Reductive group actions

Friedrich Knop^{1} and Bernhard Krötz^{2}

Abstract. In this paper, we study rationality properties of reductive group actions which are defined over an arbitrary field of characteristic zero. Thereby, we unify Luna’s theory of spherical systems and Borel-Tits’ theory of reductive groups. In particular, we define for any reductive group action a generalized Tits index whose main constituents are a root system and a generalization of the anisotropic kernel. The index controls to a large extent the behavior at infinity (i.e., embeddings). For k-spherical varieties (i.e., where a minimal parabolic has an open orbit) we obtain explicit (wonderful) completions of the set of rational points. For local fields this means honest compactifications generalizing the maximal Satake compactification of a symmetric space. Our main tool is a k-version of the local structure theorem.

1. Introduction 1 2. Notation 4 3. Invariants of quasi-elementary groups 6 4. The local structure theorem 11 5. Invariant valuations 19 6. Valuations and geometry 21 7. Toroidal embeddings 26 8. Horospherical varieties 31 9. The Weyl group 36 10. The root system 41 11. Wonderful varieties 48 12. Boundary degenerations 50 13. The weak polar decomposition 55 14. The Satake compactification 57 References 61

1. Introduction

Let G be a connected reductive group. The main focus of this paper is to study the fine large scale geometry of homogeneous G-varieties when the ground field k is an arbitrary field of characteristic zero. When k is algebraically closed the last decades have seen tremendous progress in this direction. Important milestones were the theory of toroidal embeddings in [KKMSD73] (still serving as the blueprint for all later developments), the wonderful embeddings of symmetric varieties in [DCP83], and the groundbreaking work [LV83] of Luna and Vust describing all spherical embeddings and much more. Later Brion [Bri90] discovered that the asymptotic behavior of spherical varieties is determined by a

1Emmy-Noether-Zentrum, FAU Erlangen, Cauerstraße 11, D-91058 Erlangen 2Institut für Mathematik, Uni. Paderborn, Warburger Str. 100, D-33098 Paderborn 2010 Mathematics Subject Classification. 14L30, 14M27, 20G25, 22F30. Key words and phrases. Compactifications, reductive groups, spherical varieties, root systems.

system of spherical roots. This was subsequently generalized to arbitrary G-varieties in [Kno94].

Over non-closed fields much less is known even though the work of Satake [Sat60] on compactifications of symmetric spaces (where k = ℝ) considerably predates the Luna-Vust theory. Here, mainly rationality questions are in the foreground, like which embeddings are defined over k, or which orbits carry k-rational points. One of the main obstacles is the fact that the theory over closed fields hinges crucially on the existence of a Borel subgroup. For that reason, the Luna-Vust theory extends quite easily to the case when G is split over k. This has enabled Sakellaridis and Venkatesh [SV12] to prove a Plancherel theorem for spherical varieties of split groups over p-adic fields. But even for quasi-split groups things become more difficult and for all other groups the theory has to be built up from scratch. The idea is of course to replace the Borel subgroup by a minimal parabolic subgroup P.

A first breakthrough in this direction was achieved over k = ℝ with the paper [KKS15a] where an ℝ-version of the local structure theorem of [BLV86] was proved and used to study the geometry of ℝ-spherical varieties. Later, it was successfully used in [KKSS15, KKSS14,KKS15b] for purposes of harmonic analysis.

In the present paper we go far beyond [KKS15a]. First, we replace ℝ by an arbitrary ground field of characteristic 0. Secondly, we have much more precise results on the structure of spherical embeddings and thirdly, most of our results are also valid in the non-spherical setting. Thereby we end up with a theory which unifies Luna’s theory of spherical varieties with the Borel-Satake-Tits theory [Tit59, Sat63, BT65, Tit66] of reductive groups over non-closed fields.

Since our approach deals mostly with rationality properties we will always assume that the G-variety X is k-dense, i.e., that X(k) is Zariski dense in X. Let P = MAN be a minimal parabolic k-subgroup of G. Here N is the unipotent radical, A is a maximal split torus and M is anisotropic. Then we construct a quite specific P -stable open subset X′ of X such that the quotients X_{el} := X′/N and X_{an} := X′/AN exist, called the elementary kernel (an MA-variety) and the anisotropic kernel of X (an M-variety), respectively. The latter plays, like for reductive groups, the rôle of a black box about which we have little to say.

The canonical morphism X_{el} → X_{an} is a principal bundle for a quotient A_{k}(X) of A. This torus A_{k}(X) is the analogue of A in G. Its character group Ξ_{k}(X) := Ξ(A_{k}(X)) replaces the weight lattice of a group. Accordingly, the k-rank of X is defined as

(1.1) rk_{k} X := rk Ξ_{k}(X) = dim A_{k}(X) = dim X_{el} − dim X_{an}.

The root system is much more intricate to construct. For this, we start by considering a special kind of G-invariant valuations on the function field k(X) which we call kcentral. By definition these are invariant valuations which are trivial on the subfield of AN-invariants. Each such valuation is related to a G-stable irreducible divisor D in some birational model of X. An important point is that for k-central valuations D is k-dense. This provides us with a tool to study rational points “at infinity”.

It is shown that k-central valuations are classified by elements of a certain subset Z_{k}(X) of the vector space N_{k}(X) := Hom(Ξ_{k}(X), ℚ). A central fact (pun intended) is that Z_{k}(X) is a Weyl chamber for a finite reflection group W_{k}(X) acting on Ξ_{k}(X). Suitably normalized, this defines a set Σ_{k}(X) of simple roots for a root system Φ_{k}(X) ⊂ Ξ_{k}(X) with Weyl group W_{k}(X). The elements of Σ_{k}(X) are called the k-spherical roots of X.

Now consider the algebraic closure K of k. Since everything can be done over K, as well there is also a root system Φ_{K}(X) of X. Of course, this is just the one constructed previously by Brion. Another main result of this paper is that Φ_{k}(X) is more or less the restricted root system of Φ_{K}(X) to A. This was kind of unexpected since Φ_{K}(X) is constructed using the action of minimal K-parabolic, i.e., a Borel subgroup on X. So there is no direct link between both root systems.

At this point, we have now four root systems to deal with, namely the two root systems of G (over K and k) and the two root systems of X. They are subject to very strong compatibility constraints of which we prove a few. An exhaustive list will be subject of future research. It should be mentioned that the pair (Φ_{K}(X), Φ_{k}(X)) looks very much like a Tits index of a reductive group but is in fact slightly more general.

We now come to the main goal of this paper namely analyzing the behavior of X at infinity. For this we construct for every fan F which is supported in Z_{k}(X) a normal partial compactification X(F) of X. This embedding comes with a stratification with one stratum X(C) for every cone C ∈ F. The main point is that every stratum is k-dense. Now assume that the support of F is all of the valuation cone Z_{k}(X). Then it is not true that X(F) is a complete variety. Instead it is kind of complete with respect to rational points. This means that every normal completion of X(F) adds only very few rational points.

It is easiest to make this precise when X is k-spherical. This means by definition that P has an open orbit in X. Equivalently, the action of M on the anisotropic kernel of X is transitive. So, when X is k-spherical and supp F = Z_{k}(X) then we show that \overline{X}(k) = X(F)(k) for any normal (equivariant) completion \overline{X} of X(F). Another consequence of k-sphericity is that the strata are precisely the G-orbits. In particular there are only finitely many of them. In contrast, it may happen that a completion has infinitely many orbits but the point is that almost none of them carries a k-rational point. Therefore we regard it as one of the main insights of this paper (and its precursor [KKS15a]) that in order to study rational points it is not necessary to construct true completions.

Now the main consequence of Z_{k}(X) being a Weyl chamber is that it is cosimplicial, i.e., defined by a set of linearly independent inequalities. A particularly favorable situation arises if Z_{k}(X) is strictly convex. In this case X will be called k-convex. This condition is rather mild, since every G-variety can be made k-convex by dividing by a suitable split torus. In this case Z_{k}(X) is also simplicial and one can take as a fan the set of all faces of Z_{k}(X). We call this the standard fan F_{st} of X and X_{st} := X(F_{st}) the standard embedding. Now for a k-convex, k-spherical homogeneous variety X its standard embedding has the following nice properties:

• The boundary X_{st} \ X consists of r = rk_{k} X irreducible components X_{{1}}, …, X_{{r}}.

• The orbit closures are precisely the set-theoretic intersections X_{I} := ⋂_{i∈I} X_{{i}} where I runs through all subsets of {1, …, r}.

• Each X_{I} is k-dense and of codimension |I| in X_{st}.

• Let X_{st} ⊆ \overline{X} be an equivariant embedding. Then \overline{X}(k) = X_{st}(k).

Often the standard embedding is even smooth. In this case, we show that all orbit closures X_{I} are smooth and that the intersections above are even scheme-theoretic. Following DeConcini and Procesi, [DCP83], the varieties X and X_{st} will then be called k-wonderful. Even that condition is quite mild: we show that for every k-convex variety X there is a finite abelian group E of automorphisms such that X/E is wonderful.

The paper contains a couple more topics which have not yet been mentioned. In Section 8 we classify two types of varieties. The first are spaces whose k-rank is zero. We show that they are all parabolically induced from anisotropic actions. Their importance lies in the fact that the closed stratum of a standard embedding is of rank zero. The second, more general type are the so-called k-horospherical varieties. They are characterized by Z_{k}(X) being the entire space N_{k}(X) or, equivalently, that the root system is empty. We show that all k-horospherical varieties are parabolic inductions of elementary actions. Geometrically they appear as maximal degenerations of spherical varieties. In Section 12 we consider more general degenerations. More precisely, we construct a smooth morphism 𝔛 → 𝔜 with projective base whose fibers are called the boundary degenerations of X.

In Section 13 we generalize the main result of [KKSS15] from ℝ to local fields of characteristic zero. More precisely, we prove the a weak version of the polar decomposition. Finally, we show how to obtain the classical maximal Satake compactification of a Riemannian symmetric space with our approach. Moreover, we compare the standard embedding of a variety X considered as a variety over k with the standard embedding over the algebraic closure. They are related by a birational morphism from the former to the later which is in general not an open embedding. We also give an examples in which the standard embedding over K has bad rationality properties.

Some words to the proofs: As already mentioned everything hinges on the k-Local Structure Theorem. Here we follow the proof of [KKS15a] which is itself an adaption of the proof in [Kno94]. The main difficulty is to show the existence of sufficiently many P -eigenfunctions on X. For this we replace the integration and positivity argument in [KKS15a] by an argument using Kempf’s theory of optimal 1-parameter subgroups [Kem78].

Secondly, even though the main applications are for k-spherical varieties the theory does not simplify very much by sticking to the spherical case. The reason for this is that k-spherical varieties do not stay spherical when passing to the algebraic closure. So we do need to a considerable extent the embedding theory of non-spherical varieties.

Finally, the order in which the theory evolves is different than the one in the exposition above where we first described the root system and then derived properties of embeddings. In practice we prove a fair amount of properties of the compactification first and then reinterpret them in terms of the root system.

2. Notation

In the entire paper, k is a field of characteristic zero. Its algebraic closure is denoted by K and its Galois group by G := Gal(K|k).

Our point of view will be that a k-variety is variety over K equipped with a compatible G-action. Accordingly, for us the terms “connected” and “irreducible” mean “absolutely connected” and “absolutely irreducible”. All k-varieties are, by definition, non-empty and irreducible.

Let H be a linear algebraic group defined over K. Then Ξ(H) := Hom(H, G_{m}) denotes its character group. If H is defined over k, then Ξ(H) carries a G-action. The space N (H) := Hom(Ξ(H), ℚ) denotes the space of rational cocharacters of H.

Let V be a representation of H and χ ∈ Ξ(H). Then V_{χ}^{(H)} denotes the set of χeigenvectors of V. Observe that V_{χ}^{(H)} never contains the vector 0. Thus, for the trivial character χ = 1 we have V_{1}^{(H)} = V^{H} \ {0}. We denote by V^{(H)} the union of all V_{χ}^{(H)} with χ ∈ Ξ(H). For v ∈ V^{(H)} let χ_{v} be the unique character with v ∈ V_{χ_{v}}^{(H)}.

In the whole paper, G is a connected reductive k-group. We fix a minimal parabolic ksubgroup P ⊆ G with decomposition P = MAN where N = R_{u}P is the unipotent radical, A is a maximal split subtorus and M is anisotropic. Hence MA is a Levi complement of P. Let T ⊆ MA be a maximal k-subtorus. Then A ⊆ T and restriction to A defines a surjective homomorphism

(2.1) res_{A} : Ξ(T) ↠ Ξ(A).

If Ω ⊆ Ξ(T) is any subset we adopt the notation

(2.2) res′_{A} Ω := (res_{A} Ω) \ {0}.

Let Φ = Φ(G, T) be the associated root system and

(2.3) Φ_{k} := res′_{A} Φ

the restricted root system (in general not reduced). The Weyl group of Φ is denoted by W = W (G). Choosing a Borel subgroup B with T ⊆ B ⊆ P yields a set of simple roots S ⊆ Φ and a set of simple restricted roots

(2.4) S_{k} := res′_{A} S ⊆ Φ_{k}.

On the other side, the elements of

(2.5) S^{0} := {α ∈ S | res_{A} α = 0}

are the compact simple roots which are the simple roots of M. Reflections about elements of S^{0} generate the Weyl group W^{0} ⊆ W of M.

The Galois group G acts on Ξ(T) leaving the root system Φ invariant. The restriction map res_{A} is G-invariant. There is also the so-called ∗-action of G on Ξ(T) which is constructed as follows: for each γ ∈ G there is a unique element w_{γ} ∈ W with w_{γ}γS = S. Then one defines

(2.6) γ ∗ χ := w_{γ}γ(χ).

In the following, we refer to G acting by the ∗-action as G^{∗}.

It is known that w_{γ} ∈ W^{0} for all γ ∈ G ([BT65, Proof of Prop. 6.7] or [Spr98, Proof of Prop. 15.5.3]). This means that if

(2.7) Δ^{∗} := {φ ∈ Aut Ξ(T) | φ(S) = S, φ(S^{0}) = S^{0}}

is the group of “graph automorphisms” then the G^{∗}-action is a homomorphism G → Δ^{∗} while the G-action is a lift to Δ^{∗} ⋉ W^{0} . As a consequence the ordinary and the ∗-action coincide on Ξ(T)^{W_{0}} = Ξ(T/(T ∩ M′)^{0}).

Throughout this paper we denote algebraic groups by upper case Latin letters and their corresponding Lie algebras by lower case German letters, e.g. 𝔥 is the Lie algebra of H. The unipotent radical of an algebraic group H is denoted by R_{u}H or H_{u}.

3. Invariants of quasi-elementary groups

Following Borel and Tits, [BT65, Def. 4.23], we define

3.1. Definition. A linear algebraic k-group G is anisotropic if it is connected, reductive, and its k-rank is 0, i.e., it does not admit a non-trivial split subtorus.

According to [BT65, Cor. 8.5] an equivalent condition for a group G to be anisotropic is that all elements in G(k) are semisimple and that Hom_{k}(G, G_{m}) = 1. Some authors, like, e.g., Springer in [Spr98, p. 271], drop the second condition. The ensuing notion is different but also very important for our purposes:

3.2. Definition. A linear algebraic k-group G is called elementary if it is connected and all of its k-rational elements are semisimple.

Clearly, elementary groups are reductive. The difference between an anisotropic and an elementary group is that the latter may have a non-trivial central split subtorus. Because of this reason, elementary groups are often called “anisotropic modulo center” (with reductivity tacitly being assumed). As an example, the group G = GL(1, ℍ) is an elementary but non anisotropic ℝ-group. Over K, the algebraic closure of k, only the trivial group is anisotropic while exactly the tori are elementary.

These definitions extend to non-reductive groups as follows:

3.3. Definition. A k-group H with unipotent radical H_{u} is quasi-anisotropic or quasielementary if H/H_{u} is anisotropic or elementary, respectively.

Clearly, these two concepts imply connectivity. Since any parabolic subgroup of H contains H_{u}, it follows from [BT65, Cor. 4.17] that H is quasi-elementary if and only if it does not contain a proper parabolic k-subgroup. In particular, a minimal parabolic k-subgroup of any group is quasi-elementary and its Levi component is elementary.

The most general of these four concepts is that of a quasi-elementary groups. They have a factorization

(3.1) H = MAN

where M is anisotropic, A is a split torus and N is the unipotent radical. This factorization is unique up to conjugation by N(k). The group H is quasi-anisotropic, elementary, or anisotropic if and only if A is trivial, N is trivial, or both A and N are trivial, respectively.

It follows from the definition that k-subgroups of elementary groups are elementary. In particular, they are reductive, as well.

For any connected group H let H_{el} ⊆ H be the Zariski closure of the subgroup generated by all k-rational unipotent elements. Then H_{el} is the smallest normal subgroup of H such that H/H_{el} is elementary. We call it the elementary radical of H. Concretely, the elementary radical can also be described as follows: if H = LH_{u} is the Levi decomposition of H then H_{el} = L_{n}H_{u} where L_{n} ⊆ L is the product of all non-anisotropic simple factors of L.

This has the following consequence: Let P = MAN ⊆ H be a minimal parabolic ksubgroup. Since MA contains all anisotropic simple factors and also the center of L we get

(3.2) H = MA · H_{el},

There is an analogous notion for anisotropic groups: every connected group H has a smallest normal subgroup H_{an} such that H/H_{an} is anisotropic. We call H_{an} the anisotropic radical of H. As a group it is generated by H_{u} and all split subtori. If P = MAN is a minimal parabolic subgroup then

(3.3) H_{an} = A · H_{el} and H = M · H_{an}.

The characterization of quasi-elementary groups in terms of parabolic subgroups can be extended as follows:

3.4. Proposition. For a connected k-group H, the following are equivalent:

i) H is quasi-elementary.

ii) Let F ⊆ H be any k-subgroup. Then H/F is an affine variety.

iii) H does not contain any proper parabolic k-subgroup.

Proof. Assume that H = MAN is quasi-elementary and let F ⊆ H be a k-subgroup. Consider the morphism

(3.4) π : H/F → H/F N ≅ (H/N)/(F N/N).

The group F N/N is a subgroup of the elementary group H/N = MA and therefore reductive. This implies that H/F N is an affine variety. The fiber of π is F N/F ≅ N/F ∩ N. It is the orbit of a unipotent group, hence affine, as well. This implies that π is an affine morphism on an affine variety. Hence H/F is affine.

Now assume ii). Let P ⊆ H be a minimal parabolic k-subgroup. Then ii) applied to F = P implies that H/P is both affine and complete and therefore a point. Thus P = H.

Finally, the equivalence of i) and iii) follows from [BT65, Cor. 4.17], as already mentioned above. □

3.5. Definition. A k-variety X is called k-dense if the set X(k) of k-rational points is Zariski dense in X.

Clearly, the affine space and the projective space is k-dense. If X → Y is a dominant k-morphism and X is k-dense then Y is k-dense, as well. The product X × Y of two k-dense varieties is k-dense. More generally, if X → Y is a Zariski trivial fiber bundle with fiber Z then X is k-dense if and only if both Y and Z are k-dense. This holds in particular for the total space of a line bundle over a k-dense variety.

In the remainder of this section we will study actions of quasi-elementary groups on (mostly) affine k-dense varieties. The point of departure will be the following fundamental theorem of Kempf:

3.6. Theorem ([Kem78, Cor. 4.3]). Let G be a connected reductive k-group acting on an affine k-variety X. Let x ∈ X(k) be a k-rational point and let Y ⊆ X be a closed Ginvariant k-subset with \overline{Gx} ∩ Y ≠ ∅. Then there exists a k-homomorphism λ : G_{m} → G such that lim_{t→0} λ(t)x exists and lies in Y (k).

This implies:

3.7. Corollary ([Kem78, Remark after Cor. 4.4]). Let G be an anisotropic group acting on a quasi-affine k-variety X. Then the G-orbit of any k-rational point x ∈ X(k) is closed in X.

Proof. Since X is quasi-affine, it is isomorphic to a locally closed subvariety of an affine G-variety \overline{X}. Replacing X by \overline{X}, we may assume that X is affine. Now apply Kempf’s theorem to Y := \overline{Gx} \ Gx and observe that any homomorphism λ : G_{m} → G is trivial since G is anisotropic. We conclude that Y is empty, i.e., Gx is closed. □

Rosenlicht has proved in [Ros61] that also orbits of unipotent groups in quasi-affine varieties are closed. The following can be regarded as a combination of both cases:

3.8. Theorem. Let H = MAN be a quasi-elementary group acting on a k-dense quasiaffine variety X. Then there is an affine H-variety Y with MN acting trivially, an H-equivariant morphism π : X → Y, and an H-stable open dense subset Y_{0} ⊆ Y such that for all y ∈ Y_{0} the fiber X_{y} is an MN-orbit.

Proof. Choose an open equivariant embedding of X into an affine H-variety \overline{X}. Since it suffices to prove the assertion for \overline{X} we may assume from the outset that X is affine.

Now choose generators r_{1}, …, r_{n} of k(X)^{N}, the field of N-invariant rational functions. Since N is unipotent and since the ideal

(3.5) {f ∈ k[X] | fr_{1}, …, fr_{n} ∈ k[X]}

is non-zero, it contains a non-zero N-invariant function f_{0}. By replacing r_{1}, …, r_{n} with f_{0}, f_{0}r_{1}, …, f_{0}r_{n} we may assume that all the functions r_{i} are in k[X]^{N}.

Let V ⊆ k[X]^{N} be the (finite dimensional) H-submodule of k[X]^{N} which is generated by {r_{1}, …, r_{n}}, let A ⊆ k[X]^{N} be the (finitely generated) subalgebra which is generated by V, and let Z := Spec A be the variety associated to A. Then Z is an affine Hvariety with N acting trivially. Moreover, the inclusion A ↪ k[X] induces a dominant H-morphism π_{N} : X → Z. By construction, this morphism induces an isomorphism of fields π_{N}^{∗} : k(Z) →^{∼} k(X)^{N}. Rosenlicht’s theorem [Ros56, Thm. 2] implies that the generic N-orbits are separated by rational N-invariants. This means that the generic fibers of π_{N} contain open dense N-orbits. Since orbits of a unipotent group on an affine variety are closed (by another theorem of Rosenlicht [Ros61], see also [Bor91, Prop. 4.10]), we see that the generic fibers of π_{N} are actually N-orbits.

Now consider the M-action on Z and let π_{M} : Z → Y := Z//M be the categorical quotient. Observe that Y carries an action of H with MN acting trivially. All fibers of π_{M} contain a unique closed M-orbit. It follows from Corollary 3.7 that the orbits Mz with z ∈ Z(k) are closed. The k-density of X implies that Z is k-dense, as well. This implies that the generic M-orbits in Z are closed. Thus the generic fibers of π_{M} are precisely the generic M-orbits. Now we take for π the composition π_{M} ∘ π_{N} : X → Y. □

From this, we obtain the following generalization of Corollary 3.7:

3.9. Corollary. Let H be a quasi-anisotropic group acting on a quasi-affine k-variety X. Then:

i) The orbit Hx of any x ∈ X(k) is closed in X. ii) Assume, moreover, that X is k-dense. Then the generic H-orbits are closed in X, i.e., there is an H-stable open dense subset X_{0} ⊆ X such that all H-orbits of X_{0} are closed in X.

Proof. i) It is well known that H(k) is dense in H ([Bor91, Thm. 18.3]). Therefore, the orbit closure \overline{Hx} is k-dense. When one applies the theorem to \overline{Hx}, the variety Y is necessarily a point. Thus \overline{Hx} is an H-orbit, i.e., Hx is closed.

ii) Since fibers are closed, this follows directly from the theorem. □

Next we establish a criterion for the existence of semiinvariants.

3.10. Proposition. Let H be a quasi-elementary group acting on a k-dense affine variety X. Let I ⊆ K[X] be a non-zero H-stable ideal (not necessarily defined over k). Then I contains an H-semiinvariant function (defined over k).

Proof. Let H = MAN be the factorization from (3.1) and π : X → Y be the morphism as in Theorem 3.8. The zero locus Z of I in X is an H-stable proper closed subset. Its image φ(Z) in Y cannot be dense since otherwise Z would contain the generic MNorbits and would therefore coincide with X. The ideal of all functions on Y which vanish in π(Z) is non-zero and therefore contains an A-semiinvariant function f_{0}. Then f_{0} is also H-semiinvariant since MN is acting trivially on Y. Now replace f_{0} by the (finite) product of all functions of the form ^{γ}f_{0} where γ runs through the Galois group Gal(K|k). This makes f_{0} being defined over k, as well. The pull-back f_{1} of f_{0} to X vanishes in Z. Hilbert’s Nullstellensatz asserts that then a power f = f_{1}^{N} is in I. □

3.11. Corollary. Let H = MAN be a quasi-elementary group acting on a k-dense affine variety X. Then either MN acts transitively on X or X carries a non-constant Hsemiinvariant function defined over k.

Proof. If MN acts transitively, then clearly all H-semiinvariants, being MN-invariant, are constant. Assume conversely that MN does not act transitively and let Y ⊊ X be a closed orbit. Now apply Proposition 3.10 to the vanishing ideal of Y. □

3.12. Corollary. Let H be a quasi-elementary group acting on a k-dense affine variety X. Then every H-semiinvariant rational function over k is the quotient of two Hsemiinvariant regular functions over k.

Proof. Let f ∈ k(X)^{(H)}. Then apply Proposition 3.10 to the ideal I = {h ∈ k[X] | hf ∈ k[X]}. □

For reductive groups one can prove the following refinement of Proposition 3.10.

