheme G over S , a spherical G -space over S is a flat separated algebraic space of finite presentationover S with a G -action such that the geometric fibers are spherical varieties.Notice that we have not defined reductive S -groups yet. But timeliness may notbe the only problematic aspect of Definition 1.2. Understandably, it may look overlygeneral at first sight. Nevertheless, the definition has many remarkable consequences.For example, according to this definition, the property for a flat finitely presented G -space with normal geometric fibers to be spherical is open and constructible on the basescheme. Moreover, for a flat finitely presented subgroup scheme H of G the property tobe spherical is open and closed on the base scheme. Most importantly for our purposes,the specialization of the base scheme to the spectrum of a field in Definition 1.2 yieldsa general classification of spherical G -spaces over arbitrary fields in terms of coloredfans that are stable under the Galois group. In fact, for spaces over fields, Wedhorn’sdefinition is much easier to use.Since explaining the concepts associated with spherical algebraic spaces and col-ored fans will take a bulky portion of our paper, we postpone giving the ballistics ofWedhorn’s theory to Section 4.2. Nearing the end of our lengthy introduction, let usmention that as a rather straightforward consequence of Wedhorn’s deus ex machinawe obtain a classification of “reductive monoid spaces,” the only new definition thatwe offer in this paper.The organization of our paper is as follows. In Section 2, we introduce what is re-quired to explain spherical algebraic spaces in the following order: Subsection 2.1 is onthe fundamentals; we introduce the notion of scheme as a functor. Subsection 2.2 setsup the notation for algebraic spaces. In Subsection 2.3, we talk about group schemes,which is followed by Subsection 2.4 on reductive group schemes. The Subsection 2.5 isdevoted to parabolic subgroups. In Subsection 2.6, we briefly discuss actions of groupschemes. In Section 3, we discuss affine monoid schemes and prove a souped-up versionof a result of Rittatore on the unit-dense algebraic monoids. The beginning of Section 4is devoted to the review of colored fans. In Subsection 4.1, we review the quasiprojec-tiveness criterion of Brion and mention Huruguen’s theorem on the k -forms of sphericalvarieties over perfect fields. In Subsection 4.2, we present Wedhorn’s generalization ofHuruguen’s results. In particular we talk about his colored fans for spherical algebraicspaces. In the subsequent Section 5, we introduce reductive monoid spaces and applyWedhorn’s and Huruguen’s theorems. Finally, we close our paper in Subsection 5.1 bypresenting an application of our observations to the lined closures of representations ofreductive groups, whose geometry is investigated by De Concini in [11]. Acknowledgement.
I thank the organizers of the 2017 Southern Regional Alge-bra Conference: Laxmi Chataut, J¨org Feldvoss, Lauren Grimley, Drew Lewis, Andrei avelescu, and Cornelius Pillen. I am grateful to J¨org Feldvoss, Lex Renner, SoumyaDipta Banerjee, and to the anonymous referee for their very careful reading of thepaper and for their suggestions, which improved the quality of the article. This workwas partially supported by a grant from the Louisiana Board of Regents. The purpose of this section is to set up our notation and provide some backgroundon algebraic spaces, reductive k -groups, and reductive k -monoids. We tried to givemost of the necessary definitions for explaining the logical dependencies. For standardalgebraic geometry facts, we recommend the books [13] and [17]. (It seems to us thatthe Stacks Project [30] will eventually replace all standard references.) In addition, wefind that Brion’s lecture notes in [6] are exceptionally valuable as a resource for thebackground on algebraic groups. Notation: Throughout our paper k will stand for a field, as G does for a group. k is called a perfect field if every algebraic extension of k is separable. A point that we want to make in this section is that the only way to know a groupscheme is to know all (affine) group schemes related to it. In fact, this statement is atheorem.Terminology:
Let ( X, F ) be a pair of a topological space X and a sheaf of rings F on X . If for each point x ∈ X , the corresponding stalk F x is a local ring, then thepair ( X, F ) is called a locally ringed space . An affine scheme is a locally ringed spacewhich is of the form (Spec( R ) , O Spec( R ) ), where R is a commutative ring, and Spec( R )is its spectrum endowed with the Zariski topology. The sheaf O Spec( R ) is the structuresheaf of Spec( R ). A scheme is a locally ringed space ( X, O X ) which has a covering byopen subsets X = S i ∈ I U i such that each pair ( U i , O X | U i ) is an affine scheme.There is a tremendous advantage of using categorical language while studying ascheme in relation with others. Therefore, we proceed with the identification of schemeswith their functors of points. In this regard, we will use the following standard notationthroughout: Obj ( C ) : the class of objects of a category C Mor ( C ) : the class of morphisms between the objects of C Mor(
X, Y ) : the set of morphisms from X to Y , where X, Y ∈ Obj ( C ) C o : the category opposite to C (schemes) : the category of schemes (sets) : the category of sets The functor of points of a scheme X is the functor h X : (schemes) o → (sets) thatis defined by the following assignments: . if Y ∈ Obj ( (schemes) o ), then h X ( Y ) = Mor( Y, X );2. if f ∈ Mor(
Y, Z ), then h X ( f ) is the set map h X ( f ) : h X ( Z ) −→ h X ( Y ) g f ◦ g Let h denote the following natural transformation: h : (schemes) → functors((schemes) o , (sets)) X h X (2.1)Here, functors(-,-) stands for the category whose objects are functors, and its mor-phisms are the natural transformations between functors. It follows from Yoneda’sLemma that (2.1) is an equivalence onto a full subcategory of the target category. Inother words, a scheme X is uniquely represented by its functor of points h X . Terminology:
In the presence of a morphism Y → X between two schemes X and Y , we will occasionally say that Y is a scheme over X or that Y is an X -scheme . Ifthere is no danger for confusion, we will write X ( Y ) for h X ( Y ), which is the set of allmorphisms from Y to X . If, in addition, Y is an affine scheme of the form Y = Spec( R ),then we often write X ( R ) instead of X ( Y ), and we say that X is a scheme over R .We want to show that the functor h in (2.1) behaves well upon restriction to thecategory of schemes over an affine scheme. To this end, let R be a commutative ring,and denote the category of R -schemes by ( R -schemes) . A morphism in this categoryis a commutative diagram of morphisms as in Figure 2.1. X Y
Spec( R )Figure 2.1: A morphism in ( R -schemes) . It is well known that the category ( R -schemes) is equivalent to the opposite cat-egory of commutative R -algebras, denoted by ( R -algebras) o . The highlight of thissubsection is the following result whose proof easily follows from the definitions. Proposition 1.