3.13. Proposition. Let G be a connected reductive group acting on an affine variety X. Let I ⊆ K[X] be a non-zero G-stable ideal and let x ∈ X(k) point which is fixed for G_{el} and which is not contained in the zero locus of I. Then I contains a G-semiinvariant function f ∈ k[X] with f(x) ≠ 0.

Proof. The group G factorizes uniquely as G = MAG_{el} where M is anisotropic and A is a split torus. Put H := MG_{el}. Then Hx = Mx is closed because of Corollary 3.7. Moreover, it is disjoint from Z, the zero locus of I, since x ∉ Z. Thus there is a function F ∈ K[X] which vanishes on Z and which is non-zero on x. Replacing F by the product of its Galois translates, we may assume that F is defined over k. Replacing F by a suitable power, we may further assume that F ∈ I. Since A is a split torus and I is A-stable, F decomposes as a sum of A-semiinvariants f_{χ_{i}} ∈ I. At least one of the f_{i} will not vanish in x. This f_{i} has all the asserted properties. □

If P is a minimal parabolic subgroup of G then P is quasi-elementary and all the results above apply to actions of P. In case, the P -action actually comes from a G-action one can be more specific about the character of a semiinvariant.

3.14. Lemma. Let G be a connected reductive group acting on an affine k-dense variety X and let P = MAN be a minimal parabolic k-subgroup of G. Let χ ∈ Ξ(A) be the character of a AN-semiinvariant function f ∈ k[X]. Then χ^{n} is, for some n ∈ ℤ_{>0}, the character of a P -semiinvariant function \overline{f} ∈ k[X].

Proof. We start with a couple of reduction steps. Let V_{0} ⊆ k[X] be the AN-eigenspace for the character χ. Because V_{0} ≠ 0, by assumption, it contains an irreducible MAsubmodule V_{1}. Then the G-module V := ⟨G · V_{1}⟩ ⊆ k[X] generated by V_{1} is irreducible. The inclusion V ↪ k[X] yields a G-morphism X → V^{∗}. Let X_{1} be the closure of its image and let ℓ : V^{∗} → k be the evaluation in any fixed 0 ≠ f ∈ V_{1}. Then ℓ is a linear AN-semiinvariant on V^{∗} with character χ which is non-zero on X_{1}. Clearly, it suffices to prove the assertion for X_{1}.

Now let \overline{X} := \overline{G_{m}X_{1}} ⊆ V^{∗} be the closed cone generated by X_{1}. It carries an action of \overline{G} = G_{m} ×G. Then \overline{P} = M\overline{A}N is a minimal parabolic subgroup of \overline{G} where \overline{A} = G_{m} ×A. Moreover, ℓ is an \overline{A}N-semiinvariant for the character \overline{χ} = (1, χ). Now suppose the assertion is true for \overline{X}. Then \overline{X} carries a \overline{P} -semiinvariant \overline{f} with character (1, χ)^{n} = (n, χ^{n}). Then the restriction f_{1} of \overline{f} to X_{1} is non-zero because \overline{f} ≠ 0 is homogeneous and G_{m}X_{1} is dense in \overline{X}. Then f_{1} is the desired semiinvariant.

Replacing X, G by \overline{X}, \overline{G} we are reduced to the situation that X is a closed cone in an irreducible G-module V^{∗}, that G contains all scalars and that f is the restriction of a linear function ℓ on V^{∗}.

Let N^{−} ⊆ G be the unipotent subgroup which is opposite to N. Then P N^{−} = MANN^{−} = NMAN^{−} is a dense open subset of G. Let P(X) ⊆ P(V^{∗}) be the projective variety corresponding to X. Since AN^{−} is a split solvable group and X is k-dense it has a fixed point in P(X) which means that X contains an AN^{−}-fixed line k · w ≠ 0. Suppose that ℓ(Mw) = 0. Then ℓ(NMAN^{−}w) = 0 and therefore ℓ(Gw) = 0 which is impossible since V^{∗} is irreducible. Thus there is g ∈ M(k) with ℓ(gw) ≠ 0. Replacing w by gw we may assume that ℓ(w) ≠ 0. But then A acts on kw via the character χ^{−1}.

Let X_{2} := \overline{Gw} ⊆ X which is a cone since G contains the scalars. Because of 0 ∈ X_{2}, the group MN can’t act transitively on X_{2}. Thus Corollary 3.11 implies that X_{2} carries a non-constant P -semiinvariant f̃. Suppose f̃(w) = 0. Then, from f̃(P N^{−}w) = 0 we get f̃(Gw) = 0 and therefore f̃ = 0 which is not true. Thus the restriction of f̃ to kw is an A-semiinvariant which is homogeneous of n > 0. Its character therefore equals χ^{n}. Finally, since the restriction map k[X] → k[X_{2}] splits as a G-homomorphism the function f̃ extends to a P -semiinvariant \overline{f} on X with the same character. □

3.15. Remark. This lemma replaces the integration argument in [KKS15a, Thm. 2.2] (Local Structure Theorem) in the case k = ℝ.

As mentioned in the introduction, the main philosophy of this paper is to extend the Borel-Tits theory of reductive groups to actions of reductive groups. We start by defining the analogues of elementary and anisotropic groups.

3.16. Definition. Let H be a connected k-group acting on a k-dense variety X. Then the action (or simply X) is called elementary or anisotropic if the elementary radical H_{el} or the anisotropic radical H_{an}, respectively, acts trivially on X.

Clearly, the action is elementary if and only if all k-rational unipotent elements of H act trivially on X. Likewise, the action is anisotropic if H_{u} and all split subtori of H act trivially. Let P = MAN ⊆ H be a minimal parabolic subgroup. Then it follows from (3.2) or (3.3) that an elementary H-action is the same as an MA-action which is trivial on MA ∩ H_{el}. Similarly, (3.3) implies that an anisotropic H-action is the same as an M-action which is trivial on M ∩ H_{an}. Observe that every anisotropic action is elementary.

Next we give a representation theoretic characterization of elementary actions. It will be the key for the proof of the Local Structure Theorem below.

3.17. Corollary. Assume that the action of the connected reductive group G on the kdense quasi-affine variety X is not elementary. Then there is a P -semiinvariant f ∈ k[X] whose character χ_{f} does not extend to a character of G.

Proof. According to [BT65, Prop. 8.4] every k-rational unipotent element is conjugate to an element of N. This implies that the action of N on X is non-trivial. Therefore k[X] contains a simple G-submodule V with a non-trivial N-action. Choose an AN-eigenvector f ∈ V and let χ be its character. Since N acts non-trivially, highest weight theory shows that there is a root α of A in Lie N such that ⟨χ, α^{∨}⟩ > 0. According to Lemma 3.14 there is a P -semiinvariant \overline{f} ∈ k[X] whose character is χ\overline{f} = χ^{n} for some n ∈ ℤ_{>0}. Then also ⟨χ\overline{f}, α^{∨}⟩ > 0 which implies that χ\overline{f} cannot be the restriction of a character of G to A. □

3.18. Remark. Using (3.2) it is easy to see that the converse of the corollary holds, as well.

4. The local structure theorem

Before we embark in stating and proving the Local Structure Theorem, we need to discuss a regularity property for algebraic group actions.

4.1. Definition. Let H be a linear algebraic group and X an H-variety. Then X is called locally k-linear if for every x ∈ X(k) there is an H-stable open neighborhood X_{0}, a finite dimensional representation V of H, and an H-equivariant embedding X_{0} ↪ P(V), where everything is defined over k.

If H is connected and X is normal in every k-rational point then Sumihiro’s theorem [Sum75] implies that X is locally k-linear. So the condition is rather mild since at least the normalization of any H-variety is locally k-linear. On the other side, local linearity has better functorial properties than normality since H-stable subvarieties of locally linear varieties are locally linear. In particular, all H-stable subvarieties of P(V) are locally linear.

We proceed with the main tool of this paper. Its purpose is to reduce arbitrary G-actions to elementary ones.

4.2. Theorem (The k-Local Structure Theorem). Let G be a connected reductive k-group, let X be a locally k-linear G-variety, and let Y ⊆ X be a G-stable, k-dense subvariety. Then there is a parabolic k-subgroup Q ⊆ G with Levi decomposition Q = LU and an L-stable affine k-subvariety R ⊆ X such that:

i) The intersection Y_{el} := R ∩ Y is a k-dense affine L-variety (thus, in particular, non-empty), all L-orbits in Y_{el} are closed, and the action of L on Y_{el} is elementary.

ii) The morphism

(4.1) U × R = Q ×^{L} R → X : [g, x] ↦ gx

is an open embedding.

For the proof, we’ll need the following lemma.

4.3. Lemma. Let X_{0} ⊆ X be a G-stable open k-subset with Y_{0} := Y ∩ X_{0} ≠ ∅. Then the k-Local Structure Theorem holds for (X, Y) if and only if it holds for (X_{0}, Y_{0}).

Proof. One direction is trivial: if the k-LST holds for (X_{0}, Y_{0}) then it holds for (X, Y). Assume conversely, that it holds for (X, Y) and let R ⊆ X be the L-stable slice. Put R′ := R ∩ X_{0}. Because U(R ∩ Y) is dense and open in Y we see that R meets every U-stable open subset of Y. In particular, R′ ∩ Y_{0} ≠ ∅. One readily sees that R′ ⊆ X_{0} inherits all requirements for a slice from R ⊆ X except possibly R′ being affine. To force this last property it suffices to construct an affine L-stable open subset R_{0} ⊆ R′ with R_{0} ∩ Y_{0} ≠ ∅.

To do this, we choose a k-rational point y ∈ R ∩ Y_{0}. This is possible since R ∩ Y is k-dense. Then y ∉ Z := R \ R′. Because the L-action on Y is elementary, the point y is fixed by L_{el}. Then Proposition 3.13 yields an L-semiinvariant function f on R vanishing on Z and being non-zero in y. Now R_{0} := {x ∈ R | f(x) ≠ 0} ⊆ R′ is affine open with R_{0} ∩ Y_{0} ≠ ∅. □

Proof of the k-LST. We start with a number of reductions. First, by the lemma and the assumption of local k-linearity we may assume that X ⊆ P(V) where V is a finite dimensional G-module. Next we apply the lemma to X and its closure in P(V). Then we may assume that X is closed in P(V).

Now let X̃ ⊆ V and Ỹ ⊆ V be the affine cones of X and Y, respectively. Put moreover X̃^{∗} := X̃ \ {0} and Ỹ^{∗} := Ỹ \ {0}. All these varieties carry an action of G̃ := G × G_{m}. Assume that the k-LST is true for (G̃, X̃, Ỹ). Then it holds for (G̃, X̃^{∗}, Ỹ^{∗}) by the lemma. Moreover, X̃^{∗} → X and Ỹ^{∗} → Y are principal G_{m}-bundles. It follows easily, that all assertions of the theorem descend to (G, X, Y). Thus, by replacing (G, X, Y) with (G̃, X̃, Ỹ) we may assume that X is an affine G-variety.

We proceed by arguing by induction on dim X. Assume first that the G-action on Y is already elementary. Then Q = L = G and R = X have all required properties except possibly the closedness of the G-orbits in Y. To force this, let Z ⊆ Y be the set of all y ∈ Y whose orbit Gy is not of maximal dimension. This is a proper closed G-stable subset of Y. Moreover, all G-orbits in Y \ Z have the same dimension, thus they are closed. By Proposition 3.13 we can choose a G-semiinvariant function \overline{f} on Y vanishing on Z. Since k[Y] is, as a G-module, a direct summand of k[X], the function \overline{f} extends to a G-semiinvariant function f on X. Let X_{f} be the non-vanishing set of f. Then Y_{f} = Y ∩ X_{f} ≠ ∅ and all of its G-orbits are closed. Thus, we can take Q = G and R = X_{f} and we are done.

Finally, assume that the action of G on Y is not elementary. Then Corollary 3.17 implies the existence of a P -semiinvariant function \overline{f} ∈ k[Y] whose character does not extend to G. As above, \overline{f} extends to a P -semiinvariant function f ∈ k[X]. Let X_{f} ⊆ X be its non-vanishing set. Then Y_{f} = X_{f} ∩ Y = Y\overline{f} ≠ ∅. Let \overline{Q} := \{g ∈ G | gf ∈ k^{∗}f\} be the normalizer of f in G. Since f is a highest weight vector, \overline{Q} is a parabolic subgroup which acts on the line kf by a dominant character χ_{f}. Moreover, the parabolic \overline{Q} is a proper subgroup of G because χ_{f} does not extend to a character of G. Let \overline{U} = \overline{Q}_{u} be its unipotent radical. From highest weight theory we get

(4.2) ⟨χ_{f}, α^{∨}⟩ > 0 for all roots α in \overline{𝔲} := Lie \overline{U}

The rest of the argument is almost identical to the proof of [Kno94, Thm. 2.3]. On the \overline{Q}-stable open set X_{f} one can define the \overline{Q}-equivariant morphism

(4.3) m : X_{f} → 𝔤^{∗} : x ↦ m_{x}

where

(4.4) m_{x} : 𝔤 → K : ξ ↦ \frac{ξf(x)}{f(x)} .

Choose a k-rational point y ∈ Y_{f} (possible because of k-density) and put a := m(y) ∈ 𝔤^{∗}. Then the \overline{Q}-invariance of f implies that the image of m is contained in the affine subspace

(4.5) a + \overline{𝔮}^{⊥} = a + \overline{𝔲}

where in the last equality we used an isomorphism 𝔤^{∗} ≅ 𝔤 to identify \overline{𝔮}^{⊥} with \overline{𝔲}.

Next, it follows from (4.2) that the stabilizer of a in G is a Levi subgroup \overline{L} of \overline{Q}. It follows moreover, that a + \overline{𝔲} = \overline{Q}a = \overline{Q}/\overline{L} is a single \overline{Q}-orbit. In conclusion, we got a \overline{Q}-equivariant morphism m : X_{f} → \overline{Q}/\overline{L}. This means that if we define X′ := m^{−1}(a) then

(4.6) \overline{U} × X′ = \overline{Q} ×\overline{L} X′ →^{∼} X_{f} : [q, x] ↦ qx.

is an isomorphism.

Because \overline{U} ≠ 1 we have dim X′ < dim X. Therefore, we can apply the induction hypothesis to \overline{L} acting on X′ and Y′ := X′ ∩ Y. This yields a parabolic k-subgroup Q′ ⊆ \overline{L}, a Levi decomposition Q′ = L′U′, and an L′-stable affine slice R ⊆ X′ such that

(4.7) U′ × R = Q′ ×^{L′} R → X′

is an open immersion. Now put Q := Q′ \overline{U} ⊆ \overline{L} \overline{U} ⊆ G, a parabolic subgroup of G. Then combining (4.6) and (4.7) we get that

(4.8) Q ×^{L} R = Q_{u} × R = \overline{U} × U′ × R ↪ \overline{U} × X′ ↪ X

is an open immersion. □

Since Q ×^{L} R = U × R is affine we get as an immediate consequence:

4.4. Corollary. Let Y be a G-stable k-dense subvariety of a locally k-linear G-variety X. Then there is a P -stable affine open k-subset X_{0} ⊆ X with X_{0} ∩ Y ≠ ∅.

The slice R is not unique, but we still have some uniqueness properties. First, we define for any k-dense G-variety X:

(4.9) Q_{k}(X) := {g ∈ G | gP x = P x for x in a dense open subset of X}.

Because of P ⊆ Q_{k}(X), this is a parabolic k-subgroup.

4.5. Proposition. In the setting of Theorem 4.2 assume (without loss of generality) additionally that P ⊆ Q. Then Q = Q_{k}(Y). Moreover, as an L-variety, the slice R is unique up to a unique L-equivariant birational isomorphism which is regular in a generic point of Y_{el}.

Proof. Put, for the moment, \overline{Q} = Q_{k}(Y). From (3.2) and the k-LST follows that the generic P -orbits and Q-orbits on Y coincide. This shows Q ⊆ \overline{Q}.

For the converse, let y ∈ Y_{el} be generic. Then \overline{Q}y = P y by assumption and P y = Qy by the k-LST. Thus \overline{Q}/\overline{Q}_{y} = Q/Q_{y} which implies Q_{y} = Q ∩ \overline{Q}_{y} and \overline{Q} = Q\overline{Q}_{y}. This in turn implies \overline{Q}_{y}/Q_{y} = \overline{Q}/Q. Now observe that Q_{y} = L_{y} is reductive since the L-action on Y is elementary. Therefore \overline{Q}_{y}/Q_{y} is an affine variety. On the other hand, \overline{Q}/Q is projective which implies \overline{Q} = Q.

Finally, let X_{0} = QR ⊆ X. This is an open, dense, Q-stable subset with X_{0} ∩ Y ≠ ∅ such that the orbit space X_{0}/Q_{u} exists. Moreover, R is isomorphic to this quotient. If X_{1} ⊆ X is any other such set then R is birationally equivalent to (X_{0} ∩ X_{1})/Q_{u} and therefore to X_{1}/Q_{u}. This shows the second assertion. □

One of the most important special cases is that of Y = X. We formulate it explicitly:

4.6. Corollary. Let X be a k-dense G-variety and let Q_{k}(X) = LU be a Levi decomposition. Then there exists a smooth affine L-subvariety X_{el} ⊆ X such that

i) the action of L on X_{el} is elementary, all orbits are closed, and the categorical quotient X_{el} → X_{el}//L is a locally trivial fiber bundle in the étale topology.

ii) the natural morphism U × X_{el} = Q_{k}(X) ×^{L} X_{el} → X is an open embedding.

The slice X_{el} is unique up to a unique L-equivariant birational isomorphism. More precisely, its field of rational functions can be computed as

(4.10) k(X_{el}) = k(X)^{U} = k(X)^{N}.

Proof. Apply Theorem 4.2 to the smooth part of X which is locally linear by Sumhiro’s theorem. This proves everything except for the local triviality statement and the second equality in (4.10). The former is well known consequence of Luna’s slice theorem (see [Lun73, III.2, Cor. 5]. The latter is a consequence of the fact that N = U · (L ∩ N) and that L ∩ N acts trivially on X_{el}. Hence the generic N- and U-orbits coincide on X. □

Because of the factorization (3.2) we don’t loose any information by considering X_{el} as an MA-variety. Since all A-orbits in X_{el} are closed we may form the orbit space.

4.7. Definition. The quotient X_{an} := X_{el}//A is called the anisotropic kernel of X.

The anisotropic kernel is, by construction a k-dense affine M-variety all of whose orbits are closed of the same dimension. In particular, all isotropy groups are reductive and, over K, conjugate to each other. In our theory X_{an} will play a role similar to the anisotropic kernel of a reductive group in Borel-Tits theory [BT65]. This means that the anisotropic kernel will mostly serve as a black box.

From (4.10) we see that

(4.11) k(X_{an}) = k(X)^{AN} and therefore k(X_{an})^{M} = k(X)^{P} .

This provides another way to see that the anisotropic kernel is a birational invariant for X.

Next we introduce an important numerical invariant for the action of G on X. Recall that the cohomogeneity of an action is the codimension of a generic orbit.

4.8. Definition. The k-complexity of X, denoted c_{k}(X), is the cohomogeneity for the P -action on X. It also equals the transcendence degree of k(X)^{P} (by Rosenlicht), the transcendence degree of k(X_{an})^{M}, and the cohomogeneity of the M-action on X_{an}.

The most important special case is:

4.9. Definition. A locally linear, k-dense G-variety X with c_{k}(X) = 0 is called kspherical. Equivalently, X is k-spherical if it is locally linear, k-dense and P has an open orbit.

Another characterizing property for X being k-spherical is that M acts transitively on the anisotropic kernel. Observe that in this case X_{an} is even unique up to a biregular (as opposed to a birational) isomorphism.

4.10. Remarks. i) Let B ⊆ P be a Borel subgroup. Then X is K-spherical if B has a dense orbit in X, i.e., if X is spherical in the usual sense. In this case, we call X absolutely spherical. Clearly all absolutely spherical varieties are kspherical. In particular, a symmetric variety G/H with H the fixed point group of an involution is k-spherical. The arguably most important example is that of G/K where k = ℝ and K is a maximal compact subgroup of G.

ii) A k-variety is absolutely spherical if and only if its anisotropic kernel is absolutely spherical. Over k = ℝ, the latter have been classified in works of Krämer [Krä79], Mikityuk [Mik86], and Brion [Bri87].

iii) If G is quasi-split then P = B. Thus, every k-spherical variety is absolutely spherical. Otherwise, X = G/AN is an example of a k-spherical variety which is not absolutely spherical. Its anisotropic kernel is MA/A = M/M ∩ A.

For quasi-affine varieties there is a representation theoretic criterion which was first proved over algebraically closed fields by Vinberg-Kimel’feld [VK78].

4.11. Proposition. For a quasi-affine, k-dense G-variety X the following are equivalent:

i) X is k-spherical.

ii) For every dominant character χ of P, the simple G-module with highest weight χ appears with multiplicity at most 1 in k[X].

Proof. Let M_{1}, M_{2} ⊆ k[X] two distinct simple G-submodules with highest weight χ. Let f_{i} ∈ M_{i} be a highest weight vector. Then f = \frac{f_{1}}{f_{2}} is a non-constant P -invariant rational function on X. Thus X cannot have an open P -orbit.

Conversely, assume that X does not have an open P -orbit. Then by Rosenlicht there is a non-constant P -invariant rational function on X. By Corollary 3.12, there are P - semiinvariants f_{1}, f_{2} ∈ k[X] with f = \frac{f_{1}}{f_{2}} . The character of f_{1} and f_{2} are the same, say χ. Hence the G-modules M_{i} generated by f_{i} are distinct and irreducible with highest weight χ. □

For k-spherical varieties Corollary 4.6 reads as follows:

4.12. Corollary. Let X be a k-spherical variety G-variety and Q := Q_{k}(X) = LU be a Levi decomposition. Then there is a point x ∈ X(k) such that Qx is open and affine in X and the isotropy subgroup satisfies L_{el} ⊆ Q_{x} ⊆ L.

Next we compare the complexity of a variety with that of its subvarieties.

4.13. Proposition. Let X be a locally linear k-dense G-variety and let Y ⊆ X be a Gstable k-dense subvariety. Then c_{k}(Y) ≤ c_{k}(X) with equality if and only if K(X)^{P} ⊆ O_{X,Y} (the local ring of X in Y).

Proof. We apply Theorem 4.2 to Y ⊆ X. Since all L-orbits in Y_{el} are closed, the morphism

(4.12) Y_{el}//L → R//L

is a closed embedding. Thus

(4.13) c_{k}(Y) = c_{k}(Y_{el}) = dim Y_{el}//L ≤ dim R//L ≤ c_{k}(R) = c_{k}(X).

In case of equality, (4.12) is an isomorphism. From

(4.14) K(X)^{P} ⊆ K(X)^{LU} = K(R)^{L} = K(Y_{el})^{L} = K(Y)^{LU} ⊆ K(Y).

we see that all P -invariant rational functions are regular in Y. Conversely, K(X)^{P} ⊆ K(Y) clearly implies c_{k}(Y) ≥ trdeg K(X)^{P} = c_{k}(X). □

4.14. Corollary. Let X be a k-spherical G-variety. Then every k-dense G-stable subvariety is k-spherical and X contains only finitely many of them. In particular, the number of G-orbits Y with Y (k) ≠ ∅ is finite.

Proof. The first assertion follows directly from Proposition 4.13. Suppose X contains infinitely many k-dense subvarieties. Then there is a minimal closed G-stable subvariety Y with the same property. Since Y is spherical, P and therefore G has a dense open orbit Y^{0}. But then one of the irreducible components of Y \ Y^{0} would be an even smaller counterexample. □

4.15. Corollary. Let k be a local field and X a k-spherical variety. Then the number of G(k)-orbits in X(k) is finite.

Proof. This follows from Corollary 4.15 and [BS64, Cor. 6.4]: if k is local (of characteristic 0) then for any homogeneous G-variety Y the orbit set Y (k)/G(k) is finite. □

Next we discuss the relationship between X_{el} and X_{an}. Let A_{0} ⊆ A be the kernel of the action of A on X_{el}. Then A_{k} = A_{k}(X) := A/A_{0} is acting freely on X_{el} and the quotient map

(4.15) π : X_{el} → X_{an}

is an M-equivariant principal bundle for A_{k}. We make this more precise. The character group of A_{k} consists of the characters χ_{f} ∈ Ξ(A) where f is an A-semiinvariant rational function on X_{el}. The Local Structure Theorem implies that these f correspond to AN-semiinvariant rational functions on X. From this, we derive another fundamental invariant of X:

(4.16) Ξ(A_{k}(X)) = Ξ_{k}(X) := {χ_{f} ∈ Ξ(A) | f ∈ K(X)^{(AN)}}.

Since χ_{f} = 1 if f is AN-invariant, the group Ξ_{k}(X) fits into the short exact sequence

(4.17) 1 → K(X_{an})^{∗} → K(X)^{(AN)} → Ξ_{k}(X) → 0.

4.16. Definition. Let X be a k-dense G-variety. The k-rank of X is

(4.18) rk_{k} X := rk Ξ_{k}(X) = dim A_{k}(X) = dim X_{el} − dim X_{an}.

Coming back to (4.15), the algebra k[X_{el}] decomposes as an A_{k}-module into isotypic components:

(4.19) k[X_{el}] = ⨁_{χ∈Ξ_{k}(X)} S_{χ}.

Moreover, each S_{χ} is an M-line bundle over R := S_{0} = k[X_{an}]. Since S_{χ} ⊗_{R} S_{η} = S_{χ+η} this induces a homomorphism

(4.20) Ξ_{k}(X) → Pic^{M}(X_{an}) : χ ↦ [S_{χ}]

which encodes the non-triviality of the bundle (4.15).

If X is k-spherical the situation simplifies. After fixing a point y ∈ X_{an}(k) we get X_{an} ≅ M/M_{y}. Then for any m ∈ M_{y} there is a unique a ∈ A_{k} with mx = ax for all x ∈ π^{−1}(y). This way, we get a homomorphism

(4.21) φ : M_{y} → A_{k} : m ↦ a

which is defined over k. Using φ, one can recover X_{el} as

(4.22) X_{el} ≅ M ×^{M_{y}} A_{k}

4.17. Remarks. i) Since the connected component M_{y}^{∘} is anisotropic, φ is trivial on it. Thus φ factors through the finite group π_{0}(M_{y}). In particular, φ(M_{y}) is finite.

ii) Besides Ξ_{k}(X) it seems natural to consider the group

(4.23) Ξ′_{k}(X) := {χ_{f} ∈ Ξ(A) | f ∈ K(X)^{(P)}}.

It is easy to see that Ξ′_{k}(X) is the character group of the torus A′_{k}(X) := A_{k}(X)/φ(M_{y}). We won’t have use for this group, though.

We finish this section by stating a comparison result between the theories over k and K. For this, choose a maximal torus T ⊆ MA which is defined over k. Let, moreover B be a Borel subgroup of P (and therefore of G) containing T. Unless G is quasi-split, B won’t be defined over k. The torus A is the maximal split subtorus of T. Recall the restriction map res_{A} : Ξ(T) → Ξ(A).

4.18. Lemma. Let X be a k-dense G-variety. Then

(4.24) Ξ_{k}(X) = res_{A} Ξ_{K}(X).

Proof. The inclusion “⊇” follows from the fact that every B-semiinvariant is also an AN-semiinvariant for the restricted character. We show the opposite inclusion. Since Ξ_{k}(X) = Ξ_{k}(X_{el}) it suffices to prove the statement for MA acting on X_{el}. Since all A-orbits in X_{el} are closed, every rational A-semiinvariant on X_{el} is the quotient of two regular A-semiinvariants. Thus it suffices to show that for every f ∈ K[X_{el}]^{(A)} there is \overline{f} ∈ K[X_{el}]^{(B_{0})} with χ_{f} = res_{A} χ\overline{f} where B_{0} := B∩MA. For this consider the A-eigenspace S_{χ} := K[X_{el}]_{χ}. This space is MA-stable and non-zero by assumption. Then any highest weight vector \overline{f} ∈ S_{χ} will be a B_{0}-semiinvariant with the required property. □

This result can be rephrased. By construction, the tori A_{k}(X) and A_{K}(X) are quotients of A and T, respectively. Then the lemma means:

4.19. Corollary. The torus A_{k}(X) is the image of A ⊆ T in A_{K}(X).

A slightly weaker result goes as follows. Recall the notation N (A) = Hom(Ξ(A), ℚ) and put

(4.25) N_{k}(X) = N (A_{k}(X)) = Hom(Ξ_{k}(X), ℚ).

This is a ℚ-vector space of dimension rk_{k} X.

4.20. Corollary. The space N_{k}(X) is the image of N (A) ⊆ N (T) in N_{K}(X).

4.21. Remarks. 1. Observe that T is not unique, not up to conjugacy nor even up to isomorphy. An exception is the field k = ℝ of real numbers where T is unique up to conjugation by M(ℝ).

2. The character group Ξ_{k}(X) can also be defined as

(4.26) Ξ_{k}(X) := {χ_{f} ∈ Ξ(A) | f ∈ k(X)^{(AN)}},

i.e., with semiinvariants defined over k instead of K. To see this, one can again reduce to the case that G is elementary, X is affine and f is regular. The point is now that since A is split, the character χ_{f} is defined over k even if f isn’t. Then k[X]_{χ_{f}} ⊗_{k} K ≅ K[X]_{χ_{f}} ≠ 0 implies k[X]_{χ_{f}} ≠ 0.

4.22. Example. We illustrate the theory with a simple but instructive example. Let D be a finite dimensional division algebra over k whose center is denoted by E. For n ≥ 2 let

(4.27) H_{1} := 1 × GL(n − 1, D) ⊂ GL(1, D) × GL(n − 1, D) ⊆ G := GL(n, D).

Then X := G/H_{1} can be realized as follows: let V = D^{n} be the space of column vectors with G-action on the left, g · \underline{v} := g\underline{v} and V^{⊤} be the space of row vectors with action g · \underline{u}^{⊤} := \underline{u}^{⊤}g^{−1}. Then the map b : V^{⊤} × V → D with b(\underline{u}^{⊤}, \underline{v}) = ∑_{i} u_{i}v_{i} is G-invariant. Since H_{1} is the isotropy group of (\underline{e}_{1}^{⊤}, \underline{e}_{1}), we get

(4.28) X = \{(\underline{u}^{⊤}, \underline{v}) ∈ V^{⊤} × V | b(\underline{u}^{⊤}, \underline{v}) = 1_{D}\}.

Let Q ⊆ G be the parabolic subgroup fixing the subspaves D\underline{e}_{1} ⊆ V and \underline{e}_{n}^{⊤}D ⊆ V^{⊤}. Then Q = LU with

(4.29) L = GL(1, D) × GL(n − 2, D) × GL(1, D) and U ≅ D^{2n−3} (as a space). Let X_{0} ⊆ X be the open set of (\underline{u}^{⊤}, \underline{v}) with \underline{u}_{1}, \underline{v}_{n} ∈ D^{×} = GL(1, D). Then it is easy to see that U acts freely on X_{0} and that the map

(4.30) π : X_{0} → D^{×} × D^{×} : (\underline{u}^{⊤}, \underline{v}) ↦ (u_{1}, v_{n})

is a quotient by U. This shows that Q_{k}(X) = Q and X_{el} = D^{×} × D^{×} with L-action

(4.31) (d_{1}, d_{2}, d_{3})(u_{1}, v_{n}) = (u_{1}d_{1}^{−1}, d_{3}v_{n}).

Since that action is transitive and elementary we see that X is k-spherical with rk_{k} X = 2. The anisotropic kernel is X_{an} = P GL(1, D) × P GL(1, D). On the other side, X_{an} is Kspherical if and only if D = E is a field.

5. Invariant valuations

Let X initially be any K-variety. In this section we study a particular type of valuations of the field K(X) of K-valued rational functions on X. More precisely, for us, a valuation is a map v : K(X) → ℚ ∪ {∞} satisfying

i) v(f_{1}f_{2}) = v(f_{1}) + v(f_{2}),

ii) v(f_{1} + f_{2}) ≥ min{v(f_{1}), v(f_{2})},

iii) v(f) = ∞ ⇔ f = 0,

iv) v(K^{∗}) = 0,

where f, f_{1}, f_{2} run through all elements of K(X). Note, that we also allow the trivial valuation v = 0 with v(f) = 0 for all f ≠ 0. Two valuations v_{1}, v_{2} will be called equivalent ( v_{1} ∼ v_{2}) if there is c ∈ ℚ_{>0} with v_{1} = c v_{2}.

If X is normal then any irreducible divisor D ⊂ X induces a valuation v_{D} by order of vanishing along D. More generally, let \overline{X} be a normal variety which is birational to X. Since K(\overline{X}) = K(X), any irreducible divisor D ⊂ \overline{X} induces a valuation v_{D} of K(X). Valuations which are equivalent to some v_{D} are called geometric. It is useful to allow D = X with v_{X} := 0 and call it geometric, as well. Geometric valuations are characterized by the property that the transcendence degree of their residue field is ≥ dim X − 1.

A pair (\overline{X}, D) with v ∼ v_{D} is called a model for v. The model is unique in the following sense: if X′, D′ is another model for v then the composition of birational maps \overline{X} [figure]X [figure]X′ is regular in a generic point of D, its inverse is regular in a generic point of D′ and it sends D to D′.

If X is defined over k then it is easy to see that \overline{X} can be chosen to be defined over k, as well. If additionally v is invariant for the action of the Galois group then one can achieve that D is defined over k, as well.

Now assume that a connected linear algebraic group H acts on X. Then H (or rather its group H(K) of rational points) acts on the set of valuations of X. In particular, v is H-invariant if

(5.1) v(hf) = v(f) for all h ∈ H and f ∈ K(X).

An infinitesimal version goes as follows: the Lie algebra 𝔥 of H acts on K(X) by derivations. Then a valuation v is H-invariant if and only if

(5.2) v(ξf) ≥ v(f) for all ξ ∈ 𝔥 and f ∈ K(X)

(see [LV83, 3.2. Cor. 3]).

The correspondence between divisors and valuations works also in the equivariant setting: if v is an H-invariant geometric valuation of K(X) then there is a normal model \overline{X} of X equipped with an H-action and an H-stable irreducible divisor D ⊂ \overline{X} such that v ∼ v_{D} ([Kno93, Kor. 7.2]). Of particular importance is the case when X is a homogeneous variety. Then an equivariant model of X is the same as an open embedding X ↪ \overline{X}, a so-called equivariant embedding of X. Moreover, any invariant irreducible divisor must be an irreducible component of the boundary \overline{X} \ X.

In this paper, only a very special kind of valuations is of interest for us.

5.1. Definition. An invariant valuation of a k-dense G-variety X is called k-central if it is trivial on the subfield K(X_{an}) = K(X)^{AN}. The set of all k-central valuations is denoted by Z_{k}(X).

It follows directly from the definition and the short exact sequence (4.17) that a k-central valuation induces a homomorphism

(5.3) λ_{v} : Ξ_{k}(X) → ℚ : χ_{f} ↦ v(f),

from which we get a map

(5.4) ι_{k} : Z_{k}(X) → N_{k}(X).

Over K, central valuations have been extensively studied in [Kno93]. For transferring properties from K to k we choose T ⊆ B ⊆ P as in the end of section 4. Recall (see Corollary 4.20) that N_{k}(X) can be considered as a subspace of N_{K}(X).

5.2. Proposition. Let X be a k-dense G-variety. Then:

i) The map ι_{k} in (5.4) is injective.

ii) Considering Z_{k}(X) and Z_{K}(X) as subsets of N_{k}(X) and N_{K}(X), respectively, then

(5.5) Z_{k}(X) = Z_{K}(X) ∩ N_{k}(X).

Proof. From AN ⊆ B it follows that K(X)^{AN} ⊇ K(X)^{B}. Thus, every k-central valuation is also K-central. From this, we get the commutative diagram

(5.6) \xymatrix{Z_{k}(X) \ar@{^{(}->}[r] \ar[d]^{ι_{k}} & Z_{K}(X) \ar[d]^{ι_{K}} \\ N_{k}(X) \ar@{^{(}->}[r] & N_{K}(X)} .

Hence ι_{k} is injective since ι_{K} is injective by [Kno93, Kor. 3.6].

It remains to be shown that Z_{K}(X) ∩ N_{k}(X) ⊆ Z_{k}(X). Let v be in the intersection. Then v is trivial on all B-semiinvariants f with res_{A} χ_{f} = 0, i.e., on (K(X)^{(B)})^{A}. Now we have

(5.7) (K(X)^{(B)})^{A} = ((K(X)^{AN})^{(M∩B)} = K(X_{an})^{(M∩B)}.

It follows again from [Kno93, Kor. 3.6]) that v is trivial on K(X_{an}). Thus v is k-central. □

From this we get:

5.3. Corollary. Let X be a k-dense G-variety. Then Z_{k}(X) is a finitely generated convex cone in N_{k}(X).

Proof. This follows from (5.5) and the corresponding statement for Z_{K}(X) (see [Kno93, Kor. 6.5]). □

In principle, Z_{k}(X) could be still reduced to a point. We show that it is actually fairly big. For this let

(5.8) N^{−}(A) = {a ∈ N (A) | α(a) ≤ 0 for all α ∈ S_{k}}

be the antidominant Weyl chamber with respect to the restricted root system of G.

5.4. Proposition. Let π : N (A) ↠ N_{k}(X) be the canonical projection. Then

(5.9) π(N^{−}(A)) ⊆ Z_{k}(X).

In particular, the interior of Z_{k}(X) is non-empty.

Proof. Let N^{−}(T) be the antidominant Weyl chamber of G over K. Then N^{−}(A) = N^{−}(T) ∩ N (A). By [Kno93, Kor. 5.2.4], the image of N^{−}(T) in N_{K}(X) is contained in Z_{K}(X). Thus, Z_{k}(X) contains the image of N^{−}(A). □

6. Valuations and geometry

Next, we study the generic structure of invariant divisors defined by valuations. For this, we need a refinement of Corollary 4.6 which also works “at infinity”.

To set this up, let G be a connected reductive group and let X be a normal k-dense G-variety. According to Corollary 4.6 there is an open embedding U × X_{el} ↪ X with U := Q_{k}(X)_{u}. By projection to the first factor we get a dominant rational map u : X [figure]U with the characterizing property that u(x)^{−1}x ∈ X_{el} for generic x ∈ X. This map induces a pull-back homomorphism of function fields u^{∗} : K(U) ↪ K(X).

6.1. Lemma. Let v be a G-invariant valuation of a k-dense G-variety X. Then v(u^{∗}(h)) ≥ 0 for all regular functions h ∈ K[U].

Proof. The proof depends on the explicit construction of X_{el} in the proof of Theorem 4.2. First, we constructed a G-stable open subset X_{0} ⊆ X which is embeddable in some P(V) and considered the affine cone X̃ ⊆ V over X_{0}. We claim that it suffices to prove our assertion for X̃ and the group G̃ = G_{m} × G. First of all observe that the slice X̃_{el} ⊆ X̃ is homogeneous. This implies that the projection ũ : X̃ [figure]U factors through u : X [figure]U. Secondly, it is known from [Kno93, Kor. 3.2] that the valuation v on X can be lifted to a G_{m} × G-invariant valuation ṽ on X̃. Then

(6.1) v(u^{∗}(h)) = ṽ(ũ^{∗}(h)) ≥ 0

proves the claim. Replacing X by X̃ we may assume that X is a subvariety of V.

Next, we used a regular P -semiinvariant regular function f on X. The stabilizer of the line Kf is the parabolic \overline{Q} = \overline{L} \overline{U}. Then a \overline{Q}-equivariant map m : X_{f} [figure]a + \overline{𝔲} with fiber X′ := m^{−1}(a) was defined such that \overline{ι} : \overline{U} × X′ →^{∼} X_{f}. Finally, application of the induction hypothesis yielded an open embedding ι′ : U′ × R ↪ X′.

The open embeddings \overline{ι} and ι′ allow us to define projections \overline{u} : X [figure]\overline{U} and u′ : X′ [figure]U′ onto the first factors. Then π(x) := \overline{u}(x)^{−1}x is the projection of X to the second factor X′. With this notation, u has the form

(6.2) u : X [figure]U = \overline{U} × U′ : x ↦ (\overline{u}(x), u′(π(x)))

Clearly, it is enough to prove the non-negativity of v(u^{∗}(h)) for generators h of the algebra K[U]. Thus, we may assume that h is either a function on \overline{U} or on U′. In the first case observe that \overline{u} is the composition of m : X_{f} → a + \overline{𝔲} with the isomorphisms a + \overline{𝔲} ≅ \overline{U}. The functions on the affine space a + \overline{𝔲} are generated by the linear functions ℓ_{ξ} := (ξ, ∗) with ξ ∈ 𝔤 where (·, ·) is an invariant scalar product on 𝔤. From the definition of m we get

(6.3) v(m^{∗}(ℓ_{ξ})) = v(\frac{ξf}{f}) = v(ξf) − v(f).

Now the G-invariance of v implies that the right-hand side is non-negative (see (5.2)).

Finally assume that h is a regular function on U′. We restrict the valuation v via π to X′, i.e., for any h′ ∈ K(X′) we define v′(h′) := v(π^{∗}(h′)). It easily seen that v′ is an \overline{L}-invariant valuation of X′. The induction hypothesis then yields

(6.4) v((u′ ∘ π)^{∗}(h)) = v′((u′)^{∗}(h)) ≥ 0. □

As a consequence we get the existence of very big slices. For that we have to give up the affinity of the slice and restrict to normal varieties.

6.2. Theorem. Let X be a normal k-dense G-variety and let Q_{k}(X) = LU be a Levi decomposition. Then there exists an L-stable subvariety \overline{X}_{el} ⊆ X (everything defined over k) such that:

i) the action of L on \overline{X}_{el} is elementary;

ii) the natural morphism Q_{k}(X) ×^{L} \overline{X}_{el} → X is an open embedding;

iii) the boundary ∂X_{0} = X \ X_{0} of the open set X_{0} := U · \overline{X}_{el} ⊆ X is of pure codimension one in X and none of its irreducible components is G-stable.

Proof. Let X_{0} ⊆ X be the largest open subset on which the rational map u : X [figure]U is regular. Put \overline{X}_{el} := u^{−1}(1) ⊆ X_{0}. Since u is U-equivariant, we infer that U × \overline{X}_{el} → X_{0} is an isomorphism which shows ii). Property i) follows from the fact that X_{el} is open in \overline{X}_{el}.

Thus, the main point is to show iii). Since U is affine, a point x ∈ X is in X_{0} if and only if u^{∗}(h) is regular in a neighborhood of x for all h ∈ K[U]. In a normal variety this is the case if and only if x is not contained in the pole divisor of any function u^{∗}(h). Thus, ∂X_{0} is the union of all these pole divisors and therefore is of pure codimension 1.

Finally, suppose ∂X_{0} has an irreducible component D which is G-stable (but possibly not defined over k). Since v_{D} is a G-invariant valuation, Lemma 6.1 implies v_{D}(u^{∗}(h)) ≥ 0 for all h ∈ K[U]. Thus, the generic point of D lies in X_{0} in contradiction to D ⊆ ∂X_{0}. □

From Proposition 4.5 we get:

6.3. Corollary. Let X be a k-dense G-variety and let D ⊆ X be a k-dense irreducible divisor. Then Q_{k}(D) = Q_{k}(X).

Next, we want to relate k-central valuations with toroidal embeddings. For this, fix a kcentral valuation v. It induces an element a := λ_{v} ∈ N_{k}(X) = N (A_{k}) (with A_{k} := A_{k}(X)) and let R_{v} := ℚ_{≥0}a ⊆ N (A_{k}) = N_{k}(X) be the ray generated by a. Then the theory of toroidal embeddings asserts the existence of an embedding \overline{A} = A_{k} ∪ D(a) where D(a) is a homogeneous divisor which is characterized by the fact that λ_{v_{D(a)}} ∼ a.

Another way to relate \overline{A} with a goes as follows: The space N (A_{k}) contains the group Hom(G_{m}, A_{k}) of cocharacters as a sublattice. Thus, there is a homomorphism \overline{a} : G_{m} → A_{k} with \overline{a} = c a for some c ∈ ℚ_{>0}. Then

(6.5) π_{a} : \overline{A} → \overline{A} : x ↦ lim_{t→0} \overline{a}(t)x

is a well-defined projection onto D(a).

There is also a slightly more general version of this construction. For this, let X → Y be a principal A_{k}-bundle (defined over k) and let X(a) := X ×^{A_{k}} \overline{A} → Y be the associated fiber bundle. Then X is open in X(a) and the complement is an irreducible divisor D_{a}. Let v_{0} = v_{D_{a}} be the associated valuation. Since K(X)^{A_{k}} = K(Y), the valuation v_{0} is kcentral for the action of A_{k} with λ_{v_{0}} = a. This shows, in particular, that Z_{k}(X) = N_{k}(X) and therefore, that every k-central valuation is of this type. Clearly, (6.5) still defines a projection π_{a} : X(a) → D_{a}.

Now we treat the general case:

6.4. Lemma. Let X be a normal k-dense G-variety and let D ⊆ X be an irreducible G-stable divisor such that v_{D} is k-central. Put a = λ_{v_{D}}. Then there exist a slice X_{el} ⊆ \overline{X}_{el} as in Theorem 6.2 with X_{el}(a) ⊆ \overline{X}_{el} open and D_{a} ⊆ D ∩ \overline{X}_{el}.

Proof. Let D_{el} := D ∩ \overline{X}_{el}. We have to show that the valuation v_{D_{el}} of X_{el} equals the valuation D_{v_{a}}. For any f ∈ K(X_{el}) let u^{∗}f ∈ K(X) be its pull-back via the rational map u : X [figure]X_{el}. Then

(6.6) v_{D}(u^{∗}f) = v_{D_{el}}(f)

by Theorem 6.2. Let f ∈ K(X_{el})^{A}. Then u^{∗}f ∈ K(X)^{AN} and therefore v_{D_{el}}(f) = 0 since v_{D} is k-central. This means that v_{D_{el}} is k-central for the A-action on X_{el}. Because of the injectivity of the map (5.4) it suffices to check v_{D_{el}}(f) = v_{a}(f) for A-semiinvariants. But this clear:

(6.7) v_{D_{a}}(f_{χ}) = a(χ) = λ_{v_{D}}(χ) = v_{D}(u^{∗}f_{χ}) = v_{D_{el}}(f_{χ}). □

Now we introduce another important property of invariant valuations.

6.5. Definition. Let X be a k-dense variety. A geometric valuation v of X is called k-dense if either it is trivial or the corresponding divisor D of any k-model of v is defined over k and is k-dense. In this case, we define the complexity of v as c_{k}(v) = c_{k}(D).

The next theorem characterizes k-central valuations. It implies in particular that all k-central valuations are k-dense.

6.6. Theorem. Let X be a k-dense G-variety. Then for a G-invariant valuation v of K(X) the following conditions are equivalent:

i) v is k-central

ii) v is k-dense and the restriction of v to K(X)^{P} = K(X_{an})^{M} is trivial. iii) v is k-dense with c_{k}(v) = c_{k}(X).

iv) v is k-dense and K-central.

Proof. If v is trivial then all conditions hold. So assume v ≠ 0. We claim that all conditions imply that v is geometric. For i) this follows from [Kno93, Satz 4.4] and the fact that v is K-central. For ii) and iii) this is part of the definition of k-density. Thus, without loss of generality we may assume that X is normal and that it contains a divisor D with v = v_{D}.

i)⇒ii). From AN ⊆ P we get K(X)^{P} ⊆ K(X)^{AN}. Thus, v is trivial on K(X)^{P}. Now consider the situation of Lemma 6.4. Since X is k-dense also X_{el} is k-dense. Then, because of the projection π_{a} : X_{el} → D(a) (see (6.5)), also D(a) is k-dense. Therefore D which is birational to U × D(a) is k-dense, as well.

ii)⇒i). We first treat the case that the action of G on X is elementary. Then clearly we may assume that G = MA is elementary itself. In that case, we have P = G and N = 1. Then using Corollary 4.4 we may assume without loss of generality that X and D are affine and smooth.

Now consider the categorical quotient π : X → Y := X//A. Then Z := π(D) = D//A is a closed subset of Y. If Z ⊊ Y there is an M-invariant function f_{0} ≠ 0 on Y which vanishes on Z (Proposition 3.13). Its pull-back f = π^{∗}(f_{0}) would be a G-invariant function on X with v(f) > 0 in contradiction to the assumption.

Thus π : D → Y is surjective. This already shows that v(f) = 0 for all non-zero f ∈ K[Y] = K[X]^{A}. It remains to show that K(X)^{A} is the quotient field of K[X]^{A}. This is equivalent to the generic fiber X_{y} = π^{−1}(y) of π containing a dense open A-orbit. Since X_{y} is smooth and irreducible it suffices to show that X_{y} contains an orbit of dimension r = dim X − dim Y. Recall that X_{y}, as a fiber of a categorical quotient, contains exactly one closed A-orbit. By construction, all G-orbits of D are closed. This implies that X_{y}∩D is the closed A-orbit in X_{y}. Since y is generic, its codimension is 1. Moreover, it is in the closure of every A-orbit in X_{y}. This shows that X_{y} contains an orbit of dimension r proving our claim.

Now let X be any G-variety. Then Theorem 6.2 yields an L-stable elementary slice \overline{X}_{el} ⊆ X such that D_{el} := \overline{X}_{el} ∩ D ≠ ∅. Put v_{el} := v_{D_{el}}. Since D is birational to U × D_{el} also v_{el} is k-dense. Because of K(X)^{P} = K(X_{el})^{MA}, v_{el} meets both conditions of ii). Thus, by the above, v_{el} is k-central. Then K(X)^{AN} = K(X_{el})^{A} shows that v is k-central.

The equivalence of ii) and iii) follows from Proposition 4.13. The equivalence of iv) with the other conditions follows from K(X)^{P} ⊆ K(X)^{B} ⊆ K(X)^{AN}. □

6.7. Remark. One can show that the complexity of a non-central k-dense valuation equals c_{k}(X) − 1.

For a k-spherical variety X we have K(X)^{P} = K. Hence:

6.8. Corollary. Let X be a k-spherical G-variety. Then a G-invariant valuation on X is k-central if and only if it is k-dense.

We conclude this section with a statement saying that subvarieties of maximal k-complexity are in a sense controlled by k-central valuations. Recall that a subvariety Y of X is the center of a valuation v if v ≥ 0 on the local ring O_{X,Y} and v > 0 on its maximal ideal 𝔪_{X,Y}. If v is geometric with model (X̃, D) this means that the birational map X̃ [figure]X maps D dominantly to Y. The center is unique if it exists. More generally, let X̃ → X be just a dominant morphism and v a valuation of X̃. Then Y ⊆ X is the center of v if it is the center of the restricted valuation v|_{K(X)}.