The functor h : ( R -schemes) −→ functors (( R -algebras),(sets)) , which is obtained from (2.1) by restriction to the subcategory ( R -schemes) , is anequivalence onto a full subcategory of the target category. In particular, a scheme over R is determined by the restriction of its functor of points to the category of affineschemes over R . or our purposes, the most important consequence of Proposition 1 is that an R -scheme can be thought of as a sheaf of sets on the category of R -algebras with respectto the Zariski topology. Closely related to this sheaf realization of schemes is the notionof an “algebraic space.” Concisely, and very roughly speaking, an algebraic space is asheaf of sets on the category of R -algebras with respect to the “´etale topology.” Wewill give a more precise definition of algebraic spaces in the next subsection. Next, weremind ourselves of some basic notions regarding the morphisms between schemes.1. A ring homomorphism f : R → S is called flat if the associated induced functor f ∗ : ( R -modules) → ( S -modules) is exact, that is to say, it maps short-exact sequences to short-exact sequences.In this case, the morphism f ∗ : Spec( S ) → Spec( R ) is called flat. More generally,a map f : X → Y of schemes is called flat if the induced functor f ∗ : (Quasicoherent sheaves on Y ) → (Quasicoherent sheaves on X ) is exact.2. A ring homomorphism f : R → S is called of finite presentation if S is isomorphic(via f ) to a finitely generated polynomial algebra over R and the ideal of relationsamong the generators is finitely generated. A map f : X → Y of schemes is called a map of finite presentation at x ∈ X if there exists an affine open neighborhood U = Spec( S ) of x in X and an affine open neighborhood V = Spec( R ) of Y with f ( U ) ⊆ V such that the induced ring map R → S is of finite presentation. Moregenerally, a map f : X → Y of schemes is called of locally finite presentation if itis of finite presentation at every point x ∈ X .3. A map f : X → Y of schemes is called separated if the induced diagonal morphism X → X × Y X is a closed immersion.4. A map f : X → Y of schemes is called unramified if the induced diagonal mor-phism X → X × Y X is an open immersion.5. A map f : X → Y of schemes is called ´etale if it is unramified, flat, and of locallyfinite presentation.Associated with these types of maps, we have two important topologies. These are1. The topology on the category of schemes associated with the ´etale morphisms;this topology is called the ´etale topology.
2. The topology on the category of schemes associated with the set of maps, flatand of locally finite presentation; this topology is called the fppf topology.
Let S be a scheme. An algebraic space X over S is a functor X : ( S -schemes) o −→ (sets) satisfying the following properties: i) X is a sheaf in the fppf topology.(ii) The diagonal morphism X → X × S X is representable by a morphism of schemes.(iii) There exists a surjective ´etale morphism e X → X , where e X is an S -scheme.It turns out that, in this definition, specifically in part (i), replacing the fppf topologyby the ´etale topology on the category of schemes does not cause any harm. In otherwords, the resulting functor describes the same algebraic space (see [30, Tag 076L]).Let us mention in passing that the category of schemes is a full subcategory of thecategory of algebraic spaces.We will follow the standard assumption, as in [34] (and as in [17]), that all algebraicspaces are quasiseparated over some scheme, hence they are Zariski locally quasisep-arated. This implies that our algebraic spaces are reasonable and decent in the senseof [30, Tag 03I8] and [30, Tag 03JX].Our final note in this subsection is that if k is a separably closed field, then analgebraic space over k is a k -scheme. Let R be a commutative ring. A group scheme over R is an R -scheme whose functor ofpoints factors through the forgetful functor from the category of groups to the categoryof sets. In a nutshell, an R -scheme G is called a group scheme over R if for every R -scheme S , there is a natural group structure on G ( S ) which is functorial with respectto the morphisms R → S . Terminology: If G is a (group) scheme over R , then the morphism G → Spec( R ) iscalled the structure morphism . Definition 2.2.
Let k be a field and let G be a group scheme over k . We will call G a k -group (or, an algebraic group over k ) if G is of finite type as a scheme over k . If k ′ is a subfield of k , then G is said to be defined over k ′ if G , as a scheme, and all of itsgroup operations as morphisms are defined over k ′ .Let S be a scheme, and let G be a group scheme over S . G is called affine, smooth,flat, or separated , respectively, if the structure morphism G → S is affine, smooth, flat,or separated, respectively. We have some remarks regarding these properties:1. Any k -group G is separated as an algebraic scheme. Indeed, the diagonal in G × G is closed as being the inverse image of the “identity” point of G ( k ) underthe morphism t : G × G → G defined by t ( g, h ) = gh − .2. Let k be a perfect field, and let G be a k -group scheme. If the underlying schemeof G is reduced, that is to say, the structure ring has no nilpotents, then G issmooth (see [13, Pg 287]).3. The Cartier’s theorem states that if the characteristic of the underlying field k is 0, then a k -group scheme is reduced (see [21, Pg 101]). It follows that, incharacteristic 0, all k -groups are smooth. ext, we will define a particular k -group that is of fundamental importance for thewhole development of algebraic group theory. Example 2.3.
Let V be a finite dimensional vector space over k . The general lineargroup , denoted by GL ( V ), is the group functor such that GL ( V )( S ) = the automorphism group of the sheaf of O S -modules O S ⊗ k V , where S is a k -scheme. By choosing a basis for V , we see that GL ( V )( S ) is isomorphicto the group of invertible n × n matrices with coefficients in the algebra O S ( S ), hence, GL ( V ) is represented by the open affine schemeGL n = { P ∈ A n : det P = 0 } . It follows that GL ( V ) is smooth and connected. Definition 2.4.