6.9. Lemma. Let π : X̃ → X be a dominant G-morphism where X̃ is k-dense and X is locally linear. Let Y ⊆ X be a k-dense closed G-subvariety with c_{k}(Y) = c_{k}(X̃). Then Y is the center of a k-central valuation of X̃.

Proof. The strategy is to construct a dominant rational G-map Φ : Z × A^{1} [figure] X̃ where Z is a k-dense G-variety, π ∘ Φ is defined in D_{0} := Z × 0, and π ∘ Φ maps D_{0} dominantly to Y. In that case, Φ induces an embedding k(X̃) ↪ k(Z × A^{1}) and we claim that we can take for v the restriction of v_{D_{0}} to k(X̃).

First, v is geometric and G-invariant since v_{D_{0}} is. Let (\overline{X}, D) be a model of v. Then the rational map Z ×A^{1} [figure] \overline{X} maps D_{0} dominantly to D. This implies that v is k-dense since Z is k-dense. Furthermore, since π∘Φ maps D_{0} dominantly to Y, the divisor D is mapped by π dominantly to Y, i.e., Y is the center of v in X. Finally, since c_{k}(v) ≥ c_{k}(Y) = c_{k}(X̃), the valuation v is k-central by Theorem 6.6.

To construct Φ, we first apply the Local Structure Theorem 4.2 to Y ⊆ X. With the notation of that theorem, we have dim Y_{el}//L = trdeg k(Y)^{P} = c_{k}(Y). On the other hand, we have dim R//L ≤ c_{k}(X) ≤ c_{k}(X̃). Since c_{k}(Y) = c_{k}(X̃) and since Y_{el}//L → R//L is a closed embedding, we see that actually Y_{el}//L = R//L =: S. This means that every L-orbit of R contains a unique L-orbit of Y_{el} in its closure.

Let R̃ := π^{−1}(R) ⊆ X̃ such that U × R̃ → X̃ is birational. Now we would like to apply Kempf’s Theorem 3.6 simultaneously to all points of the form π(x) with x ∈ R̃. To this end, consider the field extension F := k(R̃) of k(S) and the affine F -variety R_{F} := R ×_{S} Spec F obtained by base change. Then R_{F} contains just one closed L_{F} -orbit namely Y_{F} := Y_{el} ×_{S} Spec F. The generic point η = Spec F of R̃ can be considered as an F -rational point of R_{F}. Now we apply Kempf’s theorem to L_{F} acting on R_{F}, the point η, and the closed subset Y_{F}. Then this yields a 1-parameter subgroup λ : G_{m,F} → L_{F} (defined over F) such that the orbit map G_{m,F} → R_{F} : t ↦ λ(t)η extends to a morphism Φ′ : A_{F}^{1} → R_{F} with Φ′(0) ∈ Y_{F}.

Unraveling the definitions, the morphism λ corresponds to a rational morphism R̃ × G_{m} [figure]L : (x, t) ↦ λ_{x}(t). The orbit map Φ_{0} : R̃ × A^{1} [figure] R̃ : (x, t) ↦ λ_{x}(t)x has the property that Φ′ = π ∘ Φ_{0} : R̃ × A^{1} → R : (x, t) ↦ λ_{x}(t)π(x) is defined in t = 0 with Y_{0} := Φ′(R̃ × 0) ⊆ Y_{el}. The fact that Y_{F} is an L_{F} -orbit means that Y_{0} meets the generic L-orbit of Y_{el}. Moreover Φ_{0} is dominant because of Φ_{0}(x, 1) = x for all x ∈ R̃.

Finally, we extend Φ_{0} to X̃ by using the birational map U × R̃ [figure: ∼] X̃. For this we define Z := G × X̃ and Φ as the composition of

(6.8) G×X̃×A^{1} [figure: ∼]G×U ×R̃×A^{1} [figure: Φ_{0}]G×U ×R̃ [figure: ∼]G×X → X : (g, ux, t) ↦ guλ(x, t)x.

Then Φ is clearly equivariant and dominant. Moreover, D_{0} is mapped dominantly to Y because GY_{0} ⊇ ULY_{0} is dense in Y. □

A first application is:

6.10. Corollary. Let φ : X → Y be a dominant G-equivariant morphism between kdense G-varieties. Then φ induces a surjective map φ_{∗} : N_{k}(X) → N_{k}(Y) such that φ_{∗}Z_{k}(X) ⊆ Z_{k}(Y) with equality if c_{k}(X) = c_{k}(Y). In particular, if c_{k}(X) = c_{k}(Y) and rk_{k} X = rk_{k} Y then φ_{∗} : Z_{k}(X) → Z_{k}(Y) is bijective.

Proof. The morphism φ induces obviously an injective homomorphism Ξ_{k}(Y) ↪ Ξ_{k}(X) of which φ_{∗} is the dual, hence surjective map. Because of φ^{∗}K(Y)^{AN} ⊆ K(X)^{AN}, the restriction of a central valuation on K(X) to K(Y) is central, hence φ_{∗}Z_{k}(X) ⊆ Z_{k}(Y).

Now assume c_{k}(X) = c_{k}(Y) and let v ∈ Z_{k}(Y). Let (\overline{Y} , D) be a model of v. Since c_{k}(D) = c_{k}(Y) = c_{k}(X), Lemma 6.9 implies that D is the center of some \overline{v} ∈ Z_{k}(X). This proves φ_{∗}Z_{k}(X) ⊆ Z_{k}(Y). The last assertion is clear. □

6.11. Remark. If c_{k}(Y) < c_{k}(X) then φ_{∗} is in general not surjective on k-central valuations. Let, for example, k = ℂ, G = SL(2) and X = SL(2) with G acting on the left. Then Z_{k}(X) = ℚ_{>0} and c_{k}(X) = 1 (easy, see e.g. [LV83]). Now let Y = ℂ^{2} \ {0} = G/N. Then Z_{k}(Y) = ℚ and c_{k}(Y) = 0 (see section §8). In this case, the surjective morphism X → Y is not surjective on central valuations.

7. Toroidal embeddings

We have seen in Lemma 6.4 that the neighborhood of a k-central divisor is modeled after a toroidal embedding corresponding to a ray, i.e., a one-dimensional cone. In this section we generalize this construction to other toroidal embeddings.

Recall that the theory of toroidal embedding (see [KKMSD73] or [Oda88]) attaches to every fan F in N_{k} := N_{k}(X) a normal equivariant embedding A_{k} ↪ A(F). Recall that a fan is a finite collection of finitely generated strictly convex cones C ⊆ N_{k} satisfying certain axioms. The embedding A(F) has the property that to each C ∈ F there corresponds an orbit A(C) ⊆ A(F). Its closure \overline{A}(C) = \overline{A(C)} is

(7.1) \overline{A}(C) = ⋃_{C⊆C′∈F} A(C′)

The support supp F of F is the union of all C ∈ F.

More generally, if X → Y is a principal A_{k}-bundle then we denote by X(F) := X×^{A_{k}}A(F) the corresponding relative toroidal embedding over Y.

Now we generalize this to arbitrary k-dense G-varieties:

7.1. Theorem. Let X be a normal k-dense G-variety with LST-slice X_{el} and assume that X = G · X_{el}. Let F be a fan with supp F ⊆ Z_{k}(X). Then there is a unique equivariant embedding X ↪ X(F) such that the open embedding U × X_{el} → X extends to an open embedding U × X_{el}(F) → X(F) with X(F) = G · X_{el}(F).

Proof. Let X_{0} be any normal, but not necessarily equivariant model of X. Since each ξ ∈ 𝔤 acts as a derivation on K(X) it induces a rational vector field ξ_{∗} on X_{0}. Luna-Vust have shown ([LV83, Prop. 1.4]) that there is a normal G-variety \overline{X} containing X_{0} as an open subset if and only if for every ξ ∈ 𝔤 the vector field ξ_{∗} is regular on X_{0}. The condition \overline{X} = G · X_{0} makes this model even unique.

We claim that the Luna-Vust condition is satisfied for X_{0} = U × X_{el}(F). Since X_{0} is normal it suffices to show that ξ_{∗} is regular in codimension one. Regularity holds certainly on U × X_{el} since that set is open in the G-variety X. The complement consists of divisors D which correspond to the one-dimensional cones (i.e. rays) R ∈ F. Because, by assumption, R ⊆ Z_{k}(X), the set (U × X_{el}) ∪ D is isomorphic to an open subset of a G-variety by Lemma 6.4. Thus, ξ_{∗} is regular along D, as well, proving the claim. □

7.2. Remark. The condition X = G · X_{el} is very mild. For example it is clearly satisfied when G acts transitively on X. In general, let X′ := G · X_{el} = G · (U · X_{el}). Then X′ is a G-stable open subset to which the theorem applies.

We now study the orbit structure of X(F). For every C ∈ F let

(7.2) X_{el}(C) := X_{el} ×^{A_{k}} A(C) and \overline{X}_{el}(C) := \overline{X}_{el} ×^{A_{k}} \overline{A}(C)

be the relative A_{k}-orbit corresponding to C and its closure, respectively. The sets X_{el}(C) form a stratification of X_{el}(F) which extends to X(F):

7.3. Theorem. Let X ⊆ X(F) is in Theorem 7.1. For each C ∈ F put

(7.3) X(C) := G · X_{el}(C) and \overline{X}(C) := G · \overline{X}_{el}(C).

i) The X(C) form a stratification of X(F). More precisely, each X(C) is locally closed, the closure of X(C) is \overline{X}(C), and \overline{X}(C) is the disjoint union of all X(C′) with C ⊆ C′ ∈ F.

ii) The stratification of X(F) is compatible with that of X_{el}(F), i.e., X(C)∩X_{el}(F) = X_{el}(C) and \overline{X}(C) ∩ X_{el}(F) = \overline{X}_{el}(C) for all C ∈ F. Moreover, U × X_{el}(C) → X(C) and U × \overline{X}_{el}(C) → \overline{X}(C) are open embeddings.

iii) Each stratum X(C) is k-dense with

dim X(C) = dim X − dim C, c_{k}(X(C)) = c_{k}(X), rk_{k} X(C) = rk_{k} X − dim C Q_{k}(X(C)) = Q_{k}(X), X(C)_{el} = X_{el}(C), X(C)_{an} = X_{an}

iv) Let ⟨C⟩ ⊆ N_{k}(X) be the subspace spanned by C. Then

Ξ_{k}(X(C)) = Ξ_{k}(X) ∩ ⟨C⟩^{⊥}, N_{k}(X(C)) = N_{k}(X)/⟨C⟩

and Z_{k}(X(C)) is the image of Z_{k}(X).

Proof. We start with a general remark: because of X(F) = G · X_{el}(F) we have

(7.4) Z = G · (Z ∩ X_{el}(F)) for all G-stable subsets Z ⊆ X(F).

Let X̃(C) be closure of U × \overline{X}_{el}(C) in X(F). We claim that X̃(C) = \overline{X}(C). For this, we consider the case when dim C = 1 first. Let a ∈ C be a generator, let v ∈ Z_{k}(X) be the corresponding k-central valuation, and let (\overline{X}, D) be a model of v. Then it follows from Lemma 6.4 that the birational map \overline{X} [figure]X(F) is regular in D and maps D to U ×X_{el}(C). Because D is G-stable, this implies that also X̃(C) is G-stable. By construction, we have X̃(C) ∩ X_{el}(F) = \overline{X}_{el}(C). Thus (7.4) with Z = X̃(C) yields X̃(C) = \overline{X}(C).

Next let C ∈ F be arbitrary, let R_{1} …, R_{s} ⊆ C be its extremal rays, and put

(7.5) X′(C) := \overline{X}(R_{1}) ∩ … ∩ \overline{X}(R_{s}). which is a G-stable subset of X(F). Then

(7.6) X′(C) ∩ (U × X_{el}(F)) = ⋂_{i=1}^{s} (U × \overline{X}_{el}(R_{i})) = U × \overline{X}_{el}(C) = X̃(C) ∩ (U × \overline{X}_{el}(F)).

This implies that X̃(C) is an irreducible component of X′(C) and therefore G-stable, as well. From (7.4) with Z = X̃(C) we conclude X̃(C) = \overline{X}(C), proving the claim.

We have proved that \overline{X}(C) is closed with

(7.7) \overline{X}(C) ∩ (U × X(F)) = U × \overline{X}_{el}(C).

Next we define the G-stable subvariety

(7.8) X^{0}(C) := \overline{X}(C) \ ⋃_{C′⊈C} \overline{X}(C′)

Then (7.7) implies X^{0}(C) ∩ X(F) = X(C) and therefore X^{0}(C) = X(C) by (7.4). This shows ii) and that X(C) is an open subset of \overline{X}(C). The rest of i) follows easily with the help of (7.1). All assertions in iii) follow from ii) and the corresponding assertions for X_{el}(C). The same holds for all assertions in iv) except for the final one. That follows, e.g., from the statement over K which is proved in [Kno93, Satz 7.4.5]. □

7.4. Remarks. i) By shrinking X_{an} and thereby X(F), we can achieve that any finite number of “generic” conditions hold for all strata. For example: all strata X(C) admit an orbit space X(C)/G; stratum, orbit space and quotient map are smooth; all isotropy groups in a stratum have K-isomorphic Levi complements etc.

ii) In general, X(F) is only defined if X meets the conditions of Theorem 7.1. So, when we speak about X(F) we mean X_{0}(F) where X_{0} = G · X_{el} is an open subset depending on the choice of X_{el}. In this sense, X(F) is only unique up to a birational map which is defined in all strata. Observe that in the most important case, namely when X is homogeneous and k-spherical, X(F) is unambiguously defined embedding of X since then all strata are orbits.

iii) The dimension of the generic G-orbits in X(C) is not so easy to control. An upper bound for the codimension is c_{k}(X). Moreover we show in Section 8 that this bound is attained as soon as C contains an interior point of Z_{k}(X). This is in particular the case when dim C = rk_{k} X.

As usual, for k-spherical varieties we can be more specific:

7.5. Corollary. Let X be a homogeneous k-spherical variety and let X ↪ X(F) be the embedding corresponding to a fan F with supp F ⊆ Z_{k}(X). Then the strata X(C) are precisely the G-orbits. In particular, all G-orbits in X(F) are k-dense.

Proof. The group MA acts transitively on X_{an} and therefore on X_{el}(C). Thus, G acts transitively on X(C) = G · X_{el}(C). □

For local fields, the set X(k) is also stratified:

7.6. Proposition. Assume that k be a local field. Then the sets X(C)(k) with C ∈ F form a stratification of X(F)(k) for the Hausdorff topology.

Proof. Clearly, each set X(C)(k) is locally closed and its closure is contained in the union of all sets X(C′)(k) with C ⊆ C′ ∈ F. It remains to be shown that conversely each such set X(C′)(k) is contained in the closure of X(C)(k). Now Hilbert’s Theorem 90 implies that the fibration X_{el} → X_{an} is Zariski locally trivial by . Thus, the construction of X(F) shows that it is covered by open subsets of the form Y × A(F) where Y is some k-variety.

Toroidal theory shows, that the choice of a 1-parameter subgroup a : G_{m} → A_{k}(X) in the relative interior of C′ yields a surjective morphism

(7.9) π : A(C) → A(C′) : x ↦ lim_{t→0} a(t)x.

Moreover, π is even a homomorphism of split tori whose kernel is connected. Hence it has a section which shows that the induced map A(C)(k) → A(C′)(k) on rational points is surjective as well. This implies the claim. □

Next we study morphisms between toroidal embeddings.

7.7. Proposition. Let X ⊆ X(F) is in Theorem 7.1 and let φ : X → Y be a dominant equivariant morphism. Then φ extends to a rational map X(F) [figure]Y which is defined in all strata if and only all v in the relative interior C^{∘} of any C ∈ F have the same center in Y.

Proof. Suppose Z ⊆ Y is the common center. Let f ∈ O_{Y,Z}. Then v(φ^{∗}(f)) ≥ 0 for all v ∈ C^{0} which implies that φ^{∗}O_{Y,Z} ⊆ O_{X(F),\overline{X}(C)}. Hence φ regular in \overline{X}(C). □

7.8. Corollary. Let X ↪ (F) and Y ↪ Y (F′) be two toroidal embeddings and let φ : X → Y be a dominant equivariant morphism. Assume φ_{∗} : N_{k}(X) → N_{k}(Y) maps F to F′ (i.e., for all C ∈ F there is C′ ∈ F′ with φ_{∗}(C) ⊆ C′). Then φ extends to a rational map X(F) [figure]Y (F′) such that a stratum X(C) is mapped into Y (C′) whenever φ_{∗}(C) ⊆ C′.

We say that the embedding ι_{1} : X ↪ X_{1} dominates the embedding ι_{2} : X ↪ X_{2} if there is a (necessarily unique) morphism φ : X_{1} → X_{2} with ι_{2} = φ ∘ ι_{1}. Clearly, this defines a partial order on the set of all embeddings of X.

For toroidal embeddings this means the following: Let F_{1} and F_{2} be two fans supported in Z_{k}(X). Then F_{1} dominates F_{2} if for every C_{1} ∈ F_{1} there is an C_{2} ∈ F_{2} with C_{1} ⊆ C_{2}. Then X(F_{1}) dominates X(F_{2}) if and only if F_{1} dominates F_{2}. Observe, that for the dominance order there is always a maximal element, namely the trivial fan corresponding to the trivial embedding X ↪ X. On the other side, if X is k-convex there is also a minimal element namely the standard embedding.

Next we show that only toroidal embeddings can dominate a toroidal embedding.

7.9. Proposition. Let X be a homogeneous k-spherical variety. Let X ↪ X′ be a normal equivariant embedding dominating the toroidal embedding X(F). Then X′ = X(F′) for some fan F′ dominating F.

Proof. Let φ : X′ → X(F) be the dominating morphism and put X′_{el} := φ^{−1}(X_{el}(F). Then U × X′_{el} → X′ is an open embedding and X′ = G · X′_{el}. Moreover, since X_{an} is homogeneous, X′_{el} is of the form X_{el}×_{X_{an}} \overline{A} where \overline{A} is an embedding of A_{k}(X) dominating A(F). But then it follows from the theory of torus embeddings that \overline{A} = A(F′) for some fan dominating F. This implies in turn X′ = X(F′). □

Of particular importance is the case when the support of F is all of Z_{k}(X) in which case we call F complete (for X). It is not true in general that X(F) is complete in the algebro-geometric sense but still X(F) has the following completeness property with respect to rational points:

7.10. Theorem. Let F be a complete fan, let X(F) ↪ \overline{X} be a locally linear equivariant embedding, and let Y ⊆ \overline{X} be a k-dense, G-stable subvariety with c_{k}(Y) = c_{k}(X). Then Y ∩ X(F) ≠ ∅.

Proof. Lemma 6.9 asserts that Y is the center of some v ∈ Z_{k}(X). Since, by assumption, λ_{v} ∈ supp F it has a center in X(F) (namely \overline{X}(C) with v in the interior of C). Thus Y ∩ X(F) ≠ ∅. □

For k-spherical varieties this means that the complement of X(F) cannot contain krational points at all. Thus we get that X(F)(k) is maximal in the following sense:

7.11. Corollary. Let X be a k-spherical variety and let X(F) ↪ \overline{X} be a locally linear equivariant embedding. Then X(F)(k) = \overline{X}(k).

For local fields we get a true compactification:

7.12. Corollary. Assume that k is a local field and let X be a k-spherical variety. Then X(F)(k) is compact (with respect to the Hausdorff topology) if and only if F is complete.

Proof. First assume F to be complete. By [Sum75], there is an equivariant embedding of X(F) into a normal complete variety \overline{X}. Then X(F)(k) = \overline{X}(k) is compact. Conversely, suppose F is not complete. Then there is a fan F′ which properly contains F′. Because X(F)(k) is a proper dense (by Proposition 7.6) subset of X(F′) it cannot be compact. □

Since a fan consists of strictly convex cones, there is in general no canonical complete fan unless the valuation cone itself is strictly convex (apart from the trivial case when rk X = 1).

7.13. Definition. A k-dense G-variety is called k-convex if its valuation cone Z_{k}(X) is strictly convex. In that case, we define the standard fan F_{st} as the set of faces of Z_{k}(X) and the standard embedding X_{st} := X(F_{st}).

Observe, that Remark 7.4ii) also applies to the standard embedding, i.e., X_{st} is only an embedding and unique in a birational sense unless X is homogeneous and k-spherical. We will show in Section 8 that the condition of convexity is not very serious.

By construction, X_{st} has a unique closed stratum namely X(C) for C = Z_{k}(X). In particular, if X is k-spherical then X_{st} is a simple embedding meaning that it has a unique closed orbit. This leads to another characterization of the standard embedding.

7.14. Proposition. Let X ↪ X_{st} be the standard embedding of a homogeneous k-convex, k-spherical variety X. Let Z be a complete locally linear G-variety which contains exactly one k-dense orbit of rank zero. Then every G-equivariant k-morphism X → Z extends (uniquely) to a morphism X_{st} → Z.

Proof. Let X′ be the normalization of the closure of the graph of φ : X → Z in X_{st} × Z. Then X′ dominates X_{st} via the projection π to the first factor. Proposition 7.9 implies that X′ = X(F′) where F′ is a fan supported in Z_{k} := Z_{k}(X). The fact that Z is complete implies that π is proper, hence the support of F′ is all of Z_{k}. It remains to show that F′ = F_{st} since then π is an isomorphism and X_{st} maps to Z via the second projection.

Now suppose F′ ≠ F_{st}. Then F′ necessarily contains more than one cone of maximal dimension and therefore also a cone C of codimension 1 separating two of them. Let X_{h} := \overline{X}(C) ⊆ X(F′). Then X_{h} is of rank 1 by Theorem 7.3 iv) and consists of three orbits: the open orbit X(C) and two closed orbits Y_{i} = X(C_{i}), i = 1, 2, corresponding to the two cones C_{i} in F′ having C as a face. The morphism π maps X_{h} to Y := X(Z_{k}) ⊆ X_{st}, the closed stratum of X_{st}. Thereby the orbits Y_{1} and Y_{2} are mapped isomorphically to Y. Moreover, all fibers F_{y} of X_{h} → Y are isomorphic to the projective line P_{k}^{1}.

Now consider the morphism ψ : X_{h} ↪ X(F′) → Z. Then the images of Y_{1} and Y_{2} under ψ are both k-dense orbits of rank zero hence coincide with the unique orbit Ỹ with c_{k}(Ỹ) = 0. According to Corollary 4.4 we can choose a P -stable affine open subset Z^{0} ⊆ Z with Z ∩ Ỹ ≠ ∅. Let X_{h}^{0} ⊆ X_{h} be its preimage. Then Y_{i}^{0} := π(Y_{i} ∩ X_{h}^{0}), i = 1, 2, are non-empty open subsets of Y thus contain its open P -orbit Y^{1}. We conclude that F_{y} ⊆ X_{h}^{0} for all y ∈ Y^{1}. But then ψ(F_{y}) is a complete irreducible subset of the affine set Z^{0}, thus a point. This contradicts the fact that, by construction, the restriction of ψ to any fiber of π is quasifinite. □

Following [BT65, 12.3] we call an irreducible representation V of G (over k) strongly rational if it admits a P -eigenvector. In this case P(V)^{AN} consists of just one point. Its orbit is therefore the only one of rank 0.

7.15. Corollary. Let X = G/H be k-convex and k-spherical. Let, moreover, V be a strongly rational representation of G. Then every equivariant k-morphism G/H → P(V) extends to a morphism X_{st} → P(V).

8. Horospherical varieties

We would like to understand the closed strata of X(F) when F is a complete fan. Since by Theorem 7.3 these strata are G-varieties of k-rank 0 we will study rank-0-varieties in general. Over an algebraically closed field, a homogeneous variety of rank zero is known to be a flag variety. Over arbitrary fields, one gets a wider class of spaces which nevertheless share some properties with flag varieties.

To formulate our result, we need to introduce the process of parabolic induction which is also of independent interest.

8.1. Definition. Let Q = LU ⊆ G be a parabolic subgroup containing P and let Q^{−} = U^{−}L be the opposite parabolic subgroup.

i) Let Y be an L-variety. Consider it as a Q^{−}-variety via the projection Q^{−} ↠ L. Then G ×^{Q^{−}} Y is the parabolical induction of Y to G.

ii) Similarly, the parabolical induction of a k-subgroup H ⊆ L is the subgroup U^{−}H of G. In that case, the homogeneous variety G/U^{−}H is parabolically induced from L/H.

Most properties are preserved under parabolic induction:

8.2. Proposition. Let Q = LU ⊆ G be a parabolic subgroup containing P and Q^{−} = U^{−}L its opposite. Let Y be a k-dense L-variety and let X = G ×^{Q^{−}} Y be its parabolical induction. Then:

i) X is k-dense.

ii) Q_{k}(X) = U · Q_{k}(Y).

iii) X_{el} = Y_{el} and X_{an} = Y_{an}.

iv) c_{k}^{G}(X) = c_{k}^{L}(Y) and rk_{k}^{G}(X) = rk_{k}^{L}(Y).

v) Ξ_{k}(X) = Ξ_{k}(Y), N_{k}(X) = N_{k}(Y), and Z_{k}(X) = Z_{k}(Y).

Proof. The local structure theorem for Y yields an open embedding Q_{k}(Y)_{u} × Y_{el} ↪ Y. Combined with the open embedding U ↪ G/Q^{−} (the big cell) we get the open embeddings

(8.1) U × Q_{k}(Y)_{u} × Y_{el} ↪ U × Y ↪ X.