A group scheme G is called linear if it is isomorphic to a closedsubgroup scheme of GL n for some positive integer n . If V is a vector space, then ahomomorphism ρ : G → GL ( V ), which is a morphism of schemes, is called a linearrepresentation of G on V . In this case, V is called a G -module .Note that if a k -group scheme is linear, then it is an affine scheme. Note converselythat, every k -group scheme which is affine and of finite type is a linear group scheme(see [6, Proposition 3.1.1]).In some sense, atomic pieces of k -groups are given by the following two very special k -groups: Example 2.5.
Let + and · denote the addition and the multiplication operations onthe field k . • The additive 1-dimensional k -group, denoted by G a , is the affine line A k consid-ered with the group structure ( k, +). • The multiplicative 1-dimensional k -group, denoted by G m , is A k − { } consideredwith the group structure ( k ∗ , · ). Definition 2.6. A k -group G is called a torus if there exists an isomorphism ζ : G → G m × · · · × G m ( n copies, for some n ≥
1) over some field containing k . Assuming that G is a torus defined over k ′ , and that k ′ is a subfield of k , G is called k -split if both of G and ζ are defined over k .A modern proof of the following basic result can be found in [9, Appendix A]. Theorem 1 (Grothendieck) . Let k ′ and k be two fields such that k ′ ⊂ k . Let G be asmooth connected affine k -group. If G is defined over k ′ , then G contains a maximal k ′ -split torus T such that T ( k ) is a maximal torus of G ( k ). .4 Reductive group schemes. Almost any fact about reductive group schemes can be found in Conrad’s SGA3 replace-ment [9].
Let G be a k -group, and let ρ : G ֒ → GL ( V ) be a finite dimensional faithful linearrepresentation. An element g from G is called semisimple (respectively unipotent ) if thelinear operator ρ ( g ) on V is diagonalizable (respectively, unipotent). It is not difficultto check that these notions (semisimplicity and unipotency) are independent of thefaithful representation, and they are preserved under k -homomorphisms. Therefore,the following definition is unambiguous: Definition 2.7.
Let G be a linear algebraic k -group. A k -subgroup U of G is called unipotent if every element of U is unipotent.Next, we give the definition of a reductive group. However, we will do this in theopposite of the chronological development of the subject to emphasize the differences.So, let us start with the definition of the relative reductive group schemes. This is mostuseful for studying properties that are preserved in families over a commutative ring.Let S be a scheme, and let G be a smooth S -affine group scheme over S . Let s be apoint from S , and denote by G s the geometric fiber G × S Spec( k ( s )) of G → S . Here, k ( s ) is the residue field of s . If for each s ∈ S the fiber G s is a connected reductivegroup, then G is called a reductive S -group .So, what is a reductive k -group over an algebraically closed field? We take thisopportunity to define the ‘unipotent radical’ and the ‘radical’ of an algebraic group.Let G be a k -group (affine or not). There is a maximal connected solvable normal linearalgebraic subgroup, denoted by R ( G ( k )), and it is called the radical of G (see [6, Lemma3.1.4]). If R ( G ( k )) is trivial, then G ( k ) is called semisimple. The unipotent radical of G ( k ), denoted by R u ( G ( k )), is the maximal connected normal unipotent subgroupof G ( k ). If R u ( G ( k )) is trivial, then G ( k ) is called reductive. Clearly, semisimplicityimplies reductivity. Notice also that we have no connectedness assumption here. If thecharacteristic of k is 0, then the property of reductiveness of the identity component of G is equivalent to the semisimplicity of all linear representations of G . This equivalencefails in positive characteristics (see [9, Remark 1.1.13.]).In the rest of our paper we will focus mainly on the “relative” reductive groupschemes over fields. Definition 2.8.
Let k be a field. A k -group G is called reductive if the geometric fiber G k (which we take to be equal to G ( k )) is a connected reductive group in the sense ofthe previous paragraph. Let G be a reductive k -group. A subgroup P ( k ) of G ( k ) is called parabolic if G ( k ) /P ( k )has the structure of a projective variety. More generally, a smooth affine k -subgroup of G is called a parabolic subgroup if P ( k ) is a parabolic subgroup of G ( k ).A Borel subgroup in G ( k ) is a maximal connected solvable subgroup. More generally,a parabolic k -subgroup B of G is called a Borel subgroup if B ( k ) is a Borel subgroupin G ( k ).A fundamentally important result that is due to Borel (see [1, Theorem 11.1]) statesthat any two Borel subgroups are conjugate in G ( k ), and furthermore, for any Borel k -subgroup B ( k ) the quotient G ( k ) /B ( k ) is projective. Of course, according to theabove definition of parabolic k -subgroups, a Borel subgroup in G ( k ) is a (minimal)parabolic k -subgroup. Example 2.9.
The upper triangular subgroup T n of GL n is a Borel subgroup. As aconsequence of the Lie-Kolchin Theorem (see [1, Corollary 10.2]) any connected solvablegroup admits a faithful representation with image in T n .It may happen that G ( k ) is defined over a field but has no nontrivial Borel subgroup.This holds true, even for some classical groups, as we will demonstrate in the nextclassic example (from [1]). Example 2.10.