Then i), ii) and the first part of iii) follow immediately from Proposition 4.5. These imply all other assertions except the last one on valuation cones. Because of K(X)^{U} = K(Y), the restriction of any k-central valuation to K(Y) is k-central as well. This shows Z_{k}(X) ⊆ Z_{k}(Y). Now let v_{0} ∈ Z_{k}(Y) and let (\overline{Y} , D_{0}) be a model of v_{0}. Let D := G ×^{Q−} D_{0} ⊆ \overline{X} := G ×^{Q^{−}} \overline{Y}. Then the restriction of v_{D} to K(Y) is equivalent to v_{0} which shows the opposite inclusion. □

We proceed with the classification of varieties of k-rank 0.

8.3. Theorem. For a k-dense G-variety X the following are equivalent:

i) rk_{k} X = 0

ii) An open dense G-stable subset of X is induced from an anisotropic action.

iii) X = G · X^{AN}.

Proof. i)⇒ii) We apply the Generic Structure Theorem 4.6 to X. Then rk_{k} X = 0 means that A acts trivially on X_{el}, i.e., the action of L on X_{el} is anisotropic. Let Q^{−} = LU^{−} be the opposite parabolic of Q and let by 𝔲^{−} = Lie U^{−}. We claim that X_{el} consists of U^{−}-fixed points. For this it suffices to show that 𝔲^{−}x = 0 where x ∈ X_{el} is any smooth point. Since T_{x}X = 𝔲x ⊕ T_{x}X_{el}, all weights of A on T_{x}X are either zero or positive restricted roots. On the other hand all weights on 𝔲^{−}x are negative roots. Thus 𝔲^{−}x = 0.

From the claim, we infer that X_{el} is a Q^{−}-variety with U^{−} acting trivially. Now consider the (proper) G-morphism

(8.2) φ : \overline{X} := G ×^{Q^{−}} X_{el} → X : [g, x] → gx.

By construction, the restriction of φ to the open subset U × X_{el} is an open embedding, hence φ is birational. Let X_{0} ⊆ \overline{X} be the largest open subset on which φ is an open immersion. Since X_{0} is G-stable, it is induced from Y = X_{0} ∩ X_{el} which proves the theorem.

ii)⇒iii) If G ×^{Q^{−}} Y ⊆ X (open) then A acts trivially on Y, hence Y ⊆ Ỹ := X^{AN^{−}}. This implies that G · Ỹ is dense in X. On the other side, the morphism G ×^{Q^{−}} Ỹ → X is proper. Therefore its image G · Ỹ is also closed, so coincides with X. From the fact that AN^{−} is G(k)-conjugate to AN we get X = G · X^{AN}.

iii)⇒i) By assumption, the morphism X̃ := G×^{P} X^{AN} → X is surjective. Then rk_{k}^{G} X̃ = rk_{k}^{MA} X^{AN} = 0 implies rk_{k} X = 0 (Corollary 6.10). □

8.4. Remark. On can show that a locally linear X has k-rank zero then there even exist an open subset containing X(k) which is induced from an anisotropic action.

For homogeneous varieties we get:

8.5. Corollary. Let H ⊆ G be a k-subgroup. Then the following are equivalent:

i) rk_{k} G/H = 0.

ii) There is a parabolic subgroup Q ⊆ G with Q_{an} ⊆ H ⊆ Q.

iii) H contains a G(k)-conjugate of AN.

iv) Let φ : G/H ↪ \overline{X} be any locally linear G-equivariant open embedding. Then \overline{X}(k) = (G/H)(k).

If k is a local field then these conditions are also equivalent to

v) (G/H)(k) is compact.

vi) G(k)/H(k) is compact.

Proof. i)⇒ii) By Theorem 8.3 there is an isomorphism G/H ≅ G ×^{Q^{−}} Y where Y is an anisotropic homogeneous L-variety. From this we get a projection π : G/H → G/Q^{−}. Since G(k) acts transitively on (G/Q^{−})(k) ([BT65, Thm. 4.13 a)]) we may assume that π(eH) = eQ^{−}, i.e., H ⊆ Q^{−} and Q^{−}/H ≅ Y. Thus Q_{an}^{−} ⊆ H ⊆ Q^{−}.

ii)⇒iii) There is g ∈ G(k) with gP g^{−1} ⊆ Q ([BT65, Thm. 4.13 b)]). Then gANg^{−1} ⊆ Q_{an} ⊆ H.

iii)⇒i) Use iii)⇒i) from Theorem 8.3.

i)⇒iv) Since G/H is k-spherical (by iii)) the assertion follows from Corollary 7.11.

iv)⇒i) Suppose rk_{k} G/H > 0. Since then Z_{k}(G/H) ≠ {0} (by Corollary 5.3) there is a non-trivial k-central valuation. It belongs to a smooth embedding G/H ↪ X = G/H ∪D where D is k-dense. Then X(k) ⊈ (G/H)(k).

The equivalence of iii), v), and vi) is due to Borel-Tits [BT65, Prop. 9.3]. □

Our main application is to toroidal embeddings of k-spherical varieties:

8.6. Corollary. Assume X is a homogeneous k-spherical variety and let X ↪ X(F) be the embedding corresponding to a fan F with supp F = Z_{k}(X). Then all closed orbits are isomorphic to G/H_{0} where H_{0} = M_{0} · Q_{k}(X)_{an}^{−} with X_{an} ≅ M/M_{0}.

Our next goal is to understand varieties with Z_{k}(X) = N_{k}(X). For this, we start with the more general problem of studying

(8.3) Z_{k}^{0} = Z_{k}(X) ∩ (−Z_{k}(X)),

the largest subspace contained in Z_{k}(X). The importance of this spaces lies in the fact that Z_{k}^{0}, when non-zero, prevents X to have a standard embedding as in Definition 7.13.

8.7. Definition. Let X be a k-dense G-variety. An equivariant K-automorphism of K(X) is is called k-central if K(X)^{(AN)} ⊆ K(X)^{(φ)}.

Since every B-semiinvariant is also AN-semiinvariant, a k-central automorphism is also K-central. The latter have been studied in detail in [Kno93, Sec. 8] and [Kno96, Sec. 5]. It was shown that the group of K-central automorphism is the set of K-rational points of a subgroup 𝔄_{K}(X) ⊆ A_{K}(X). More precisely, a central automorphism φ is related to an element a_{φ} ∈ A_{K}(X) by the formula

(8.4) φ(f) = χ_{f}(a_{φ})f for all f ∈ K(X)^{(B)}.

If φ is k-central then it must act trivially on K(X)^{AN}. This means that φ is k-spherical if and only if χ(a_{φ}) = 1 whenever res_{A} χ = 0. From this we see that k-central automorphisms are the points of

(8.5) 𝔄_{k}(X) = 𝔄_{K}(X) ∩ A_{k}(X)

which is a k-subgroup of A_{k}(X).

Central automorphisms are related to central valuations in the following way:

8.8. Theorem. Let X be a k-dense G-variety. Then N (𝔄_{k}(X)) = Z_{k}^{0}(X).

Proof. From [Kno93, Satz 8.2] we get N (𝔄_{K}(X)) = Z_{K}^{0}(X). Now use (5.5) and (8.5). □

8.9. Corollary. A k-dense G-variety X is k-convex if and only if 𝔄_{k}(X) is finite.

From this we derive a criterion for the existence of a standard embedding:

8.10. Corollary. Let X = G/H be homogeneous and assume that Aut^{G}(X) = N_{G}(H)/H does not contain a non-trivial central split torus (e.g., if H is of finite index in its normalizer). Then X is k-convex.

Proof. The connected component 𝔄_{k}(X)^{0} is a split torus sitting in the center of Aut^{G}(X) (see [Kno96, Cor. 5.6]). Thus 𝔄_{k}(X)^{0} = 1. □

For k-spherical varieties one can be more specific:

8.11. Proposition. Let X = G/H be a homogeneous k-spherical variety. Then Aut^{G} X = N_{G}(H)/H is an elementary group whose connected split center equals 𝔄_{k}(X)^{0}. In particular X is k-convex if and only if N_{G}(H)/H is anisotropic.

Proof. The action of an automorphism is uniquely determined by its restriction to X_{el}. Thus Aut^{G} X is a subquotient of MA and therefore elementary. Its largest split subtorus acts automatically by k-central automorphisms. Thus it equals 𝔄_{k}(X)^{0}. The last assertion follows from Theorem 8.8. □

Theorem 8.8 also gives a way to make any G-variety k-convex by taking the quotient by 𝔄_{k}(X). More generally, the following holds:

8.12. Lemma. Every k-dense G-variety X contains an open G × 𝔄_{k}(X)-stable subset X_{0} for which the orbit space X_{0}/𝔄_{k}(X) exists. Moreover, for any k-subgroup 𝔄 ⊆ 𝔄_{k}(X) the quotient X′ := X_{0}/𝔄 has the following properties:

Ξ_{k}(X′) = {χ ∈ Ξ_{k}(X) | res_{𝔄} χ = 0}, A_{k}(X′) = A_{k}(X)/𝔄,
X′_{el} = X_{el}/𝔄, X′_{an} = X_{an}, c_{k}(X′) = c_{k}(X), rk_{k}(X′) = rk_{k} X − dim 𝔄
N_{k}(X′) = N_{k}(X)/N (𝔄), Z_{k}(X′) = Z_{k}(X)/N (𝔄)

Proof. By [Kno96, Cor. 5.4], there is an open G-stable subset of X on which the birational action of 𝔄_{k}(X) is regular. Then the existence of X_{0} in a consequence of Rosenlicht’s theorem.

The first two properties of X′ follow from (8.4). That formula also implies the compatibility of the 𝔄-action on X_{el} with the action induced from that of A_{k}(X). This shows X′_{el} = X_{el}/𝔄 which implies all other assertions except the last one on that valuation cone. That follows from its K-counterpart [Kno93, Satz 8.1 (4.)]. □

8.13. Corollary. Let X be a k-dense G-variety and let 𝔄 ⊆ 𝔄_{k}(X) be an open k-subgroup. Then there is a G × 𝔄-stable open dense subset X_{0} ⊆ X such that X_{0}/𝔄 has a standard embedding.

Now we study another important class of G-varieties which generalize rank-0-varieties:

8.14. Definition. A k-dense G-variety X is called k-horospherical if Z_{k}(X) = N_{k}(X).

One reason for the importance of k-horospherical varieties is the fact that a stratum X(C) of an embedding X(F) is k-horospherical if and only of C contains an inner point of Z_{k}(X). This follows immediately from Theorem 7.3 iv).

Theorem 8.8 implies that k-horospherical varieties are also characterized by the equality 𝔄_{k}(X) = A_{k}(X) which means that such a variety carries an action of A_{k}(X).

8.15. Theorem. For a k-dense G-variety X the following are equivalent:

i) X is k-horospherical.

ii) An open dense G-stable subset of X is induced from an elementary action.

iii) X = G · X^{N}.

Proof. i)⇒ii) Let X, X_{0} and X′ be as in Lemma 8.12 with 𝔄 = A_{k}(X). Since rk_{k} X′ = 0, Theorem 8.3 implies that there is an open subset X′_{1} ⊆ X′ which is induced from an anisotropic action: X′_{1} = G ≅^{Q^{−}} Y′. Let Y ⊆ X_{0} be the preimage of Y′. Because Y → Y′ is an A_{k}(X)-bundle the action on Y is elementary with X_{1} = G ×^{Q^{−}} Y being open in X_{0}.

The proof of the implications ii)⇒iii)⇒i) are the same as that of the analogous statements of Theorem 8.3. □

For homogeneous varieties this translates into:

8.16. Corollary. Let H ⊆ G be a k-subgroup and X = G/H. Then the following are equivalent:

i) X is k-horospherical. ii) There is a parabolic Q ⊆ G with R_{el}Q ⊆ H ⊆ Q.

iii) H contains a G(k)-conjugate of N.

iv) X is k-spherical and H is normalized by a split torus A_{0} with dim A_{0} = rk_{k} X and |A_{0} ∩ H| < ∞.

Proof. The proof of the equivalence of the first three parts analogous to that of Corollary 8.5. Now assume that they hold. Then G/H is k-spherical since G/N is. Moreover, it is well known that the split torus A_{k}(X) ⊆ Aut_{k} X = N_{H}(X)/H can be lifted to an isogenous torus A_{0} ⊆ N_{G}(H). Whence iv). Conversely, when X is k-spherical the action of A_{0} on X is automatically k-central and locally effective. Thus 𝔄_{k}(X) = A_{k}(X) and X is k-horospherical. □

We derive from this a characterization of k-spherical varieties:

8.17. Proposition. For a homogeneous k-dense G-variety X the following are equivalent:

i) X is k-spherical.

ii) The number of k-dense G-orbits in any normal equivariant embedding X ↪ \overline{X} is finite.

Proof. One implication follows from Corollary 4.14. For the converse choose a valuation v in the interior of Z_{k}(X) and let X ↪ \overline{X} = X ∪ D be the associated embedding (with D = X if v = 0). It follows from Theorem 7.3 that D is horospherical with c_{k}(D) = c_{k}(X). Moreover, D contains by assumption only finitely many k-dense orbits. Since D is k-dense itself one of these orbits would be open. Then Corollary 8.16 implies that D is k-spherical. Hence c_{k}(X) = c_{k}(D) = 0. □

9. The Weyl group

Our next goal is to study the valuation cone Z_{k}(X). Our strategy is to compare it with Z_{K}(X) which is known to be the Weyl chamber for a reflection group ([Bri90, Kno93]). But first, we compare the the canonical parabolic subgroups Q_{k}(X) and Q_{K}(X).

9.1. Proposition. Let X be a k-dense G-variety. Then the conjugacy class of Q_{K}(X) is defined over k and Q_{k}(X) = Q_{K}(X)P.

Proof. The first part is clear since X is defined over k. For the second, we apply Corollary 4.6 in two steps. First, we get a Levi subgroup L_{k} ⊆ Q_{k}(X) and an L_{k}-stable slice X_{el} ⊆ X such that Q_{k}(X) ×^{L_{k}} X_{el} → X is an open embedding. Considering X_{el} as a K-variety we get in a second step a parabolic K-subgroup \overline{Q} ⊆ L_{k} with Levi subgroup L_{K} and an L_{K}-stable slice \overline{X}_{el} ⊆ X_{el} such that \overline{Q} ×^{L_{K}} \overline{X}_{el} → X_{el} is an open embedding. Combined we get an open embedding

(9.1) Q̃ ×^{L_{K}} \overline{X}_{el} → X with Q̃ := R_{u}Q_{k} · \overline{Q}.

So uniqueness (Proposition 4.5) implies that Q_{K}(X) = Q̃. Next, let \overline{P} be the image of P in L_{k}. Then \overline{P} contains all anisotropic simple factors of L_{k} (see (3.2)). On the other hand, all isotropic factors act trivially on X_{el} and are thus contained in \overline{Q}. Combined, this implies L_{k} = \overline{Q} · \overline{P} and therefore

(9.2) Q_{k}(X) = R_{u}Q_{k}(X) · \overline{Q} · \overline{P} = Q_{K}(X)P. □

Observe that Q_{k}(X) equals the product Q_{K}(X)P as opposed to just the subgroup generated by it. We translate Proposition 9.1 into a combinatorial statement. Recall the set S^{0} ⊆ S of compact roots. These are, by definition, the simple roots of M. On the other hand, the parabolic Q_{k}(X) is defined over k and therefore corresponds to a subset S_{k}^{(p)}(X) ⊆ S_{k} of k-parabolic roots. Replacing k by K, we also get a set S^{(p)}(X) ⊆ S of K-parabolic simple roots corresponding to Q_{K}(X). These sets are related as follows:

9.2. Corollary. i) S^{(p)}(X) is G^{∗}-stable (see (2.6) for the notation).

ii) S_{k}^{(p)}(X) = res′_{A} S^{(p)}(X) (see (2.2) for the notation).

iii) Let C be a connected component of S^{(p)}(X)∪S^{0} in the Dynkin diagram of S. Then C ⊆ S^{(p)}(X) or C ⊆ S^{0}.

Proof. i) just means that the conjugacy class Q_{K}(X) is defined over k. ii) is then equivalent to Q_{k}(X) being generated by Q_{K}(X) and P. Finally, iii) holds because Q_{k}(X) is even the product of Q_{K}(X) and P. □

As mentioned above, it is known that Z_{K}(X) is the fundamental domain for a finite reflection group W_{K}(X) (see [Kno94, Thm. 7.4]). Following the methods of [Kno94], we show an analogous fact for Z_{k}(X). For this, we need the following geometric result:

9.3. Lemma. Let X be a normal k-dense G-variety. Then there is a Borel subgroup B ⊆ P, a k-subgroup L_{K} ⊆ G, and a point x ∈ X(k) such that:

i) L_{K} is a Levi subgroup of Q_{K}(X) where Q_{K}(X) is the canonical parabolic subgroup of X over K which contains B.

ii) Let T ⊆ L_{K} be a maximal k-torus. Then there is a commutative diagram as follows

(9.3) [figure: A; ⊆; T; ⊆; L_{K}; φ; A_{k}(X); ⊆; A_{K}(X); ∼; T x; ⊆; X]

where φ is the orbit map g ↦ gx. With the induced k-structure of A_{K}(X) all maps are defined over k.

iii) The action of the little Weyl group W_{K}(X) on A_{K}(X) extends to the closure \overline{T x} in X.

iv) Let a : G_{m} → A_{K}(X) be a 1-parameter K-subgroup which we consider as an element of N_{K}(X). Let w ∈ W_{K}(X) with wa ∈ Z_{K}(X) and let v be the K-central valuation corresponding to wa. Then x_{0} := lim_{t→0} a(t)x exists in X if and only if v has a center Y in X and in that case x_{0} ∈ Y.

Proof. First, by an affine cone construction as in the proof of Theorem 4.2 it is easy to see that we may assume X to be quasiaffine. Then X will be non-degenerate in the sense of [Kno94, §3].

We recall some facts from [Kno94]. Choose a Borel subgroup B ⊆ P which determines a canonical K-parabolic Q_{K} = Q_{K}(X) ⊇ B. Let U_{K} := Q_{K,u} be its unipotent radical. Let π : T_{X}^{∗} → X be the cotangent bundle and m : T_{X}^{∗} → 𝔤^{∗} its moment map. For α ∈ T_{X}^{∗} let x_{α} := π(α) ∈ X and L_{α} := C_{G}(m(α)). Then in [Kno94] it was shown that

(9.4) m^{−1}(𝔲_{K}^{⊥}) = {α ∈ T_{X}^{∗} | α(𝔲_{K}x_{α}) = 0}

contains an open Q_{K}-stable subset C such that π : C → X is dominant and for all α ∈ C the pair L_{K} = L_{α}, x = x_{α} has all the properties i)–iv) of the theorem except for:

• L_{K} and T might not be defined over k and therefore

• the left square of diagram (9.3) may not exist.

More precisely, i) holds by [Kno94, Thm. 2.3 and §3]. For ii) see [Kno94, §4]. Assertion iii) follows from [Kno94, Cor. 6.3] and iv) from [Kno94, Thm. 7.3].

The two remaining properties above will also hold if α would would be a k-rational point of T_{X}^{∗}. We claim that such α is indeed possible to find after possibly changing B by a conjugate in P.

Do see this, we first we apply Corollary 4.6 to X. This way, we get a decomposition Q_{k} := Q_{k}(X) = L_{k}U_{k} and an L_{k}-subvariety X_{el} ⊆ X such that U_{k} × X_{el} → X is an open embedding. Next we apply Corollary 4.6 to the L_{k}-variety X_{el} but now considered as a variety over K. We get a decomposition Q̃_{K} := Q_{K}(X_{el}) = L̃_{K} Ũ_{K} ⊆ L_{k} and an L̃_{K}-subvariety X̃ ⊆ X_{el} such that Ũ_{K} × X̃ → X_{el} is an open embedding. Since then also U_{k} × Ũ_{K} × X̃ → X is an open embedding, uniqueness (Proposition 4.5) implies that Q_{K} = Q̃_{K}U_{k} and U_{K} = U_{k}Ũ_{K}.

Now consider the restriction map

(9.5) π^{−1}(X_{el}) = T_{X}^{∗}|_{X_{el}} → T_{X_{el}}^{∗}.

It has an L_{k}-equivariant section by extending α ∈ T_{X_{el}}^{∗} by 0 on 𝔲_{k}x_{α}. In other words, we get a k-isomorphism

(9.6) ι : {α ∈ T_{X}^{∗} | x_{α} ∈ X_{el}, α(𝔲_{k}x_{α}) = 0} → T_{X_{el}}^{∗}

Now let

(9.7) C_{el} := C ∩ π^{−1}(X_{el}) ⊆ {α ∈ T_{X}^{∗} | x_{α} ∈ X_{el}, α(𝔲_{K}x_{α}) = 0}.

Then C_{el} → X_{el} is dominant since C → X is dominant and U_{k} ·X_{el} is dense in X. Because of 𝔲_{K} = 𝔲_{k} ⊕ 𝔲̃_{K}, the morphism ι maps the set C maps isomorphically to an open subset of

(9.8) {α ∈ T_{X_{el}}^{∗} | α(𝔲̃_{K}x_{α}) = 0}

or, more precisely, to the irreducible component C̃ which maps dominantly to X_{el}. Now we can apply [Kno94, Theorem 3.2] to the L_{k}-variety X_{el} and conclude that that the morphism

(9.9) L_{k} × C̃ → T_{X_{el}}^{∗} :

is dominant. Because of L_{k} = (P ∩ L_{k}) · (L_{k})_{el} (see (3.2)) and because (L_{k})_{el} acts trivially on X_{el} we see that also

(9.10) (P ∩ L_{k}) × C_{el} → T_{X_{el}}^{∗} : (g, α) ↦ gι(α)

is dominant. Now recall that X, and therefore X_{el} and T_{X_{el}}^{∗} are k-dense. Thus, we can find g ∈ P ∩ L_{k} and α ∈ C_{el} such that gα ∈ T_{X_{el}}^{∗}(k). By replacing B with gBg^{−1} we may assume that g = 1. Then α is a k-rational point in C_{el}, hence in C. Observe that because of g ∈ P ∩ L_{k} none of P, Q_{k}, and L_{k} change.

Now both L_{K} := L_{α} and x := x_{α} are defined over k. Then we can choose a maximal torus T ⊆ P ∩ L_{K} which automatically contains the center of L_{k} and therefore A. From this and the fact that x ∈ X_{el} we get the left square in ii). □

9.4. Remark. An A-orbit Ax as above is called a k-flat. Accordingly the T -orbit T x is a K-flat. In general, there are multiparameter families of flats which are not conjugate to each other. Observe that the set of flats which have the last two properties iii) and iv) depends on the global structure of X (and not just its generic structure). For example, one could make any specific flat “bad” by removing the orbit Gx_{0} from X (unless Y has an open orbit) or blowing it up. This problem does not occur for spherical varieties.

Now we are in the position to show that also Z_{k}(X) is the Weyl chamber of a reflection group. For this we define

(9.11) N(N_{k}) := {w ∈ W_{K}(X) | wN_{k}(X) = N_{k}(X)}

to be the normalizer of N_{k}(X) in W_{K}(X),

(9.12) C(N_{k}) := {w ∈ W_{K}(X) | w|_{N_{k}(X)} = id_{N_{k}(X)}},

its centralizer, and

(9.13) W_{k}(X) := N(N_{k})/C(N_{k}),

the restricted Weyl group.

9.5. Theorem. Let X be a k-dense G-variety. Then Z_{k}(X) is a fundamental domain for the action of W_{k}(X) on N_{k}(X).

Proof. First of all, because of Z_{k} ⊆ Z_{K} no two different elements of Z_{k}(X) are W_{k}(X)- conjugate because the same is true for Z_{K}(X) and W_{K}(X). It remains to show, that every element of N_{k}(X) is W_{k}(X)-conjugate to an element of Z_{k}(X).

We claim that it suffices to show that every v ∈ N_{k}(X) is W_{K}(X)-conjugate to an element of Z_{k}(X). To see this, define

(9.14) N_{k}^{∘} := N_{k}(X) \ ⋃_{w∈W_{K}(X)\N(N_{k})} wN_{k}(X).

This is an open Zariski dense subset of N_{k}(X) with the property that w ∈ W_{K}(X), v ∈ N_{k}^{∘}, and wv ∈ N_{k}(X) imply w ∈ N(N_{k}). Thus, if for every v ∈ N_{k}^{∘} there is w ∈ W_{K}(X) with wv ∈ N_{k}(X) then automatically w ∈ N(N_{k}). The claim follows by continuity.

So let v ∈ N_{k}^{∘}, let v_{0} ∈ Z_{K} be the unique element in the W_{K}(X)-orbit of v, and let (\overline{X}, D) be a model of v_{0} (not necessarily defined over k). Let further a : G_{m} → A_{k} be a cocharacter in the ray spanned by v. Then according to Lemma 9.3, the limit x_{0} = lim_{t→0} a(t)x exists and lies in D. In particular, D contains the k-rational point x_{0}. We claim that that this implies that D is in fact k-dense. For this let Z be its Zariski closure of D(k) in D and put \overline{X}′ := \overline{X} \ Z and D′ := D \ Z. If Z ≠ D then (\overline{X}′, D′) would be still a model for v_{0} but this time with D′(k) = ∅. But this is impossible by the argument above applied to \overline{X}′ instead of \overline{X}.

Thus v_{0} is k-dense and K-central, hence v_{0} ∈ Z_{k} by Theorem 6.6. □

Recall that a facet of a convex cone is a face of codimension one.