Let k be a field whose characteristic is not 2, and let V be k -vectorspace. Let Q be a nondegenerate quadratic form on V , and let F denote its symmetricbilinear form. We assume that Q is isotropic , that is to say, there exists a nonzerovector v ∈ V such that Q ( v ) = 0. A subspace is called isotropic if it contains anisotropic vector; a subspace is called anisotropic if it contains no nonzero isotropicvector; a subspace is called totally isotropic if it consists of isotropic vectors only. A hyperbolic plane is a two-dimensional subspace E of V with a basis { e, f } with respectto which the restriction of F has the form F ( x e + x f, y e + y f ) = x y + x y .By the Witt’s Decomposition Theorem we know that the dimension q of a maximaltotally isotropic subspace is an invariant of Q . More precisely, it states that V contains q linearly independent hyperbolic planes H , . . . , H q , and V is an orthogonal direct sumof the form V ∼ = V o ⊕ q M i =1 H i , where V o is an anisotropic subspace. Let Q o denote the restriction of Q to V o .For i = 1 , . . . , q we choose a basis { e i , e n − q + i } for H i in such a way that the followingidentities are satisfied: F ( e i , e i ) = F ( e n − q + i , e n − q + i ) = 0 and F ( e i , e n − q + i ) = 1 . Here n is the dimension of V . Let { e j : j = q + 1 , . . . , n − q } denote a basis for V o .For each pair e i , e n − q + i ( i = 1 , . . . , q ) of basis elements and x ∈ k we have a linear map s i ( x ) : V ( k ) → V ( k ) defined by s i ( x ) e i = xe i , s i ( x ) e n − q + i = x − e n − q + i , and s i ( x ) f = f for f ∈ H ⊥ i . Clearly, the s i ( x )’s are semisimple and generate a diagonal torus, denoted by T , whoseelements expressed in the basis { e , . . . , e n } of V are of the formdiag( a , . . . , a q , , . . . , , a − , . . . , a − q ) , where a i ∈ k. f G denotes SO ( Q ), the k -group consisting of linear automorphisms of V that preservesthe quadratic form Q and of determinant 1, then S := T ∩ G is in fact a maximal torusof G . Note that the centralizer of S in G , denoted by Z G ( S ), is isomorphic to theproduct S × SO ( Q o ).The group G is a reductive k -group, and since S is a maximal torus, by definition, Z G ( S ) is a Cartan subgroup of G . Therefore Z G ( S ) is contained in a minimal parabolic k -subgroup P of G . In fact, Z G ( S ) is equal to the Levi component of P . Observe that,if n > q + 2, then SO ( Q o ) is not commutative, hence Z G ( S ) is not contained by aBorel subgroup. We conclude that if n > q + 2, then G does not have any Borelsubgroups.For connected semisimple k -groups, where k is a perfect field, the question of theexistence of Borel subgroups has a nice answer. Theorem 2 (Ono) . Let k ′ and k be two fields such that k ′ ⊂ k and k ′ is perfect. If G is a connected semisimple k -group, then G posses a Borel subgroup B defined over k ′ if and only if a maximal torus T of G is k ′ -split. In this case, all Borel subgroupscontaining T ( k ′ ) are conjugate by the elements of the group N G ( T )( k ′ ), where N G ( T )denotes the normalizer group of T in G . Proof.
See [23].Finally, we finish this section by mentioning another important related result.
Theorem 3 (Chevalley) . The parabolic subgroups of any connected linear algebraic k -group are connected, and moreover, the normalizer of a parabolic subgroup in G ( k )is equal to P ( k ). Proof.
See [1, Section 11].
Let G be a group scheme over S . An action of G on an S -scheme X is an S -morphism a : G × S X → X such that for any S -scheme T the morphism a ( T ) : G ( T ) × X ( T ) → X ( T ) is an actionof G ( T ) on X ( T ). In this case, X is called a G -scheme. If the action map a is clearfrom the context we will denote a ( g, x ) by g · x , where g ∈ G, x ∈ G ( S ).Let X be a G -scheme with respect to an action a : G × X → X . If X is an affinescheme, then for any scheme S and g ∈ G ( S ) we have an O S -algebra automorphismdefined as follows: ρ ( g ) : O S ⊗ k Γ( X, k ) −→ O S ⊗ k Γ( X, k )1 ⊗ f ( − )) ⊗ ( f ◦ a ( g − , − )) , where Γ( X, k ) is the global section functor applied to X . These automorphisms patchup to give a linear representation ρ : G → GL (Γ( X, k )) . (2.11) emark . If X is a G -scheme of finite type, then the representation (2.11) decom-poses Γ( X, k ) into a union of finite dimensional G -submodules ([6, Proposition 2.3.4]).Furthermore, if X is affine, then there exists a closed G -equivariant immersion of X into a finite dimensional representation of G ([6, Proposition 2.3.5]).We close this section by mentioning an important theorem of Brion on projectiveequivariant embeddings of algebraic groups. Let G be a k -group, and let H ⊆ G bea subgroup scheme. An equivariant compactification of the homogenous space G/H isa proper G -scheme X equipped with an open equivariant immersion G/H → X withschematically dense image. Theorem 4 (Brion) . Let G be a k -group, and let H ⊆ G be a subgroup scheme. Then G/H has an equivariant compactification by a projective scheme.
Proof.
See [6, Section 5.2].Some remarks are in order:1. The homogenous space
G/H of Theorem 4 is quasiprojective; this is a well-knowntheorem for algebraic groups in the classical sense. But notice here that Brionproves the result for not necessarily reduced algebraic groups; of course, if theground field is of characteristic zero, then Cartier’s theorem implies that G isreduced.2. If in addition G is smooth, then G/H has an equivariant compactification by anormal projective scheme.3. If the characteristic of k is 0, then every homogenous space has a smooth projec-tive equivariant compactification.4. Over any imperfect field, there exists smooth connected algebraic groups havingno smooth compactification (see [6, Remark 5.2.3] for an example). For an algebro-geometric introduction to the theory of (not necessarily affine) monoidand semigroup schemes we recommend Brion’s lecture notes [5] (see also [13, ChapterII] ).Let us define a monoid scheme by relaxing the condition of invertibility in thedefinition of group schemes. More precisely, let R be a commutative ring. A monoidscheme over R is an R -scheme whose functor of points factors through the forgetfulfunctor from the category of monoids to the category of sets. Definition 3.1.
Let k be an algebraically closed field. An algebraic monoid over k ,also called a k -monoid , is a monoid scheme over k whose underlying scheme is separatedand of finite type. This book is one of the few if not the only book in algebraic geometry that acknowledges monoid schemesas part of the theory of group schemes. emark . Our definition of k -monoids is somewhat more general than the one thatis used by Brion in [5] since we do not assume reducedness of the underlying schemestructure.It is not difficult to see that the category of k -groups forms a full subcategory ofthe category of k -monoids. We are mainly interested in the affine k -monoids but letus first mention a general result.Let M be a k -monoid, and let G = G ( M ) denote its unit-group. Then M is called unit-dense if G is dense in M . A (weaker) form of the following theorem was firstproven by Rittatore. Here, we will make use of Brion’s proof from [5, Section 3]. Theorem 5.
Let M be a unit-dense irreducible k -monoid, and let G denote the unit-group of M . If G is affine, then so is M . Proof. If M is reduced, then the result follows from [5, Theorem 2]. If M is notreduced, then we pass to a normalization f M which factors through the reduction M red .This follows from the fact that M is irreducible (hence Noetherian). Therefore, wecan apply [30, Lemma 28.52.2]. But a normalization is a finite morphism, therefore M red → M is a finite morphism. Since M red is affine (once again by [5, Theorem 2]),and since the image of an affine scheme under a finite morphism is affine, M is an affinescheme as well. We start with reviewing the classification schematics of the spherical embeddings overan algebraically closed field. Throughout this section k will denote an algebraicallyclosed field and we assume that all varieties are defined over k . As usual, let G be areductive group, let B be a Borel subgroup, and let T be a maximal torus contained in B . If K is an algebraic group, we will denote by X ∗ ( K ) the group of characters of K .Note that X ∗ ( B ) = X ∗ ( T ). This is because the commutator subgroup of B coincideswith the unipotent radical R u ( B ), and B = R u ( B ) ⋊ T .Let Y denote G/H , where H is a spherical subgroup of G . Quotients of affinegroups, in particular Y , have the structure of a quasiprojective variety. Recall that Y is a spherical G -variety if and only if B has only finitely many orbits with respect tothe left multiplication action on Y . By a theorem of Brion, this is equivalent to thestatement that B has an open orbit in Y . Thus, it should come as no surprise that the B -invariant rational functions on Y are among the main players in this game.The space of B -semiinvariant rational functions on Y is denoted by k ( Y ) ( B ) . Inother words, k ( Y ) ( B ) = { f ∈ k ( Y ) : b · f = χ ( b ) f for all b ∈ B and for some character χ of B } . (4.1)The gist of the classification schematics for spherical varieties will take place inside thevector space Hom Z ( X ∗ ( B ) , Q ). Indeed, it is easy to check that k ( Y ) ( B ) is a subgroup f k ( Y ) and that the assignment k ( Y ) ( B ) −→ X ∗ ( B ) (4.2) f χ f , where χ f is a character as in (4.1) is an injective group homomorphism. We will denotethe image of (4.2) by Ω Y , and we will denote the Q -vector space associated with thedual of Ω Y by Q Y . In other words, • Ω Y := { χ ∈ X ∗ ( B ) : b · f = χ ( b ) f for all b ∈ B and for some f ∈ k ( Y ) ( B ) } ; • Q Y := Hom Z ( Ω Y , Q ).We occasionally refer to Ω Y as the character group of the homogenous variety Y sincein the special case, where Y = G × G/ diag( G ) ∼ = G viewed as a spherical G × G -variety,the character group of Y is isomorphic to the ‘ordinary’ character group, Ω Y ∼ = X ∗ ( B ).We look closely at the divisors and their invariants on Y . A function ν : k ( Y ) → Q is called a Q -valued discrete valuation on k ( Y ) if for every a, b ∈ k ( Y ) ∗ , we have:1. ν ( k ( Y ) ∗ ) ∼ = Z and ν ( k ) = { } ;2. ν ( ab ) = ν ( a ) + ν ( b );3. ν ( a + b ) ≥ min { ν ( a ) , ν ( b ) } provided a + b = 0.We notice here that every Q -valued discrete valuation ν on k ( Y ) defines a function on Ω Y . More precisely, there is a map ρ : { Q -valued discrete valuations on k ( Y ) } −→ Q Y ν ρ ν such that ρ ν is the map that sends an element χ = χ f of Ω Y to ν ( f ), where f isthe B -semiinvariant that specifies χ (so we wrote χ = χ f ). Indeed, since χ is defineduniquely by f (up to a scalar), ρ ν is well-defined. Moreover, it is easy to check that ρ ν ( χ f χ g ) = ρ ν ( χ f ) + ρ ν ( χ g ), hence ρ ν ∈ Q Y . • A valuation ν is called G -invariant if ν ( g · a ) = ν ( a ) for every g ∈ G and a ∈ k ( Y ).We will denote the set of all Q -valued G -invariant discrete valuations on Y by V Y .It is not completely obvious, but nevertheless true, that the restriction ρ | V Y : V Y −→ Q Y is an injective homomorphism of abelian groups.The reason for which we started looking at discrete valuations defined on Y inthe first place is because many important discrete valuations come from irreduciblehypersurfaces. From now on, we will refer to irreducible hypersurfaces as prime divisors.For any spherical action of G on Y , we will consider the set of all B -stable prime divisorsin Y . More generally, if Y ′ is a normal spherical G -variety, then a color of Y ′ is definedas a B -stable, but not G -stable, prime divisor. In our case, since G acts transitivelyon Y = G/H , any B -stable prime divisor in Y is non- G -stable. The set of all B -stable, but not G -stable, prime divisors in Y is denoted by D Y .The elements of D Y are called the colors of Y .Let us point out that on a noetherian integral separated scheme, which is regular incodimension one, the local ring associated with each prime divisor is a discrete valuationring (DVR). In particular, since our Y is a smooth variety, the local ring of a color of Y at its generic point is a DVR, and we have the map e ρ : D Y → Q Y , (4.3)which is defined as the composition of ρ with the map that assigns a color to thecorresponding discrete valuation. Definition 4.4.