9.6. Corollary. The group W_{k}(X) is generated by reflections about the facets of Z_{k}(X).

Proof. Indeed, only reflection groups have closed fundamental domains. □

Fundamental domains of a finite reflection group are very special:

9.7. Corollary. The valuation cone Z_{k}(X) is cosimplicial, i.e., it is defined by a set of linearly independent linear inequalities.

9.8. Corollary. Let X(F) be the embedding corresponding to a fan F. Then there is an M-stable open dense subset X_{an}^{0} ⊆ X_{an} such that for all y ∈ X_{an}^{0}(k) the following holds: let F (y) ⊆ X_{el} be the A_{k}(X)-orbit over y and 𝔮(y) its closure in X. Then 𝔮(y) is isomorphic to the toroidal A_{k}(X)-embedding which corresponds to the fan

(9.15) W_{k}F := {wC | w ∈ W_{k}(X), C ∈ F}.

In particular, \overline{A} carries an action of W_{k}(X) and is complete if and only if F is complete.

Proof. The same proof as that for [Kno94, Cor. 6.2] shows that there is an open subset X_{an}^{0} such that for all y ∈ X_{an}^{0}(k) the embeddings 𝔮(y) are isomorphic to each other. This common A_{k}(X)-embedding corresponds to a fan F′ supported in N_{k}(X). Lemma 9.3iv) implies that F′ is W_{k}(X)-stable. The construction of X(F) shows that A_{k}(F) is an open subset of A_{k}(F′) which means that F is a subfan of F′, wence W_{k}F ⊆ F. Equality follows from the fact that Z_{k}(X) is a fundamental domain for W_{k}(X). □

Another consequence of Lemma 9.3 is:

9.9. Proposition. Let X be a k-convex, k-spherical, and quasiaffine G-variety. Then X is affine and homogeneous.

Proof. Let X_{0} ⊆ X be the open G-orbit and let X ↪ \overline{X} be an equivariant affine embedding. Suppose X_{0} ≠ \overline{X}. Then Kempf’s Theorem 3.6 implies that \overline{X} \ X_{0} contains a k-rational point y. Lemma 6.9 shows that Y := \overline{Gy} is the center of a k-central valuation v ∈ Z_{k}(X). Now let A_{k}x ⊆ A_{K}x ⊆ X_{0} be a flat as in Lemma 9.3 and let \overline{A} be the closure of A_{k}x in \overline{X}. Then \overline{A} meets Y by part iv) of Lemma 9.3. Therefore, the normalization of \overline{A} is a non-trivial affine torus embedding and therefore which given by a non-zero strictly convex cone C ⊆ N_{k}. Moreover, C is W_{k}-stable by part iii). Such a cone can only exist when the fixed point set N_{k}^{W_{k}} is non-zero. But that set equals Z_{k}^{0} which is 0, by assumption. This contradiction shows that X = X_{0} = \overline{X} is homogeneous and affine. □

Here is an application:

9.10. Corollary. Let X = G/H be a k-spherical G-variety. Assume that N_{G}(H)/H is anisotropic (e.g. finite) and that H is unimodular. Then H is reductive.

Proof. The condition on N_{G}(H)/H means that X is k-convex (Proposition 8.11). Then also G/H^{0} is k-convex (Corollary 6.10) and k-spherical. Thus, we may assume without loss of generality that H is connected.

Now consider 𝔥 = Lie H and let τ be a generator of the line ∧^{d}𝔥 (with d = dim 𝔥). Let N_{τ} be its isotropy group in N_{G}(H). Then H being unimodular means that H ⊆ N_{τ}. Since, by assumption, N_{G}(H)/H is anisotropic also its subgroup N_{τ}/H is anisotropic.

Therefore it acts (locally) trivially on the fibers of X_{el} → X_{an} which entails rk_{k} G/H = rk_{k} X_{el} = rk_{k} X_{el}/N_{τ} = rk_{k} G/N_{τ}. From Corollary 6.10 we get Z_{k}(G/H) = Z_{k}(G/N_{τ}) and, in particular, that G/N_{τ} is k-convex, as well.

We now consider τ as a point of ∧^{d}𝔤. The orbit map then yields an embedding

(9.16) ϱ : G/N_{τ} ↪ ∧^{d}𝔤 : gN_{τ} ↦ gτ

which means that G/N_{τ} is quasiaffine. Since it is also k-spherical and k-convex we get from Proposition 9.9 that G/N_{τ} is affine which implies that N_{τ} is reductive. But then its normal subgroup H is reductive, as well. □

10. The root system

In this section, X will be a fixed k-dense G-variety. Accordingly, if there is no danger of confusion we drop the reference to X in most notation:

(10.1) S^{(p)}, S_{k}^{(p)}, Ξ_{k}, A_{k}, N_{k}, Z_{k}, W_{k}

where k can also be replaced by K.

After assigning a k-Weyl group to X we now construct an integral root system for it. The simple roots should vanish on the facets of the Weyl chamber Z_{k} and are therefore unique up to a positive scalar. So, we face an issue of normalization. The most obvious is:

10.1. Definition. Let X be k-dense G-variety.

i) A primitive k-spherical root of X is a primitive element σ ∈ Ξ_{k} such that Z_{k} ∩ {σ ≥ 0} is a facet of Z_{k}. In other words, σ is the unique shortest element of Ξ_{k} which is an outward normal vector to a facet of Z_{k}.

ii) The set of primitive k-spherical roots is denoted by Σ_{k}^{pr} := Σ_{k}^{pr}(X). It is in 1:1correspondence with the facets of Z_{k}.

iii) The primitive k-root system of X is

(10.2) Φ_{k}^{pr} = Φ_{k}^{pr}(X) := W_{k} · Σ_{k}^{pr} ⊆ Ξ_{k}.

Corollary 9.7 implies that Σ_{k}^{pr} is linearly independent and that the valuation cone has the presentation

(10.3) Z_{k} = {a ∈ N_{k} | σ(a) ≤ 0 for all σ ∈ Σ_{k}^{pr}}.

Recall the set S_{k} ∈ Ξ(A) of restricted simple roots of G. Then Proposition 5.4 has the following reformulation:

10.2. Lemma. Every σ ∈ Σ_{k}^{pr} is a linear combination σ = ∑_{α∈S_{k}} c_{α}α with c_{α} ∈ ℚ_{≥0}.

The set of α ∈ S_{k} with c_{α} > 0 is called the support of σ (denoted supp σ).

A weight lattice for a root system Φ with Weyl group W is a lattice Ξ containing Φ with (1 − s_{σ})Ξ ⊆ ℤσ for all σ ∈ Φ. Here, s_{σ} denotes the reflection about σ. If Φ is a reduced root system then this is equivalent to W acting trivially on Ξ/⟨Φ⟩_{ℤ}. This follows easily from ℚσ ∩ ⟨Φ⟩_{ℤ} = ℤσ. Yet another criterion is the existence of a (unique) coroot σ^{∨} ∈ Hom(Ξ, ℤ) with

(10.4) s_{σ}(χ) = χ − ⟨χ|σ^{∨}⟩σ for all χ ∈ Ξ.

We record:

10.3. Proposition. Φ_{k}^{pr} is a reduced root system with Weyl group W_{k}. The set Σ_{k}^{pr} is a system of simple roots for Φ_{k}^{pr}. The valuation cone Z_{k} is the antidominant Weyl chamber with respect to Σ_{k}^{pr}. The lattice Ξ_{k} is a weight lattice for Φ_{k}^{pr}.

Proof. Everything follows from the definitions and the fact that Z_{k} is a fundamental domain for W_{k} once we know that Ξ_{k} is W_{k}-stable. Over K this is [Kno94, Thm. 4.2]. For k it now follows from (4.24). □

Of course, all constructions above are valid with k replaced by K. In this case, several other normalizations of spherical root are in use (see, for example, the account in [VS13]) but the primitive one is most common. Therefore, over K we are going to drop the superscript ^{pr} and write Σ_{K} = Σ_{K}(X) := Σ_{K}^{pr} and Φ_{K} = Φ_{K}(X) := Φ_{K}^{pr}. In the following, we will focus on the relationship between Σ_{K} and Σ_{k}^{pr}.

For any σ ∈ Ξ_{K} let \overline{σ} := res_{A} σ be the restriction to A and put

(10.5) Ξ_{K}^{0} = Ξ_{K}^{0}(X) = \{σ ∈ Ξ_{K}(X) | \overline{σ} = 0\}.

Elements of

(10.6) Σ_{K}^{0} = Σ_{K}^{0}(X) := Σ_{K} ∩ Ξ_{K}^{0} = \{σ ∈ Σ_{K} | \overline{σ} = 0\}.

will be called (k-)compact spherical roots. They comprise the set From Lemma 10.2 (over K) it follows that the compact spherical roots can be recovered from the compact simple roots:

(10.7) Σ_{K}^{0} = {σ ∈ Σ_{K} | supp(σ) ⊆ S^{0}}.

We will see later (Corollary 10.16) that Σ_{K}^{0} is the set of K-spherical roots of X_{el}. Let

(10.8) W_{K}^{0} = W_{K}^{0}(X) = ⟨s_{σ} | σ ∈ Σ_{K}^{0}⟩ ⊆ W_{K}

be the subgroup of W_{K} which is generated by all reflections about compact spherical roots. Using notation (2.2), we define the set of restricted spherical roots as

(10.9) Σ_{k} = Σ_{k}(X) := res′_{A} Σ_{K} = \{\overline{σ} | σ ∈ Σ_{K}, \overline{σ} ≠ 0\}.

Our goal is to show that Σ_{k} is another system of simple roots for W_{k}.

10.4. Remark. In general, Σ_{k} and Σ_{k}^{pr} are different. This already happens in the group case X = H × H/H with H = U(1, 1) and k = ℝ. In that case, the restricted simple root \overline{σ} is not primitive. Thus Σ_{k} = \{±\overline{σ}\} while Σ_{k}^{pr} = \{±\frac{1}{2} \overline{σ}\}.

Next we study the action of the Galois group G = Gal(K|k). Recall the situation of Lemma 9.3, in particular the maximal k-torus T containing the maximal k-split torus A and a point x ∈ X(k) such that the orbit T x is a flat. Then the following objects will be considered:

(10.10) [figure: Σ_{K}; ⊆; Ξ_{K}; ⊇; Ξ_{k}; ⊇; S; ⊆; Ξ(T); Ξ(A)]

The Galois group G acts on Ξ(T) and trivially on Ξ(A) . Since T x ≅ A_{K} is defined over k, the sublattices Ξ_{K} and Ξ_{k} are G-stable.

Now recall the ∗-action of G on Ξ(T). It is defined as

(10.11) γ ∗ χ = w_{γ} ^{γ}χ

where w_{γ} is the unique element of W (G) with w_{γ} ^{γ}S = S. It is known that w_{γ} ∈ W^{0} = ⟨s_{α} | α ∈ S^{0}⟩ (see §2). This implies that the restriction res_{A} is also G^{∗}-invariant.

The subgroup Ξ_{K} ⊆ Ξ(T) is G^{∗}-stable, as well. In fact, if f is a B-semiinvariant function then ^{n_{γ}γ}f is also B-semiinvariant with character γ ∗ χ_{f}. Here, n_{γ} ∈ N_{G}(T) is a lift of w_{γ}.

We define a G-action on valuations by ^{γ}v(f) := v(^{γ^{−1}}f). Then it is easy to see that the injection ι_{K} : Z_{K} ↪ N_{K} is G^{∗}-equivariant. Indeed, because of the G-invariance of v we have

(10.12) ι_{K}(^{γ}v)(χ) = ^{γ}v(f_{χ}) = v(^{γ^{−1}}f_{χ}) = v(^{n_{γ^{−1}}γ^{−1}}f_{χ}) = v(f_{γ^{−1}∗χ}) = (γ ∗ ι_{K}(v))(χ)

This shows that Σ_{K} is a G^{∗}-stable subset of Ξ_{K}. In particular, Φ_{K} and W_{K} are G^{∗}-stable.

10.5. Lemma. Let γ ∈ G. Then w_{γ}Ξ_{K} = Ξ_{K} and restriction of w_{γ} to Ξ_{K} is an element of W_{K}^{0}.

Proof. The first assertion follows from the fact that Ξ_{K} is both γ- and γ∗-stable.

Next we show that there is an element \overline{w}_{γ} ∈ W_{K} with w_{γ}v = \overline{w}_{γ}v for all v ∈ N_{K}. For this, choose v ∈ Z_{K} with model (\overline{X}, D). Without loss of generality we may assume that \overline{X} does not contain any other G-stable divisor D′ ≠ D with v_{D′} ∈ Z_{K}. Consider the closure \overline{T x} of the flat T x in \overline{X}. Then Lemma 9.3 implies that \overline{T x} equals the toroidal embedding A_{K}(F) where F is the fan consisting of {0} and all rays ℚ_{≥0}wv with w ∈ W_{K}. Now we twist \overline{X} by an element γ ∈ G. Then the embedding X ↪ ^{γ}\overline{X} = X ∪^{γ}D corresponds to the valuation γ ∗ v ∈ Z_{K}. Using the fact that T x is G-stable and applying Lemma 9.3 to γX we get

(10.13) γ ⋃_{w} ℚ_{≥0}wv = ⋃_{w} ℚ_{≥0}w(γ ∗ v)

Since G preserves the primitive generators of the rays we see that the orbit of v is mapped to the orbit of γ ∗ v:

(10.14) γ(W_{K}v) = W_{K}(γ ∗ v)

Every v ∈ N_{K} is W_{K}-conjugate to an element of Z_{K}, hence (10.14) holds for all v ∈ N_{K}. Now using the fact that W_{K} is normalized by G^{∗} we obtain

(10.15) w_{γ}W_{K}v = (w_{γ}γ)(γ^{−1}W_{K}v) = γ ∗ W_{K}(γ^{−1} ∗ v) = W_{K}v

Thus for every v there is \overline{w}_{γ} ∈ W_{K} with w_{γ}v = \overline{w}_{γ}v. The element \overline{w}_{γ} might depend on v. It is unique for v in the interior Z^{∘} of Z_{K} and there it depends continuously on v. Thus \overline{w}_{γ} is constant for v ∈ Z^{∘}, hence constant everywhere by linearity.

Because of ^{γ}σ = \overline{w}_{γ}^{−1}γ ∗ σ we see that Φ_{K} is G-stable and that \overline{w}_{γ} is the unique element of W_{K} with

(10.16) \overline{w}_{γ} ^{γ}Σ_{K} = Σ_{K}

We claim that \overline{w}_{γ} ∈ W_{K}^{0}. Let Φ^{0} = W_{K}^{0}Σ_{K}^{0} be the subroot system generated by Σ_{K}^{0}. It follows from Lemma 10.2 that Φ^{0} = {σ ∈ Φ_{K} | res_{A} σ = 0}. Since res_{A} is G-invariant this implies that Φ_{K}^{0} is G-stable. Thus, for every γ ∈ G there is a unique element w_{γ}^{0} ∈ W_{K}^{0} with

(10.17) w_{γ}^{0} ^{γ}Σ_{K}^{0} = Σ_{K}^{0}.

For any non-compact σ ∈ Σ_{K} let σ̃ := w_{γ}^{0} ^{γ}σ. Since res_{A} is G- and W_{K}^{0}-invariant we get res_{A} σ̃ = res_{A} σ. This and Z_{k} ⊆ Z_{K} imply that the spherical root σ̃ has at least one negative value on Z_{K} and is therefore a positive root. Combined, this shows that w_{γ}^{0} ^{γ}Σ_{K} consists entirely of positive roots and is therefore equal to Σ_{K}. Uniqueness implies \overline{w}_{γ} = w_{γ}^{0} ∈ W_{K}^{0}. □

The following was proved in the course of the proof:

10.6. Corollary. The root system Φ_{K} is a G-stable subset of Ξ_{K} and W_{K} is normalized by G.

We also get that the ordinary and the ∗-action of G are the same modulo compact roots:

10.7. Corollary. Let γ ∈ G and χ ∈ Ξ_{K}. Then ^{γ}χ − γ ∗ χ ∈ ⟨Σ_{K}^{0}⟩_{ℤ}.

Proof. Because of \overline{w}_{γ} ∈ W_{K}^{0} we have

(10.18) ^{γ}χ − γ ∗ χ = (1 − \overline{w}_{γ})(^{γ}χ) ∈ ⟨Σ_{K}^{0}⟩_{ℤ} □

As a consequence we get that N_{k} can be computed from Σ_{K}^{0} and the G^{∗}-action on Ξ_{K}:

10.8. Corollary. The subspace N_{k} of N_{K} is defined by the equations

(10.19) {σ = 0 | σ ∈ Σ_{K}^{0}} ∪ {χ − γ ∗ χ = 0 | χ ∈ Ξ_{K}, γ ∈ G}.

Proof. Because of N_{k} = N_{K}^{G}, the space N_{k} is defined by the equations χ −^{γ}χ = 0 with χ ∈ Ξ_{K}, γ ∈ G. Moreover, all equations σ = 0 for σ ∈ Σ_{K}^{0} hold on N_{k} by definition. Modulo those equations, we have

(10.20) χ −^{γ}χ = χ − γ ∗ χ

proving the assertion. □

Observe that the weight lattice Ξ_{k} can also be computed since it is the image of Ξ_{K} in N_{k}^{∗}. The equations for N_{k} can be made a bit more precise. We start with unique W_{k}-stable decomposition N_{k} = N_{k}^{0} ⊕ N_{k}^{1} with N_{k}^{0} := N_{k}^{W_{k}} = Z_{k}^{0} and such that Σ_{k} forms a basis of (N_{k}^{1})^{∗}. Then

(10.21) N_{k}^{0} = (N_{K}^{0})^{G} = (N_{K}^{0})^{G^{∗}}.

Moreover, it suffices to let χ in (10.19) run through a set of generators. Thus, the subspace N_{k}^{1} ⊆ N_{K}^{1} is defined by the equations σ = 0 for all compact spherical roots and σ − τ = 0 for all pairs of non-compact spherical roots which are in the same G^{∗}-orbit. This shows:

10.9. Corollary. The set Σ_{k} ⊆ Ξ_{k} is linearly independent. Moreover, \overline{σ} = \overline{τ} ≠ 0 for σ, τ ∈ Σ_{K} if and only if σ and τ are in the same G^{∗}-orbit.

From this, we obtain:

10.10. Corollary. There is a map Σ_{k}^{pr} → ℤ_{>0} : σ ↦ n_{σ} such that Σ_{k} = {n_{σ}σ | σ ∈ Σ_{k}^{pr}}.

Proof. It follows from (5.5) that Z_{k} is also defined by the inequalities \overline{σ} ≤ 0 with σ ∈ Σ_{K}. Since they are linearly independent, they form a minimal set of inequalities. Consequently, every \overline{σ} is an integral multiple of an element of Σ_{k}^{pr}. □

Since elements of Σ_{K} and Σ_{k} correspond to facets of Z_{K} and Z_{k}, respectively, we obtain:

10.11. Corollary. Let F ⊆ Z_{K} be a facet and \overline{F} := F ∩ Z_{k}. Then either \overline{F} = Z_{k} or \overline{F} is a facet of Z_{k}.

A formula for the fundamental coweights due to Borel-Tits [BT65, Thm. 6.13] also generalizes almost verbatim. Let ω_{σ}^{∨} ∈ N_{k}^{1} be the dual basis of Σ_{k} such that

(10.22) Z_{k} = N_{k}^{0} + ∑_{σ∈Σ_{k}} ℚ_{≤0} ω_{σ}^{∨}.

10.12. Corollary. For any non-compact σ ∈ Σ_{K} the following formula holds:

(10.23) ω∨ ; \overline{σ} = ∑_{τ∈G∗σ} ω_{τ}^{∨}

Now we proceed with the construction of a root system. For this, we define

(10.24) Φ_{k} := W_{k}Σ_{k} = W_{k} res′_{A} Σ_{K}.

generalizing (10.2). One can also consider the restriction of the K-root system

(10.25) Φ_{k}^{res} := res′_{A} Φ_{K} = res′_{A} W_{K}Σ_{K}

but this is, in general, not a root system:

10.13. Example. Let k = ℝ and G = Sp(4, 2). Then rk_{ℝ} G = 1 with S^{0} = {α_{1}, α_{3}}. Let H ⊆ G be a real form of item 4′ of [Was96, Table C]. More precisely, let N′ ≅ G_{a}^{3} be the commutator subgroup of N. Then H := MAN′ ⊆ G. Put X = G/H. From Wasserman’s table we get

(10.26) Σ_{ℂ} = {σ_{1} := α_{1} + α_{3}, σ_{2} := α_{2}}.

Thus, Σ_{ℂ} generates a root system of type G_{2} with the long root σ_{1} being compact. In particular σ_{1} + nσ_{2} ∈ Φ_{ℂ} for n = 1, 2, 3. Therefore Σ_{ℝ} = \{\overline{σ}_{2}\} but

(10.27) Φ_{ℝ}^{res} = \{±\overline{σ}_{2}, ±2\overline{σ}_{2}, ±3\overline{σ}_{2}\}

which is not a root system.

Nevertheless we have:

10.14. Theorem. Let X be a k-dense G-variety. Then

i) (Φ_{k}, Ξ_{k}) is an integral root system. Its Weyl group is W_{k} and Σ_{k} is a system of simple roots.

ii) n_{σ} ∈ {1, 2} for all σ ∈ Σ_{k}^{pr} (see Corollary 10.10).

iii) Φ_{k} consists precisely of the indivisible elements of Φ_{k}^{res}, i.e., those σ ∈ Φ_{k}^{res} such that \frac{1}{n}σ ∉ Φ_{k} for any n ∈ ℤ_{≥2}.

Proof. i) Let R_{K} := ⟨Σ_{K}⟩ and R_{k} := res_{A} R_{K} = ⟨Σ_{K}⟩ be the root lattices. Since R_{K} is W_{K}-stable and therefore N(N_{k})-stable, we see that R_{k} is W_{k}-stable. Since the elements of Σ_{k} are primitive in R_{k} by Corollary 10.9, we conclude that Φ_{k} is a root system for W_{k} and that Σ_{k} is a set of simple roots.

It remains to be shown that Ξ_{k} is a set of integral weights for Φ_{k}. Since Ξ_{k} is W_{k}-stable this means that W_{k} acts trivially on Ξ_{k}/R_{k}. Clearly, that holds over K. The assertion now follows from the fact that Ξ_{k}/R_{k} is an quotient of Ξ_{K}/R_{K}. ii) Let σ ∈ Σ_{k}^{pr} with σ̃ = n_{σ}σ ∈ Σ_{k}. Since σ ∈ Ξ_{k} it follows from i) that

(10.28) \frac{2}{n_{σ}} = ⟨\frac{1}{n_{σ}} σ̃, σ̃^{∨}⟩ = ⟨σ, σ̃^{∨}⟩ ∈ ℤ.

iii) Clearly Φ_{k} ⊆ Φ_{k}^{res} ⊆ res_{A} R_{K} = R_{k}. Since all elements of Φ_{k} are primitive in R_{k} they are all indivisible in Φ_{k}^{res}. Conversely, let σ ∈ Φ_{K} such that \overline{σ} = res_{A} σ ∈ Φ_{k}^{res} is indivisible. Let H = \{\overline{σ} = 0\} ⊆ N_{k} be the hyperplane defined by \overline{σ}, choose a ∈ H off any other reflection hyperplane, and an element \overline{w} ∈ W_{k} with \overline{w}a ∈ Z_{k}. Then F := \overline{w}H ∩ Z_{k} is of codimension 1 in Z_{k}. Now lift \overline{w} to an element w ∈ N(N_{k}). Then F is the intersection of Z_{k} with Z_{K} ∩ {wσ = 0}. The latter is a face of Z_{K}, hence F is a facet of Z_{k}. Let \overline{τ} ∈ Σ_{k} correspond to F. Then \overline{w}\overline{σ} is a multiple and therefore equal to \overline{τ}. We conclude \overline{σ} = \overline{w}^{−1}\overline{τ} ∈ Φ_{k}. □

Summarizing, to any K-variety X with G-action one can associate the following combinatorial objects:

(10.29) S, Ξ(T), Ξ_{K}, Σ_{K}, S^{(p)}

If X is a k-dense G-variety one additionally has a subset S^{0} ⊆ S of compact roots and a G^{∗}-action on all objects. By restriction to a maximal split k-torus (computable from the data above by Corollary 10.8) one gets the restricted objects

(10.30) S_{k}, Ξ(A), Ξ_{k}, Σ_{k}, S_{k}^{(p)}.

We compute the spherical roots in a special case:

10.15. Proposition. Let X be a k-dense G-variety, let F be a fan supported in Z_{k}(X). For any C ∈ F consider the stratum Y := X(C) of X(F). Then

Σ_{k}(Y) = Σ_{k}(X) ∩ ⟨C⟩^{⊥},
Σ_{K}(Y) = Σ_{K}(X) ∩ ⟨C⟩^{⊥} = {σ ∈ Σ_{K}(X) | res_{A} σ ∈ Σ_{k}(Y) ∪ {0}}.