The colored cone of Y = G/H is the pair ( C Y , D Y ) where C Y is thecone in Q Y that is generated by ρ ( V Y ) and e ρ ( D Y ).So far what we have are some ‘birational invariants’ that are defined solely for Y = G/H , and we have not given any indication of how they are related to its embeddings.To see how all these basic ingredients come together to play a role, next, we introducethe notion of a colored fan. This will give us a generalization of the combinatorialclassification of toric varieties.Let Y be a G -equivariant embedding of Y . Let D Y denote the set of B -stable, butnot G -stable, prime divisors of Y . Clearly, D Y is a subset of D Y . Since Y is the open G -orbit in Y , we have k ( Y ) = k ( Y ). In particular, there is an extension of (4.3) to amap ρ : D Y → Q Y . Let π : D Y → D Y denote the partially defined map π ( S ) = S ∩ Y , whenever S ∩ Y is an element of D Y .Let us mention in passing that both of the sets D Y and D Y contain a finite number ofelements since Y is spherical. Note also that the set of G -invariant discrete valuationson Y is equal to V Y . Y ′ : a G -orbit in Yν S : the discrete valuation in k ( Y ) associated with a prime divisor S of Y V Y ′ ֒ → Y : the set of G -invariant valuations in k ( Y ) of the form ν S with Y ′ ⊂ S ⊂ Y D Y ′ ֒ → Y : the set of colors D ∈ D Y such that ∃ S ∈ D Y with Y ′ ⊂ S and D = π ( S ) C Y ′ ֒ → Y : the cone in Q Y that is generated by the images ρ ( V Y ′ ֒ → Y ) and e ρ ( D Y ′ ֒ → Y ) • The pair ( C Y ′ ֒ → Y , D Y ′ ֒ → Y ) is called the colored cone of the G -orbit Y ′ . A face of( C Y ′ ֒ → Y , D Y ′ ֒ → Y ) is a pair of the form ( C , D ), where C is a face of C Y ′ ֒ → Y suchthat(i) C ∩ ( C Y ′ ֒ → Y ) = ∅ ;(ii) e ρ − ( C ) ∩ D Y ′ ֒ → Y = ∅ .Any colored cone satisfies the following defining properties: The cone C Y ′ ֒ → Y is generated by e ρ ( D Y ′ ֒ → Y ) and a finite number of vectors of theform ρ ( ν S ), where S ∈ V Y ′ ֒ → Y . C2.
The relative interior of C Y ′ ֒ → Y has a nonempty intersection with the set ρ ( V Y ′ ֒ → Y ). C3. C Y ′ ֒ → Y is strictly-convex, that is to say, C Y ′ ֒ → Y ∩ ( −C Y ′ ֒ → Y ) = { } . C4. e ρ ( D Y ′ ֒ → Y ).Now we are ready to introduce the combinatorial objects which parametrize the G -equivariant embeddings of Y . Definition 4.5.
The following (finite) set is called the colored fan of Y : F Y := { ( C Y ′ ֒ → Y , D Y ′ ֒ → Y ) : Y ′ is a G -orbit in Y } . The colored fans satisfy the following defining properties:
F1.
Every face of a colored cone in F Y is an element of F Y . F2.
For every G -invariant valuation ν in V Y , there exists at most one colored cone( C , D ) in F Y such that v ∈ C .It is easy to make abstract versions of colored fans. Let V be a finite dimensionalvector space over Q . Starting with a subset V of V and a finite set D together witha set map e ρ : D → V , we define a colored fan associated with ( V, V , D , e ρ ) as a finitecollection of pairs ( C , E ), where C is a cone in V and E is a subset of D satisfying theproperties F1 , F2 , C1 − C4 . Of course, V plays the role of ρ ( V Y ) in Q Y and e ρ : D → V plays the role of e ρ : D Y → Q Y .Let H and H ′ be two closed subgroups in G such that the homogenous varieties Y := G/H and Z := G/H ′ are spherical. Let ϕ : Y → Z be a morphism of varieties. If ϕ is G -equivariant, then the resulting map on thecharacter groups ϕ ∗ : X ∗ ( Z ) → X ∗ ( Y ) is injective, hence, the ‘dual’ linear map ϕ ∗ : Q Y → Q Z is surjective. Furthermore, we have ϕ ∗ ( V Y ) = V Z . Let D cϕ denote the set ofcolors of Y that are mapped into Z dominantly, D cϕ := { D ∈ D Y : ϕ ( D ) is dense in Z } . In other words, D cϕ is the set of colors of Y which are too big, so, we may ignore(!)them in the combinatorial setup. We set D ϕ := D Y − D cϕ . Definition 4.6.
Let ϕ : Y → Z be a G -equivariant morphism between two sphericalhomogenous G -varieties. Let Y and Z denote two equivariant embeddings of Y and Z ,respectively. Let Y ′ and Z ′ be two G -orbits in Y and Z , respectively. The map ϕ is saidto be a morphism between the colored cones ( C Y ′ ֒ → Y , D Y ′ ֒ → Y ) and ( C Z ′ ֒ → Z , D Z ′ ֒ → Z ) ifwe have . ϕ ∗ ( C Y ′ ֒ → Y ) ⊆ C Z ′ ֒ → Z , and2. ϕ ( D ϕ ∩ D Y ′ ֒ → Y ) ⊆ D Z ′ ֒ → Z .The map ϕ is said to be a morphism between the colored fans F Y and F Z if for everycone ( C , D ) in F Y there exists a cone ( C ′ , D ′ ) in F Z such that ϕ : ( C , D ) → ( C ′ , D ′ ) isa morphism of cones.The following result, which is proven by Knop in [16], is a generalization of theclassification result of Luna and Vust for simple embeddings. Theorem 6 (Knop) . Let Y be a spherical homogenous G -variety, and let B be a Borelsubgroup in G . The assignment Y F Y is a bijective correspondence between theisomorphism classes of G -equivariant embeddings of Y and the isomorphism classes ofcolored fans associated with ( Q Y , V Y , D Y , e ρ ). In fact, this assignment is an equivalencebetween the category of equivariant embeddings of Y and the category of colored fansassociated with ( Q Y , V Y , D Y , e ρ ). Remark . As we mentioned before the theorem of Knop, the role of colored fansfor simple embeddings was already known. In fact, Luna and Vust had shown in [20]that the colored cone ( C Z֒ → Y , D Z֒ → Y ), where Z ֒ → Y is the closed orbit of Y , uniquelydetermines Y . Remark . It is not difficult to check that all definitions pertaining to the coloredcones make sense (definable) if we use a separably closed field instead of an algebraicallyclosed field.