Proof. Theorem 7.3 iv) implies Σ_{k}^{pr}(Y) = Σ_{k}^{pr}(X)∩⟨C⟩^{⊥}. For k = K this means Σ_{K}(Y) = Σ_{K}(X) ∩ ⟨C⟩^{⊥}. From this we deduce Σ_{k}(Y) = Σ_{k}(X) ∩ ⟨C⟩^{⊥} by restriction to A. Finally let σ ∈ Σ_{K}(X) and put \overline{σ} := res_{A} σ. Then σ ∈ Σ_{K}(Y) if and only if σ ∈ ⟨C⟩^{⊥} if and only if \overline{σ} ∈ ⟨C⟩^{⊥} (since C ⊆ N (A)) if and only if \overline{σ} ∈ Σ_{k}(X) ∪ \{0\}. □

From this we get:

10.16. Corollary. Let X be a k-dense G-variety. Then

Ξ_{K}(X_{an}) = Ξ_{K}^{0}(X), Σ_{K}(X_{an}) = Σ_{K}^{0}(X), S^{(p)}(X_{an}) = S^{(p)}(X)
Ξ_{K}(X_{el}) = Ξ_{K}(X), Σ_{K}(X_{el}) = Σ_{K}^{0}(X), S^{(p)}(X_{el}) = S^{(p)}(X)

Proof. Let C ⊆ Z_{k}(X) be a strictly convex cone of maximal dimension. Then X(C) is induced from X_{an} and the data for X_{an} follow from Theorem 7.3 and Proposition 10.15. The first equation for X_{el} follows from the k-LST, the two others from the anisotropic case. □

10.17. Remark. Since all orbits in X_{an} are affine varieties there are strong constraints on the pair (Σ_{K}^{0}, S^{(p)}). For example, if X is absolutely spherical then X_{an} is an affine homogeneous K-spherical variety. These have been classified by [Krä79], [Mik86], and [Bri87]. See [BP15] for the calculation of the corresponding spherical roots.

The pair (S_{k}, Σ_{k}) looks very much like the spherical system of a complex spherical variety. This indeed very often the case but not always.

10.18. Examples. i) Let k = ℝ and G = SU(n, n) with n ≥ 2. Then the restricted root system of G is of type C_{n} with simple roots β_{i} = \overline{α}_{i} = \overline{α}_{2n−i} for i = 1, …, n. Consider the subgroup H := U(n, n − 1) and X = G/H. Then X is absolutely spherical of real rank 1 with Σ_{ℂ} = {σ} where

(10.31) σ := α_{1} + … + α_{2n}.

Thus Σ_{ℝ} = \{\overline{σ}\} with

(10.32) \overline{σ} = 2β_{1} + … + 2β_{n−1} + β_{n}.

But \overline{σ} is not a spherical root of any spherical variety over ℂ (see, e.g., [Was96, Table 1] for a list of possible roots). Observe that rescaling of β_{n} to \frac{1}{2}β_{n} converts S_{ℝ} to type B_{n} and \overline{σ} to the spherical root of SO(2n+1)/SO(2n). Therefore, one might wonder whether the normalization of S_{ℝ} is chosen correctly. The next example invalidates this objection.

ii) Take k = ℝ and let G be the quasi-split group of type E_{6} (type EII). Its restricted simple roots are

(10.33) β_{1} = \overline{α}_{2}, β_{2} = \overline{α}_{4}, β_{3} = \overline{α}_{3} = \overline{α}_{5}, β_{4} = \overline{α}_{1} = \overline{α}_{6}

(Bourbaki notation [Bou68]) which span a root system of type F_{4}. According to Berger, [Ber57], G contains a subgroup H which is isogenous to SO(6, 4) · SO(2). Its spherical roots over ℂ are

(10.34) σ_{1} = α_{1} + α_{3} + α_{4} + α_{5} + α_{6}, σ_{2} = α_{2} + α_{4} + \frac{1}{2}α_{3} + \frac{1}{2}α_{5}

(see [BP15]). Their restrictions to A are

(10.35) \overline{σ}_{1} = β_{2} + 2β_{3} + 2β_{4}, \overline{σ}_{2} = β_{1} + β_{2} + β_{3}.

They generate a root system of type B_{2}. The root \overline{σ}_{1} is again not a spherical root of any ℂ-variety, but worse, Wasserman’s tables [Was96] show that \overline{σ}_{1}, \overline{σ}_{2} are not the spherical roots of any F_{4}-variety over ℂ even if one allows rescaling the β_{i} or \overline{σ}_{j}

One can also look at the triple

(10.36) Ξ_{K}, Σ_{K}, Σ_{K}^{0}

together with the G^{∗}-action, called the spherical index of X, which very much looks and indeed very often is a Satake-Tits diagram. But there are counterexamples: Let X = G/H be as in Example 10.13. Then Σ_{ℂ} is of type G_{2} and one of the roots is compact. This never occurs in the group case even for any ground field (see [Tit66, p. 61]).

Nevertheless, as in the group case ([BT65, 4.9]) the same major constraint for the structure of indices is still valid:

10.19. Lemma. The spherical index (Ξ_{K}, Σ_{K}, Σ_{K}^{0}) is invariant under the opposition map ι = −w_{0} (where w_{0} ∈ W_{K} is the longest element).

Proof. The G^{∗}-action stabilizes both W_{K} and Σ_{K}. This implies that G^{∗} commutes with w_{0} and therefore with ι. It remains to be shown that ιΣ_{K}^{0} = Σ_{K}^{0}. Let Ξ^{0} := kern res_{A} ⊆ Ξ_{K}. Then N(N_{k}) (the normalizer of N_{k} in W_{K}) acts on Ξ^{0}. Let \overline{w}_{0} ∈ W_{k} be the longest element and let n_{0} ∈ N(N_{k}) be a lift. Then Z_{k} = −\overline{w}_{0}Z_{k} ⊆ −n_{0}Z_{K}. Let F ⊆ Z_{K} be the face corresponding to Σ_{K}^{0}. Since Z_{k} contains an interior point of F we get F ⊆ Z_{K} ∩(−n_{0}Z_{K}). Thus there is w^{0} ∈ W_{K}^{0} with −w^{0}n_{0}Z_{K} = Z_{K}. By uniqueness we get w_{0} = w^{0}n_{0}. Hence,

(10.37) ιΣ_{K}^{0} = −w^{0}n_{0}Σ_{K}^{0} ⊆ Σ_{K} ∩ Ξ^{0} = Σ_{K}^{0}. □

11. Wonderful varieties

In this section, we use our results on root systems to investigate smoothness properties of the standard embedding of a k-convex variety. As a direct consequence of the fact that the valuation cone is simplicial (Corollary 9.7) we get:

11.1. Theorem. Let X be a k-convex G-variety with rk_{k} X = r. Then the standard embedding X_{st} contains r irreducible divisors X_{1}, …, X_{r} (namely the codimension-1-strata) such that the strata are exactly the (set theoretic) intersections X_{I} := ⋂_{i∉I} X_{i} where I runs through all subsets of {1, …, r}.

A more canonical way to parametrize strata is by their set of spherical roots:

11.2. Corollary. Let X be a k-convex G-variety and let X_{st} be its standard embedding. Then there is a bijection J ↔ X^{J} of subsets J of Σ_{k}(X) and strata X^{J} of X_{st} with

(11.1) Σ_{k}(X^{J}) = J.

Proof. Each divisor X_{i} corresponds to an extremal ray R_{i} of Z_{k}(X). Thus there is precisely one σ_{i} ∈ Σ_{k}(X) with σ_{i}(R_{i}) ≠ 0 and

(11.2) Σ_{k}(X_{I}) = {σ_{i} | σ_{i}(R_{j}) = 0 for all j ∈ I} = {σ_{i} | i ∉ I}.

Thus, if we set

(11.3) J := {i = 1, …, r | σ_{i} ∉ I}

then X^{I} := X_{J} has the desired property. □

11.3. Remark. One application is that given any subset I of k-spherical roots there is a variety, namely X^{I}, with that set as a set of spherical roots.The variety X^{I} is called. This variety id therefore called the localization of X in Σ.

Together with the invariance under the opposition map (10.19), the existence of X^{I} puts enormous constraints on the possible subsets Σ_{K}^{0} ⊆ Σ_{K}. For example, if Σ_{K} is of inner type A_{n} then Σ_{K} \ Σ_{K}^{0} must be of the form {α_{d}, α_{2d}, …, α_{n+1−d}} with d | n + 1.

We now address the problem of when the standard embedding is actually smooth. Clearly, this is the case if and only if the torus embedding A_{k}(F_{st}) is smooth. A well known criterion of toroidal geometry asserts that this is the case if and only if the cone Z_{k}(X) is spanned by an integral basis. Dualizing, this leads to:

11.4. Definition. A k-dense G-variety is called k-wonderful if Σ_{k}^{pr}(X) is a ℤ-basis of Ξ_{k}(X).

Observe that a k-wonderful variety is automatically k-convex and therefore has a standard embedding. Thus we have:

11.5. Proposition. Let X be a k-dense G-variety. Then X is k-wonderful if and only if X has a smooth standard embedding. In this case the intersections of Theorem 11.1 can be taken to be scheme theoretic.

For reductive groups, the difference between the weight lattice and the root lattice is related to the center of the group. Something similar is valid in general with the rôle of the center being taken over by the group 𝔄_{k} := 𝔄_{k}(X) of k-central automorphisms of X (Definition 5.1). This group is a subgroup of A_{k} = A_{k}(X) which means that the restriction map Ξ_{k} → Ξ(𝔄_{k}) is surjective. Let Γ_{k}(X) be its kernel, i.e., Γ_{k}(X) = Ξ(A_{k}/𝔄_{k}). In the following discussion we tacitly assume that the orbit space X/𝔄_{k} exists. This is certainly the case when X is homogeneous. In general it can be achieved by replacing X by a suitable dense open subset. Here is the main result of this section:

11.6. Theorem. Let X be a k-dense G-variety. Then the quotient X/𝔄_{k}(X) is kwonderful.

The theorem will be proved simultaneously with Proposition 11.8 below.

11.7. Remarks. i) A typical example of a non-wonderful k-variety is X = SL(n, ℂ)/ SO(n, ℂ), n ≥ 3, over k = K = ℂ. Then Ξ_{k}(X) is 2 times the weight lattice of SL(n, ℂ) which is not a root lattice.

ii) Recall that 𝔄_{k}(X) is finite if and only if X is k-convex (Corollary 8.9). Thus for any k-convex variety X there is finite abelian group E of automorphisms (namely E = 𝔄_{k}(X)) such that X/E is k-wonderful. With a bit of luck one has E(k) = 1 in which case X(k) can be considered as a subset of (X/E)(k). This is for example the case when X is a Riemannian symmetric space (over k = ℝ).

iii) A very important special case is that of a k-dense G-variety X with 𝔄_{k}(X) = 1. Then X is k-wonderful. Since Aut^{G} G/H = N_{G}(H)/H, the condition 𝔄_{k}(X) = 1 is in particular satisfied when X = G/H is homogeneous with H = N_{G}(H) selfnormalizing. Observe that even in the absolutely spherical case there examples with 𝔄_{k}(X/𝔄_{k}(X)) ≠ 1. So Theorem 11.6 cannot be reduced to the case 𝔄_{k} = 1.

Put

(11.4) Σ_{k}^{aut}(X) := Σ_{k}^{pr}(X/𝔄_{k}).

Then another way to phrase the theorem is that Σ_{k}^{aut} is a ℤ-basis of Γ_{k} = Ξ_{k}(X/𝔄_{k}). Moreover Σ_{k}^{aut} is a system of simple roots for the root system

(11.5) Φ_{k}^{aut}(X) := W_{k}Σ_{k}^{aut}(X) = Φ_{k}^{pr}(X/𝔄_{k}).

with Z_{k}(X) as antidominant Weyl chamber and W_{k}(X) as Weyl group. We record some more properties:

11.8. Proposition. Let X be a k-dense G-variety. Then:

i) Ξ_{k}(X) is a weight lattice for Φ_{k}^{aut}(X). In particular, there is a map Σ_{k}^{pr}(X) → {1, 2} : σ ↦ n_{σ}^{aut} with Σ_{k}^{aut} = {n_{σ}^{aut}σ | σ ∈ Σ_{k}^{pr}(X)}.

ii) Γ_{k}(X) = res_{A} Γ_{K}(X) and Σ_{k}^{aut} = res′_{A} Σ_{K}^{aut}.

iii) If X is K-wonderful then X is also k-wonderful.

Proof. For k = K, Theorem 11.6 is the main result of [Kno96]. From that we derive the general case.

Let X_{k} := X/𝔄_{k}(X). It is easy to see that the group 𝔄_{K}(X) and its action on X are defined over k. Therefore also the variety X_{K} := X/𝔄_{K}(X) is defined over k. Because of 𝔄_{k} ⊆ 𝔄_{K} these spaces are connected by surjective morphisms

(11.6) X ↠ X_{k} ↠ X_{K}.

We have by definition Ξ_{k}(X_{k}) = Γ_{k} and Ξ_{K}(X_{K}) = Γ_{k}. The equality 𝔄_{k} = 𝔄_{K} ∩ A_{k} (eqn. (8.5)) implies that A_{k}/𝔄_{k} → A_{K}/𝔄_{K} is injective. Passing to character groups this means that

(11.7) res_{A} : Γ_{K} → Γ_{k}

is surjective which already shows the first half of ii). From

(11.8) Γ_{K} = Ξ_{K}(X_{K}) ↠ Ξ_{k}(X_{K}) ⊆ Ξ_{k}(X_{k}) = Γ_{k}

we get Ξ_{k}(X_{K}) = Γ_{k}. Now let

(11.9) Σ′_{k} := res′_{A} Σ_{K}^{aut}(X) = res′_{A} Σ_{K}(X_{K}) = Σ_{k}(X_{K})

which is a linearly independent subset of Γ_{k}. The main result of [Kno96], Cor. 6.5, asserts that Σ_{K}(X_{K}) is a ℤ-basis of Γ_{K}. It follows that Σ′_{k} generates Γ_{k}. Thus it also a ℤ-basis of Γ_{k} which therefore must equal Σ_{k}^{pr}(X_{k}) = Σ_{k}^{aut}(X). This proves Theorem 11.6 and the second half of ii).

An even simpler argument proves iii): If X is K-wonderful then Σ_{K}(X) is a ℤ-basis of Ξ_{K}(X). Restriction to A shows that Σ_{k}(X) generates and therefore is a ℤ-basis of Ξ_{k}(X). Since then Σ_{k}(X) = Σ_{k}^{pr}(X), it follows that X is k-wonderful, as well.

Finally, it is known that W_{K} acts trivially on Ξ_{K}/Γ_{K} (by [Kno96, Cor. 6.5c)]). It follows that W_{k} acts trivially on the quotient Ξ_{k}/Γ_{k}. This means that Ξ_{k} is a weight lattice for Φ_{k}^{aut} and proves i). □

11.9. Remark. All elements of Σ_{K}^{aut}(X) are integral multiples of elements of Σ_{K}(X) = Σ_{K}^{pr}(X). It follows by restriction that all elements of Σ_{k}^{aut}(X) are integral multiples of elements of Σ_{k}(X). This implies n_{σ} ≤ n_{σ}^{aut} where n_{σ} was defined in Corollary 10.10

12. Boundary degenerations

Let, for simplicity, X = G/H be a homogeneous k-spherical variety with character group Ξ = Ξ_{k}(X) and let 𝔄 := 𝔄_{k}(X) ⊆ A_{k} = A_{k}(X) be its group of central automorphisms. Then Ξ(A_{k}/𝔄) = ℤΣ where Σ = Σ_{k}^{aut}(X).

The group 𝔄 acts freely on X. The quotient Y := X/𝔄 has the property that Σ_{k}(Y) = Σ and Ξ_{k}(Y) = ℤΣ. Thus, Y admits a smooth standard embedding 𝔜 := Y_{st}.

Now put Z := (X ×X)/𝔄 where 𝔄 acts diagonally. Then Z is a k-spherical G×G-variety with

(12.1) Ξ_{k}(Z) = {(χ, η) ∈ Ξ ⊕ Ξ | χ + η ∈ ℤΣ}.

Thus there is a short exact sequence

(12.2) 0 → Ξ →^{Δ_{−}} Ξ_{k}(Z) →^{β} ℤΣ → 0

with Δ_{−}(χ) = (χ, −χ) and β(χ, η) = χ + η. In particular, β induces an injection

(12.3) β^{∗} : N_{k}(Y) ↪ N_{k}(Z).

Since Σ_{k}(Z) = (Σ × 0) ∪ (0 × Σ) we have β(Σ_{k}(Z)) = Σ and therefore

(12.4) C_{bd} := β^{∗}(Z_{k}(Y)) ⊆ Z_{k}(Z).

Let F_{bd} be the fan of faces of C_{bd} and let 𝔛 := Z(F_{bd}) be the corresponding embedding of Z.

12.1. Theorem. Let X be a homogeneous k-spherical variety.

i) There are unique morphisms

(12.5) p : 𝔛 → 𝔜, δ : 𝔜 → 𝔛

extending the first projection Z → Y : (x_{1}, x_{2})/𝔄 ↦ x_{1}/𝔄 and the diagonal morphism Y ↪ Z : x/𝔄 ↦ (x, x)/𝔄, respectively.

ii) The spaces 𝔛, 𝔜 and the morphism p are smooth while δ is a section of p. In particular, δ is a closed embedding and p is surjective.

iii) For any y ∈ 𝔜(k) the fiber 𝔛_{y} := p^{−1}(y) is a homogeneous k-spherical G-variety with base point δ(y). Its spherical data are

(12.6) Ξ_{k}(𝔛_{y}) = Ξ_{k}(X), S^{(p)}(𝔛_{y}) = S^{(p)}(X), Σ_{K}(𝔛_{y}) = Σ_{K}(Gy).

Moreover, 𝔛_{y} is a k-form of X or horospherical if and only if Gy is the open or the closed orbit of 𝔜, respectively.

iv) Let F be a fan supported in Z_{k}(X) and ΔF ⊆ Z_{k}(X) × Z_{k}(X) = Z_{k}(X × X) the diagonal fan. Then there is a Cartesian diagram

(12.7) \xymatrix{(X × X)(ΔF) \ar[r] \ar[d]^{q} & 𝔛 \ar[d]^{p} \\ X(F) \ar[r]^{π} & 𝔜}

where π and q extend the morphism X → Y and the first projection X × X → X, respectively.

v) For any C ∈ F let x ∈ X(C)(k). Put y := φ(x) ∈ 𝔜 and z := δ(y) ∈ 𝔛_{y}. Then there is a short exact sequence

(12.8) 1 → G_{z} → G_{x} → A_{0} → 1

where A_{0} is the connected subgroup of A_{k}(X) with N (A_{0}) = ⟨C⟩_{ℚ} ⊆ N_{k}(X).

Proof. i) The projection p_{1} : Z → Y induces the map

(12.9) N (p_{1}) : N_{k}(Z) → N_{k}(Y)

of which β^{∗} is a section. Thus N (p) maps C_{bd} isomorphically onto Z_{k}(Y) which shows that p_{1} extends to a morphism p (Corollary 7.8). Using the local structure theorem, the extendability of the diagonal map is reduced to that between torus embeddings

(12.10) (A/𝔄)(F_{st}) [figure]((A × A)/𝔄)(F_{bd})

which is clear by construction. Then p ∘ δ = id_{𝔜} follows by continuity.

ii) We already know that 𝔜 is smooth. Since β has a section also C_{bd} is spanned by a partial basis of Hom(Ξ_{k}(Z), ℤ) which means that 𝔛 is smooth, as well.

Clearly p is smooth in all image points of the section δ. Now, the fact that C_{bd} → Z_{k}(Y) is bijective means that p induces a bijection between the set of G × G-orbit in 𝔛 and G × 1-orbits in 𝔜. This implies that 1 × G acts transitively on all fibers of p. This shows that p is smooth everywhere.

iii) We already know that 𝔛_{y} is G-homogeneous with base point δ(p). Assume that Gy is the closed orbit of 𝔜. Then p^{−1}(Gy) is the closed orbit of 𝔛 which is horospherical since C_{bd} meets the interior of Z_{k}(Z). Thus there exists an N × N fixed point z ∈ p^{−1}(Gy). But then z is an N-fixed point of the fiber 𝔛_{p(z)} and is therefore horospherical, as well. Being also homogeneous this means that 𝔛_{p(z)} is spherical which means that P has dense orbit. That being an open condition we conclude that 𝔛_{y} is spherical for y in an open neighborhood of p(z). By G × 1-equivariance we conclude that 𝔛_{y} is k-spherical for all y ∈ 𝔜(k). The calculation of the spherical data is deferred to part v).

iv) First consider the diagram of open subsets:

(12.11) \xymatrix{X × X \ar[r] \ar[d] & Z \ar[d] \\ X \ar[r] & Y}

where the vertical maps are projections onto the first factor. Because X → Y is a principal 𝔄-bundle we have X ×_{Y} X ≅ X × 𝔄. Then

(12.12) X ×_{Y} Z = (X ×_{Y} X × X)/𝔄 ≅ (X × 𝔄 × X)/𝔄 = X × X

shows that the diagram (12.11) is Cartesian. Next we show that the maps in (12.11) extend to embeddings. For this it suffices (again by Corollary 7.8) that the corresponding maps of N -spaces

(12.13) \xymatrix{N (X) × N (X) \ar[r] \ar[d] & (N (X) × N (X))/N (𝔄) \ar[d] \\ N (X) \ar[r] & N (X)/N (𝔄)}

map fans to each other. But that is clear since C_{bd} is nothing else than the image of

(12.14) Z_{k}(Y) ↪ N (X)/N (𝔄) ↪^{diag} (N (X) × N (X))/N (𝔄).

It remains to show that

(12.15) u : (X × X)(F) → V := X(F) ×_{𝔜} 𝔛

is an isomorphism. An argument as above shows that the fibers of q are the 1 × G-orbits. Semicontinuity implies that all fibers of q are irreducible of the same dimension. This shows that u is birational, surjective and quasi-finite. The target V is smooth over X(F) hence normal. Zariski’s main theorem implies that u is an open embedding hence an isomorphism.

v) Consider first the map of G × G-orbits

(12.16) Z_{1} := (X × X)(ΔC) → Z_{2} := (X × X)(C × C) = X(C) × X(C).

It follows from the local structure theorem and toroidal theory that this is a principal bundle with a structure group A_{0} with Ξ(A_{0}) = ΔC^{⊥} ∩ Ξ^{2}/(C^{⊥} × C^{⊥}) ∩ Ξ^{2}. Projection to the first component yields an isomorphism Ξ(A_{0}) = res_{⟨C⟩} Ξ. Thus, A_{0} ⊆ A_{k} is connected with N (A_{0}) = ⟨C⟩. For isotropy group this means

(12.17) 1 → (G × G)_{(x,z)} → G_{x} × G_{x} →^{η_{1}·η} A_{0} → 1.

We have η_{1} = η_{2} by symmetry which implies that η_{1} is surjective. Then G_{z} = ker η_{1} and the assertion follows.

Finally we compute the spherical data of 𝔛_{y} = G/G_{z}. Then clearly S^{(p)}(G/G_{z}) = S^{(p)}(G/G_{x}) = S^{(p)}(X(C)) = S^{(p)}(X(C)). The same argument yields Σ_{k}(G/G_{z}) = Σ_{k}(G/G_{x}) = Σ(y). Finally Ξ(𝔛_{y}) = Ξ since every AN-semiinvariant rational function can be extended to 𝔛.

It follows that 𝔛_{y} is horospherical if and only if Σ(y) = ∅ if and only if Gy is closed in 𝔜. On the other hand, for any y ∈ Y let u ∈ X(K) be a lift. Then X → 𝔛_{y} : x/𝔄 ↦ (u, x)/𝔄 is a K-isomorphism. □

12.2. Remark. The varieties 𝔛_{y} are called the boundary degenerations of X. The last point v) gives another interpretation: it means that the isotropy groups G_{x′} ⊆ G have a well-defined limit G_{z} ⊆ G whenever x′ ∈ X tends to some point x ∈ X(F). Moreover, this limit subgroup is a normal subgroup of G_{x}, the quotient being the split torus A_{0}. Moreover For that reason, the spaces G/G_{x′} = 𝔛_{π(x)} is a boundary degenerations of X.

The fibration 𝔛 → 𝔜 could be called the big boundary degeneration of X since because of the G-action on the base many fibers 𝔛_{y} are isomorphic to each others. It is possible to cut down on the amount of isomorphic fibers by restricting the base to a subspace. For this, let Ã := A_{k}/𝔄 = A_{k}(𝔜) and 𝔜̃ ⊆ 𝔜_{el} ⊆ 𝔜 the affine closure of a flat. Now define 𝔛̃ := 𝔛 ×_{𝔜} 𝔜̃. Then

(12.18) p̃: 𝔛̃ → 𝔜̃

is also a smooth deformation of X which we call the small boundary degeneration of X. There is an action of G × Ã on 𝔛̃ for which p̃ is equivariant.

Next we give an infinitesimal version of the boundary degeneration. For this let Grass(𝔤) be the Grassmann variety of linear subspaces of Lie G. Then for every homogeneous G-variety X there is a canonical morphism Φ : X → Grass(𝔤) mapping x ∈ X to the isotropy subalgebra 𝔤_{x}. If x ∈ X(k) is a base point with 𝔥 := 𝔤_{x} then Φ is just the orbit map Φ(gx) = Ad g 𝔥.

12.3. Corollary. Let X be a homogeneous k-spherical variety and let F be a fan supported in Z_{k}(X).

i) The morphism Φ : X → Grass(𝔤) extends uniquely to a morphism

(12.19) \overline{Φ} : X(F) → Grass(𝔤).