It is useful to know when an equivariant embedding of a spherical homogenous varietyis affine, projective, or more generally quasiprojective. Such criteria are found by Brionin [4]. Here we only give Brion’s criterion for quasiprojectiveness.
Theorem 7 (Brion) . Let F Y be the colored fan of an equivariant embedding Y of aspherical homogenous G -variety Y . In this case, Y is quasiprojective if and only if foreach colored cone C Z := ( C Z֒ → Y , D Z֒ → Y ) in F Y there exists a linear form, denoted by ℓ Z , on Q Y such that the following two conditions are satisfied:1. If C Z = ( C Z֒ → Y , D Z֒ → Y ) and C Z ′ = ( C Z ′ ֒ → Y , D Z ′ ֒ → Y ) are two elements from F Y ,then the restrictions of the corresponding linear forms onto C Z֒ → Y ∩ C Z ′ ֒ → Y arethe same.2. If C Z = ( C Z֒ → Y , D Z֒ → Y ) and C Z ′ = ( C Z ′ ֒ → Y , D Z ′ ֒ → Y ) are two distinct elementsfrom F Y , and if a vector χ ∈ Q Y lies in the intersection of the interior of C Z֒ → Y with the image ρ ( V Y ), then ℓ Z ( χ ) > ℓ Z ′ ( χ ).Following Huruguen, we call a colored fan whose cones satisfy the requirementsof Theorem 7 a quasiprojective colored fan . We know from (the remarks following)Theorem 4 that there are plenty of quasiprojective colored fans, especially over perfectfields.With this precise definition of quasiprojectiveness at hand, now we are able to stateHuruguen’s result. heorem 8 (Huruguen) . Let k be a perfect field, let G be a connected reductivegroup that is defined over k , and let Y ( k ) be an embedding of a spherical homogenousspherical G -variety Y defined over k . We assume that the fan of Y is Γ -stable. In thiscase, Y admits a k -form if and only if for every cone C Z := ( C Z֒ → Y , D Z֒ → Y ) in F Y , thecolored fan consisting of the cones ( σ ( C Z֒ → Y ) , σ ( D Z֒ → Y )), σ ∈ Γ as well as all of itsfaces are quasiprojective. Proof.
See [15, Theorem 2.26].
We will start with giving a brief summary of Wedhorn’s work on the classificationof spherical spaces. For all unjustified claims (and for some definitions) we refer thereader to [34] and to the references therein.
Definition 4.9.
Let k be a field, and let G be a reductive k -group. Recall that thisamounts to the requirement that G k is a connected reductive group. According to [34,Remark 2.2], an algebraic space X over k with an action of G is G -spherical if X k is aspherical G k -variety.Let k denote a fixed algebraic closure of k , let k s denote the separable closureof k , and let us denote by Γ the Galois group of the extension k s /k . (Here, weare intentionally vague about our choices because it does not matter which separableclosure we choose.) In the sequel we will look at continuous and linear actions of Γ onsome structures. When we speak of a continuous action of Γ on a set X , we will treat X with the discrete topology. The important point here is that if X is a finite set, or,if the action of Γ is linear on some finite dimensional vector space X , then the actionis continuous if and only if it factors through some finite discrete quotient of Γ . Thisfact should alleviate a possible pain of confronting a large absolute Galois group suchas Γ of Q s / Q .If Y is a spherical G -space, then there exists a homogenous spherical G -space Y such that Y is a spherical embedding of Y . This actually amounts to the statement that Y is the unique open minimal G -invariant subspace of Y . By definition, a G -invariantsubspace in an algebraic space Y is minimal if there exists no proper non-empty G -invariant subspace of Y . Theorem 9 (Wedhorn) . Let G be a reductive k -group, and let Y be a spherical G -scheme viewed as an equivariant embedding of the spherical homogenous scheme G/H .If k is separably closed, then the assignment Y Y k induces a bijection between theisomorphism classes of spherical embeddings of G/H over k and the isomorphism classesof spherical embeddings of G/H over k .Notice that the bijection between isomorphism classes that is mentioned in Theo-rem 9 is essentially the application of the base change functor from k to k . In general,this does not give an equivalence of categories. A straightforward example is producedby the left translation action of G = G m on Y = G m . Luckily, since the definition ofcolored fans works over separably closed fields, and since we have faithfully flat descent pon restriction, the classification reduces to the classification over algebraically closedfields. The caveat is that one needs to consider all G -invariant minimal subschemes ofthe spherical space. Corollary 1.
Let G be a reductive k -group, and let Y be a spherical G -space viewedas an equivariant embedding of the spherical homogenous space Y := G/H . If k isseparably closed, then the assignment Y ( C Y ′ ֒ → Y , D Y ′ ֒ → Y ) Y ′ , where Y ′ runs overall minimal G -invariant subschemes of Y , is an equivalence between the category ofequivariant embeddings of Y (over k ) and the category of colored fans associated with( Q Y , V Y , D Y , e ρ ).Of course, the theorem and its corollary that we just presented here give us some-thing new (compared to Knop’s theorem) only when the characteristic of k is nonzero.Now we proceed with the general case and assume that G is a reductive k -group. Let Y be a spherical homogenous G -variety, and let Y be a spherical embedding of Y .Both of Y and Y are assumed to be defined over k . Note that Borel subgroups alwaysexist over separably closed fields, whence we fix a Borel subgroup B in G despite thefact that B may not have any k -rational points. This is where we start to notice adeparture from Huruguen’s work.There is a natural action of the Galois group Γ on the space of B -semiinvariants k s ( Y ) ( B ) . In particular, Γ acts on the k s -vector space Ω Y ks continuously and linearly.Moreover, it acts continuously on the valuation cone V Y ks as well as on the set of colors D Y ks , and the maps ρ : V Y ks → Q Y ks and e ρ : D Y ks → Q Y ks are Γ -equivariant. • A colored fan F Y ks is said to be Γ -invariant if its colored cones are permuted bythe action of Γ . Theorem 10 (Wedhorn) . Let G be a reductive k -group, and let Y be a spherical G -space viewed as an equivariant embedding of the spherical homogenous G -space Y := G/H , which is defined over k . Then the assignment Y k s ( C Y ′ ֒ → Y ks , D Y ′ ֒ → Y ks ) Y ′ ,where Y ′ runs over all minimal G -invariant subschemes of Y k s , induces an equivalencebetween the category of equivariant embeddings of Y over k and the category of Γ -invariant colored fans associated with ( Q Y ks , V Y ks , D Y ks , e ρ ). In this section we will consider the reductive monoids that are defined over arbitraryfields. We will show how Wedhorn’s theorems are applicable to the algebraic monoidsetting.
Definition 5.1. A reductive k -monoid is a k -monoid whose unit-group is a reductive k -group in the sense of Definition 2.8.In particular, according to our Definition 5.1, the unit-group of a reductive k -monoid is a connected reductive monoid, conforming with our tacit assumption fromthe introduction as well as with that of [29]. emark . Let M be a reductive monoid defined over an algebraically closed field,and let G denote its unit-group. The following results are recorded in [29]:1. G is dense in M ;2. M is affine;3. the reductive monoids are exactly the affine G × G -embeddings of reductivegroups;4. the commutative reductive monoids are exactly the affine embeddings of tori;5. the isomorphism classes of reductive monoids with unit-group G are in bijectionwith the strictly convex polyhedral cones of Q G generated by all of the colors anda finite set of elements from V G .Now we propose a definition for ‘monoid algebraic spaces.’ Probably this definitionexists in the literature, however, we could not locate it. For our monoid space definition,once again, we will relax the definition of a group algebraic space (as given in StacksProject Tag 043G). Definition 5.3.
Let S be a scheme, and let B be an algebraic space that is separatedover S . • A monoid algebraic space over B is a pair ( M, m ), where M is a separated al-gebraic space over B and m : M × B M → M is a morphism of algebraic spacesover B with the property that, for every scheme T over B , the pair ( M ( T ) , m ) isa monoid. • A morphism ψ : ( M, m ) → ( M ′ , m ′ ) of monoid algebraic spaces over B is amorphism ψ : M → M ′ of algebraic spaces such that, for every T /B , the inducedmap ψ : M ( T ) → M ′ ( T ) is a homomorphism of monoids. Definition 5.4. A reductive monoid space over a scheme S is a monoid algebraic space M over S such that M → S is flat, of finite presentation over S , and for all s ∈ S thegeometric fiber M s is a reductive k -monoid.Clearly, if a monoid algebraic space M over a field k is a scheme, then M is a k -monoid in the sense of Definition 3.1 but the converse is not true. Indeed, in [15]Huruguen has found an example of a smooth toric variety of dimension 3 that is splitover a quadratic extension of k , having no k -forms. This pathological example showsthat even for the purposes of classifying reductive monoids over an arbitrary field oneneeds to venture into the category of algebraic spaces.Extending Rittatore’s classification to reductive monoid spaces, we record the fol-lowing observations which are simple corollaries of Wedhorn’s theorems combined withRittatore’s results.Recall that the reductive monoids with unit-group G are G × G -equivariant embed-dings of G . When we speak of ‘colors’ in this context, we always mean the colors of G as a G × G -spherical k -group. heorem 11. Let k be a field, and let G denote a reductive k -group. Let M be areductive monoid space with G as the group of invertible elements.1. If k is separably closed, then the assignment M ( C Y ′ ֒ → M , D G ) Y ′ , where Y ′ runsover all minimal G × G -invariant subschemes of M , is an equivalence between thecategory of reductive monoid spaces over k and the category of strictly convexcolored polyhedral cones of Q G generated by all of the colors of G and a finiteset of elements from V G .2. Let k s be a separable closure of k . If k is properly contained in k s , then thefollowing categories are equivalent:(a) the category of reductive monoid spaces over k with unit-group G ;(b) the category of Γ -invariant strictly convex colored polyhedral cones of Q G ks generated by all of the colors of G k s and a finite set of elements of V G ks .Here, Γ is the Galois group of the extension k s /k .It is now desirable to know exactly which reductive monoid schemes over a fieldhave a k -form. Theorem 12.
Let k be a perfect field, let M be a reductive monoid defined over k with unit-group G , and assume that G is defined over k . In this case, M has a k -formif and only if its colored fan, which is a strictly convex polyhedral cone, is stable underthe action of absolute Galois group of k ⊂ k . Proof.
By Theorem 5, we know that M is affine, therefore, its colored fan is automat-ically quasiprojective. Now our result follows from Theorem 8. k -forms of lined closures. For the next application we restrict our attention to the field of complex numbers, andwe assume that the reader is familiar with the highest weight theory.It is well known that any complex irreducible affine monoid M admits a faithfulfinite dimensional rational monoid representation. In other words, there exists a finitedimensional vector space V and an injective monoid homomorphism ρ : M → End ( V ) , which is a morphism of varieties (see [25]). We notice, in the light of Remark 2.12,that this fact holds true more generally for all irreducible affine k -monoids, where k is an arbitrary field. In particular, ( V, ρ | G ) is a faithful rational representation of theunit-group G , and M ∼ = ρ ( G ) in End ( V ). In this case, we will write M = M V . (5.5)Now, let V λ denote the irreducible representation of G with highest weight λ . The saturation of λ , denoted by Σ λ , is the set of all dominant weights that are less than orequal to λ , Σ λ := { µ : µ is dominant and µ ≤ λ } . et V Σ λ denote the representation ⊕ µ ∈ Σ V µ , and let M λ denote the reductive monoiddefined by V Σ λ as in in (5.5). In a similar manner, we will denote M V λ by M λ . (Theseare special cases of the “multi-lined closure” construction of Li and Putcha in [19].)Clearly, both of the monoids M λ and M λ are reductive.In [11], De Concini analyzed the geometric properties of M λ in relation with thatof M λ , and he proved the following theorem. Theorem 13 (DeConcini) . M λ is a normal variety with rational singularities.2. M λ is the normalization of M λ .3. M λ and M λ are equal if and only if λ is minuscule, that is to say, Σ λ = { λ } .We finish our paper with a theorem whose proof will be given somewhere else. Theorem 14.
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