Moreover, this \overline{Φ} factors uniquely through 𝔜, i.e., there is a G-morphism Φ′ : 𝔜 → Grass(𝔤) which is independent of F such that \overline{Φ} = Φ′ ∘ π:

(12.20) \xymatrix{X \ar@{->>}[d] \ar@{^{(}->}[r] & X(F) \ar[d]_{π} \ar[dr]^{\overline{Φ}} \\ Y \ar@{^{(}->}[r] & 𝔜 \ar[r]^{Φ′} & Grass(𝔤).}

ii) For C ∈ F let 𝔞_{0} := ⟨C⟩_{ℚ} ⊗_{ℚ} k ⊆ N_{k}(X) ⊗_{ℚ} k = Lie A_{k}(X) and let x ∈ X(C). Then there is a short exact sequence of Lie algebras

(12.21) 0 → \overline{Φ}(x) → 𝔤_{x} → 𝔞(C) → 0.

iii) Let F = \overline{Ay} ⊆ 𝔜 be a flat. Then the restriction of Φ′ to F is a finite morphism.

Proof. i) The morphism Φ′ is simply y ↦ 𝔤_{δ(y)}. Then \overline{Φ} := Φ′ ∘ π is clearly an extension of Φ.

ii) This follows from part v) of Theorem 12.1.

iii) From N_{G}(H^{0}) = N_{G}(𝔥) and Z_{k}(G/H) = Z_{k}(G/H^{0}) it follows that 𝔄^{0} is also the maximal split subtorus of N_{G}(𝔥)/H^{0}. This shows that N_{A}(𝔥)^{0} ⊆ 𝔄H = G_{y} and therefore that the restriction of Φ′ to Ay is quasi-finite. Now let F →^{φ} F̃ →^{ψ} Grass(𝔤) be the Stein factorization, i.e., φ has connected and ψ has finite fibers. We have seen above that φ is in fact birational and we have to show that it is an isomorphism. The fan corresponding to the toroidal embedding F is F := W_{k}F_{st}(Y). Let F̃ be the fan corresponding to F̃. Suppose they are not equal. Then there is a maximal cone C ∈ F with C ∉ F̃. By W_{k}- symmetry we may assume that C = Z_{k}(Y). Since F is a refinement of F̃ there is a C̃ ∈ F̃ with C ⊊ C̃. Thus C has a facet C_{0}, corresponding to a root σ ∈ Σ_{k}(Y), which is not a facet of C̃. It follows that the A_{k}(Y)-orbits corresponding to C and C_{0} are mapped to the same orbit in F̃. But that is not possible since their images in Grass(𝔤) lie in different G-orbits: since the boundary degenerations have different sets of spherical roots (namely ∅ and {σ}, respectively) they are not even isomorphic as G-varieties. This completes the proof that F → F̃ is an isomorphism. □

12.4. Remark. When k is algebraically closed it is known that Φ′ is a closed embedding (see [Los09] for the last step) which means that \overline{Φ} cannot be factored any further. This is not true in general since 𝔄_{k}(X) may not constitute the full automorphism group of X. Nevertheless, it might be possible that the restriction of Φ′ to a flat F is always a closed embedding.

Following [Bri90], one can use the morphism Φ′ to construct a distinguished basis for 𝔥. More precisely, let X = G/H be a homogeneous k-spherical variety and let Q = LU be an adapted parabolic such that x_{0} := eH ∈ X_{el} and let y_{0} be its image in Y. Let L_{0} := L_{x_{0}} = L ∩ H such that X_{el} = L/L_{0}. Because of 𝔩 = 𝔩_{0} + 𝔪 + 𝔞 there is an AL_{0}- stable subspace 𝔩̃ ⊆ 𝔪 ⊕ 𝔞 such that 𝔩 = 𝔩_{0} ⊕ 𝔩̃ (whence 𝔞 ∩ 𝔩̃ ≅ 𝔞_{k}(X)). We also choose an L_{0}-stable splitting 𝔥 = 𝔩_{0} ⊕ 𝔥̃. Then the k-LST implies

(12.22) 𝔤 = 𝔲 ⊕ 𝔩̃⊕ 𝔩_{0} ⊕ 𝔥̃

Now let y_{0} ∈ Y be the image of x_{0}, let y_{1} ∈ 𝔜 be the fixed point in \overline{Ay_{0}} ⊆ 𝔜_{el}, and let x_{1} := δ(y_{1}). Then 𝔤_{x_{1}} = 𝔩_{0} ⊕ 𝔲^{−} and therefore

(12.23) 𝔤 = 𝔲 ⊕ 𝔩̃⊕ 𝔩_{0} ⊕ 𝔲^{−}

This shows that 𝔥̃ is the graph of an L_{0}-equivariant homomorphism Υ : 𝔲^{−} → 𝔲 ⊕ 𝔩 inside 𝔲 ⊕ 𝔩 ⊕ 𝔲^{−} which implies

(12.24) 𝔥 = 𝔩_{0} ⊕ {Υ(ξ) + ξ | ξ ∈ 𝔲^{−}}.

To make this more precise, let Φ_{𝔲} be the set of restricted roots α ∈ Φ_{k} with 𝔤_{α} ⊆ 𝔲. Then for all α, β ∈ Φ_{𝔲} consider the component

(12.25) Υ_{αβ} : 𝔤_{−α} ↪ 𝔲^{−} →^{Υ} 𝔲 ⊕ 𝔩̃ ↠ 𝔤_{β}

For β = 0 we put

(12.26) Υ_{α0} : 𝔤_{−α} ↪ 𝔲^{−} →^{Υ} 𝔲 ⊕ 𝔩̃ ↠ 𝔩̃.

The follow statement can be seen as a vanishing theorem for components:

12.5. Corollary. Let

(12.27) M := {α + β | α ∈ Φ_{𝔲}, β ∈ Φ_{𝔲} ∪ {0}, and Υ_{αβ} ≠ 0}.

Then M and Σ_{k}(X) generate the same convex cone in Ξ(A) ⊗ ℚ. In particular, M ⊆ Ξ_{k}(X) ⊗ ℚ and

(12.28) Z_{k}(X) = {a ∈ N_{k}(X) | σ(a) ≤ 0 for all σ ∈ M}.

Proof. Consider the closure \overline{V} of Ad A(𝔥) in Grass(𝔤) it contains 𝔤_{x_{1}} as a fixed point. Let V ⊆ \overline{V} be the A-stable open affine neighborhood of this fixed point. Its normalization is the affine toroidal embedding corresponding to the cone C which is dual to ℚ_{≥0}M (see the proof of [Bri90, Thm. 2.3]). On the other hand, this cone equals Z_{k}(Y) according to Corollary 12.3 iii). Thus, the dual cones are the same, as well. □

13. The weak polar decomposition

We present an application of our theory to local fields.

13.1. Proposition. Let k be a local field and let X be a homogeneous G-variety of rank 0. Then G(k) acts transitively on X(k).

Proof. Let X = G/H. Then, by Corollary 8.5, there is a k-parabolic Q = LU ⊆ G with Q_{an} ⊆ H ⊆ Q. Let G_{0} be the largest anisotropic factor of L. Then Q = G_{0}Q_{an} and therefore the action of G_{0} on X_{0} := Q/H is transitive. Since G(k) acts transitively on (G/Q)(k) ([BT65, Thm. 4.13 a)]) it suffices to show that G_{0}(k) acts transitively on X_{0}(k). Thus, replacing (G, X) by (G_{0}, X_{0}) we are reduced to the case that G is anisotropic.

Then G(k) and X(k) are compact. Now suppose G(k) does not act transitively on X and let X_{1} ⊆ X(k) be an orbit. Since G(k) has only finitely many orbits (Corollary 4.15) and all of them are closed the complement X_{2} = X(k) \ X_{1} is closed and nonempty. The variety X is also an affine variety (see Proposition 3.4). Therefore, the Stone-Weierstraß theorem holds for regular functions on X(k) (see [Die44, Kap50]). Thus, approximating the characteristic function of X_{1} we obtain f ∈ k[X] with |f(x) − 1| < \frac{1}{2} for x ∈ X_{1} and |f(x)| < \frac{1}{2} for x ∈ X_{2}. Let \overline{f}(x) := ∫_{G(k)} f(gx)dg. Then still |\overline{f}(x) − 1| < \frac{1}{2} for x ∈ X_{1} and |\overline{f}(x)| < \frac{1}{2} for x ∈ X_{2} which implies that \overline{f} is not constant. On the other hand, since G(k) is Zariski-dense in G and \overline{f} is G(k)-invariant it is also G-invariant. Thus it must be constant since G acts transitively on X. □

We generalize this to k-spherical varieties of arbitrary rank. Let | · | be an absolute value on the local field k. Let X be a k-dense G-variety and A_{k} := A_{k}(X). Then we call

(13.1) A_{k}^{+} := {x ∈ A_{k}(k) | |σ(x)| ≤ 1 for all σ ∈ Σ_{k}(X)}

the compression domain in A_{k}. Then the following weak form of a polar decomposition holds.

13.2. Theorem. Let k be a local field, let X be a homogeneous k-spherical variety and let x_{0} ∈ X(k) such that A_{k}x_{0} ⊆ X is a flat. Then there is a compact subset Ω ⊆ G(k) with

(13.2) X(k) = Ω · A_{k}^{+}x_{0}.

Proof. Let F be a complete fan Z_{k}(X) which we also choose to be smooth, i.e., each cone in F is generated by a partial integral basis. Let \overline{A}_{k} := A(F) and \overline{X} := X(F). Then both \overline{A}_{k} and \overline{X} are smooth.

Let C ∈ F be of maximal dimension. Then the corresponding orbit A_{k}(C) consists of a single fixed point a_{C}. Let C be defined by inequalities β_{1} ≤ 0, …, β_{r} ≤ 0. If we assume that the β_{i} are primitive in Ξ_{k}(X) they even form an integral basis (by smoothness). A neighborhood of a_{C} in \overline{A}_{k}(F) is isomorphic to the affine space A^{r} where A_{k} acts with the characters β_{i}. Thus, if we put

(13.3) A_{C}^{+} := {x ∈ A_{k}(k) | |β_{i}(x)| ≤ 1 for all i = 1, …, r}

then its closure \overline{A}_{C}^{+} in \overline{A}(k) contains D^{r} ⊆ A^{r}(k) where where D ⊆ k is the unit disk. This implies that \overline{A}_{C}^{+} is a neighborhood of a_{C} in \overline{A}.

Since the β_{i} form an integral basis of Ξ_{k}(X) one can express the spherical roots in terms of the β_{i}:

(13.4) σ = ∑_{i=1}^{r} ν_{i}(σ)β_{i} with ν_{i}(σ) ∈ ℤ.

From the inclusion C ⊆ Z_{k}(X) we get ν_{i}(σ) ≥ 0 and therefore A_{C}^{+} ⊆ A_{k}^{+}. Thus we have proved: the closure \overline{A}_{k}^{+} of A_{k}^{+} in \overline{A}_{k}(k) is a neighborhood of every fixed point a_{C}.

Let φ : \overline{A}_{k} → \overline{X} be the embedding extending the orbit map a → ax_{0}. Then x_{C} := φ(a_{ℂ}) is a representative of the closed orbit X(C). Now the facts that X_{el} → X_{an} is locally trivial (Hilberts Satz 90) and that M(k) acts transitively on X_{an}(k) (Proposition 13.1) imply that M(k)φ(\overline{A}_{k}^{+}) is a neighborhood of every x_{C} in \overline{X}_{el}(k). Let Ω_{0} ⊆ U(k) be a compact neighborhood of 1 and Ω_{1} := Ω_{0}M(k). Then the local structure theorem implies that Ω_{1} is a compact subset of G(k) such that Ω_{1}φ(\overline{A}_{k}^{+}) is a neighborhood of each point x_{C} in \overline{X}(k).

Next we show that for every x ∈ \overline{X}(k) there is g ∈ G(k) such that gx lies in the open interior Ω^{0} of Ω_{1}φ(\overline{A}_{k}^{+}). This clearly implies the theorem because of compactness of \overline{X}(k) it can be covered by finitely many translates gΩ^{0}.

Since U\overline{X}_{el} is (Zariski-)open in \overline{X} and meets every orbit there is g ∈ G(k) with gx ∈ U\overline{X}_{el}. But then there is u ∈ U(k) such that ugx ∈ \overline{X}_{el}. Then again since M(k) acts transitively on X_{an}(k) one can find m ∈ M(k) and a ∈ \overline{A}_{k}(k) with mugx = φ(a). Finally, since every A_{k}-orbit of \overline{A}_{k} contains a fixed point a_{C} in its closure there is a homomorphism λ : G_{m} → A_{k} such that lim_{t→0} λ(t)a = a_{C} for some C (also in the Hausdorff topology). A positive power λ^{n} can be lifted to a homomorphism λ̃ : G_{m} → A. Since then lim_{t→0} λ̃(t)φ(a) = φ(lim_{t→0} λ(t)^{n}a) = x_{C} there is t ∈ G_{m}(k) such that λ̃(t)mugx ∈ Ω^{0}. □

13.3. Remarks. i) One can strengthen the weak polar decomposition in the following way: Let K ⊆ G(k) be a maximal compact subgroup in case k is archimedian and a compact open subgroup otherwise. Then Ω can be chosen to be of the form Ω = F K where F ⊆ G(k) is a finite subset. Indeed, if K is open then every compact subset of G(k) is a subset of finitely many translates of K. In the archimedian case see the argument in [KKSS15]. ii) For k = ℝ one has A_{k}^{+} = A_{2} exp 𝔞_{k}^{+} where A_{2} ⊆ A_{k}(k) is the subgroup of 2-torsion elements and 𝔞_{k}^{+} is the compression cone

(13.5) 𝔞_{k}^{+} = {ξ ∈ Lie A_{k}(k) | dσ(ξ) ≤ 0 for all σ ∈ Σ_{k}}.

We keep the notation of §12 on boundary degenerations. For local fields there is a simplification. More precisely, a given k-spherical variety X might in general have nonisomorphic boundary degenerations with the same spherical root system. This is due to the fact that the k-rational points of a G-orbit of 𝔜 may decompose into several G(k)- orbits. This does not happen for the small degeneration 𝔛̃ → 𝔜̃ since Ã ≅ G_{m}^{r} and 𝔜̃ ≅ A^{r} with r = rk_{k} 𝔜. Thus, the Ã(k)-orbits in 𝔜̃(k) are parametrized by subsets of Σ_{k} with the fixed point corresponding to the empty set. By the weak polar decomposition every y ∈ 𝔜(k) is G(k)-conjugate to an element of 𝔜̃(k). Thus we get:

13.4. Corollary. Let k be a local field and X a k-spherical variety. Then every boundary degeneration 𝔛_{y} is isomorphic to a fiber of 𝔛̃ → 𝔜̃. In particular, two boundary degenerations 𝔛_{y_{1}} and 𝔛_{y_{2}} are isomorphic to each other if and only if Gy_{1} = Gy_{2} ⊆ 𝔜 if and only if Σ_{k}(𝔛_{y_{1}}) = Σ_{k}(𝔛_{y_{2}}) ⊆ Σ_{k}(X).

14. The Satake compactification

In this final section we compare our compactifications with compactifications already appearing in the literature.

Suppose the k-dense variety X is even K-convex. Then one can obtain another embedding by regarding X as a K-variety, take its standard embedding and observe that it is defined over k. The thus obtained variety, we denote it by X_{st}^{K}, is another canonical embedding of X. The advantages of X_{st}^{K} are that it is independent of k and that it is well studied. For symmetric varieties it has been introduced by DeConcini-Procesi in [DCP83] and more generally for K-spherical varieties by Luna-Vust [LV83]. If k = ℝ and X_{st}^{ℂ} is smooth then the set X_{st}^{ℂ}(ℝ) of real points is clearly a compact G(ℝ)-manifold containing X(ℝ) as a dense open subset.

A major disadvantage of X_{st}^{K} is that it doesn’t work well when X is not absolutely spherical because then X_{st}^{K} may contain infinitely many orbits and is therefore only defined up to certain birational isomorphisms. Also its boundary is not so easy to control. In particular, the union of all k-dense orbits may not be open (see Example 14.2 iii) further down).

The relation of the two standard embeddings is:

14.1. Proposition. Let X be a k-spherical variety which we assume to be K-convex. Then its K-standard model X_{st}^{K} can be chosen to be defined over k. Moreover, X is also k-convex and the identity on X extends to a birational morphism

(14.1) ι_{X} : X_{st}^{k} → X_{st}^{K}.

The following are equivalent:

i) ι_{X} is injective on k-rational points.

ii) ι_{X} is an open embedding.

iii) G^{∗} acts trivially on Σ_{K}(X) \ Σ_{K}^{0}(X) (see (10.6)).

When these conditions hold and X is k-spherical then ι_{X} is bijective on k-rational points.

Proof. The k-standard embedding corresponds to the cone Z_{k}. The same holds for K. Thus the existence of ι_{X} follows from Z_{k} ⊆ Z_{K}. From toroidal theory follows that ι_{X} is an open embedding if and only if Z_{k} is a face of Z_{K}. This holds if and only if the equations for N_{k} do not contain an equation of the form σ = γ ∗ σ (see Corollary 10.8). This shows the equivalence of ii) and iii). Now assume that Z_{k} is not a face of Z_{K}. Then there exists an ray R of Z_{k} which is not an extremal ray of Z_{K}. let C be the smallest face of Z_{K} containing R. Then ι_{X} maps the stratum X^{k}(R) of X_{st}^{k} onto the stratum X^{K}(C) of X_{st}^{K}. The fibers over k-rational points are tori whose dimension is dim C − dim R > 0. This shows that ι_{X} is not injective on k-rational points. This shows the equivalence of i) and ii).

Finally, assume that Z_{k} is a face of Z_{K} and that X is k-spherical. Then the orbits of X_{st}^{K} containing a k-rational point are precisely those which correspond to a face of Z_{K} which has a an element of Z_{k} in its relative interior. These are precisely the faces of Z_{k}. Thus the image ι_{K} contains all k-rational points of X_{st}^{K}. □

14.2. Examples. i) Condition iii) above is certainly satisfied if G is an inner form since then already the G^{∗}-action on S is trivial. It even suffices that G^{∗} acts trivially on S \ S^{0} which, in the real case, means that the Satake diagram of G does not contain any arrows (like all compact groups or SO(p, q) with p − q > 2 and p − q ≡ 2 mod 4).

ii) Even if G^{∗} acts non-trivially on S \ S^{0} its action on Σ_{K} \ Σ_{K}^{0} may be trivial. That happens, e.g., for all Riemannian symmetric (see below) or the spaces U(p_{1} + p_{2}, q_{1} + q_{2})/U(p_{1}, q_{1}) × U(p_{2}, q_{2}) with k = ℝ.

iii) The simplest example where the two standard embeddings differ is when k = ℝ and X := SU(2, 1)/μ_{3}SO(2, 1) where μ_{3} is the center of G (it is thrown in to make X_{st}^{ℂ} smooth). Then X_{ℂ} is the space of smooth quadrics in P^{3}(ℂ) and X_{st}^{ℂ} is the so-called space of complete quadrics. It is smooth and consists of four orbits: the open orbit X, two orbits D_{1}, D_{2} of codimension 1, and a closed orbit Y. The group SU(2, 1) is quasisplit and G^{∗} swaps the two simple roots of S. Thus, the two divisors D_{1} and D_{2} are also swapped by conjugation which means, in particular, they do not contain any real points. For the real points of the entire space this means X_{st}^{ℂ}(ℝ) = X(ℝ) ∪ Y (ℝ). So all G(ℝ)-orbits are either open or closed and the closed one, Y (ℝ), which is of codimension 2. On the other side, the closed orbit of X_{st}(ℝ) is of codimension 1. This shows that

(14.2) ι_{X} : X_{st}(ℝ) → X_{st}^{ℂ}(ℝ)

is surjective but not injective. In fact, X_{st} is the blow-up of X_{st}^{ℂ} in Y with the proper transforms of D_{1} and D_{2} removed. This example also shows that the union of the two k-dense orbits, namely X ∪ Y, is not open but only constructible.

14.3. Remark. In [BJ06], Borel and Ji describe various constructions for compactifying symmetric and locally symmetric spaces. In particular, §II.10 deals with the compactification of semisimple (aka. non-Riemannian or pseudo-Riemannian) symmetric spaces, i.e. real points of X = G/H with G real semisimple and H the fixed point group of an involution, by X_{st}^{ℂ}(ℝ). Unfortunately, the exposition is not quite accurate. In particular Thm. II.10.5 is wrong as stated. Part iv) of the theorem asserts for example that the boundary is of codimension 1 which is not true in general as Example 14.2 iii) above shows. The error seems to occur in the transition of (II.10.1) to (II.10.2) where unjustifiedly “σ” is replaced by “o”. Coincidally, the discussion from eq. (II.10.2) to the end of the section furnishes a description of our X_{st}(ℝ) instead of X_{st}^{ℂ}(ℝ). In particular, [BJ06, Thm. II.10.5] becomes correct when interpreted as a description of the G(ℝ)-orbits in X_{st}(ℝ).

For future reference we briefly make the connection of our theory to the classical theory of symmetric spaces of non-compact type. For this let k = ℝ, let G be a semisimple group, let G := G(ℝ)^{0} be the connected components of the real points, let K ⊆ G be a maximal compact subgroup, and let K ⊆ G be its Zariski closure in G. Then X = G/K is a symmetric space and X := G/K is an ℝ-variety with X ⊆ X(ℝ).

14.4. Proposition. Let X = G/K be a symmetric space. Assume that G is of adjoint type. Then the closure of X in X_{st}(ℝ) is the maximal Satake compactification of X.

Proof. The spherical roots of X are simply the restricted roots of G ([Vus74]). In particular, the G^{∗}-action on them is trivial. It follows that X has isomorphic closures in X_{st}(ℝ) and X_{st}^{ℂ}(ℝ). The closure in the latter is the maximal Satake compactification by [BJ06, I.17.9, II.14.10]. □

It is well-known that the maximal Satake compactification of a symmetric space has the structure of manifold with corners. Recall, that a manifold with corners is a space X with C^{∞}-structure which is locally modeled after ℝ_{≥0}^{k} × ℝ^{n−k} with fixed n and varying k. See, e.g., [Joy12], for a nice introduction to manifolds with corners. The points corresponding to 0^{k} × ℝ^{n−k} ⊆ ℝ_{≥0}^{k} × ℝ^{n−k} are said to have depth k. Let X_{k} := {x ∈ X | dp x = k}. A boundary component is the closure of a connected component of X_{1}.

In this generality, pathologies may occur like selfintersections of boundary components with the teardrop ([Joy12, Fig. 2.1]) being the simplest example. For that reason, Jänich [Jän68] introduced the concept of an ⟨N⟩-manifold which is a manifold with corners together with an assignment of a number in {1, …, N} to every boundary component such that in every point all boundary components meeting there have a different number (see [Jän68, Joy12] for a precise definition). Clearly, the set {1, …, N} can be replaced by any set Σ. In that case, we talk about a Σ-manifold.

Now let X be a Σ-manifold and Y a boundary components of type σ. Then Y is a Σ\{σ}- manifold. More generally, let Y_{0} be a connected component of X_{k} for some k. Then Y_{0} is contained in precisely k boundary components and let the elements of Σ_{0} ⊆ Σ be their types. Then the closure of Y of Y_{0} has the structure of a Σ \ Σ_{0}-manifold. All subsets of the form Y_{0} form a stratification which we call the boundary stratification of X.

One can construct Σ-manifolds by cutting manifolds along hypersurfaces. More precisely, let X′ be a manifold (without boundary) and for each σ ∈ Σ let D_{σ} ⊂ X′ be a smooth hypersurface (possibly disconnected or empty). Assume that the D_{σ} intersect each other transversally. For each x ∈ X′ let dp(x) be the number of D_{σ}’s containing x and let X′_{0} be set of points of depth 0, i.e., the complement of all D_{σ}. Then let X be the manifold with corners obtained by cutting X′ along the D_{σ}. More precisely, a point of X is a pair (x, η) with x ∈ X′ and η is a connected component of X′_{0} near x, i.e., an element of the inverse limit lim_{⟵} π_{0}(U ∩ X′_{0}) where π_{0} denotes the set of connected components and U

runs through all open neighborhoods of x in X′. This limit has 2^{dp x} elements, one for each orthant of a local chart in x. Thus, the map

(14.3) π : X → X_{0} : (x, η) ↦ x is proper, surjective, and the fiber over x has cardinality 2^{dp x}. Furthermore, each boundary component of X maps to a specific hypersurface D_{σ} which makes X into a Σ-manifold.

Now we apply this construction to a homogeneous ℝ-spherical variety X or, more precisely, to its real points X(ℝ). Choose any smooth fan F whose support is Z_{ℝ}(X) and let X′ be the set of real points of X(F). Then X′ is a smooth compactification of X(ℝ). Its boundary is a union of smooth divisors indexed by the set Σ of 1-dimensional cones in F. The manifold X with corners obtained by cutting along the boundary is therefore a compact Σ-manifold. It is equivariant with respect to G(ℝ). One distinguishing feature of X is that each connected component contains exactly one G(ℝ)^{0}-orbit. If that orbit is a (Riemannian) symmetric space then the component is precisely its maximal Satake compactification. Observe that if X is k-wonderful one can choose F to be the standard fan. In that case Σ can be identified with Σ_{ℝ}(X).

14.5. Proposition. Let G := G(ℝ)^{0}. Then the boundary strata of X are precisely the G-orbits.

Proof. The boundary strata are connected and G-stable. Moreover, the action of G on each of them is locally transitive. Hence, they are G-orbits. □

14.6. Examples. In all examples we have k = ℝ and G = SL(2, ℝ). All examples have ℝ-rank 1 and are not horospherical except in the last one. Hence X = G/H is compactified by adding a closed orbit Y.

i) H = SO(2). Then G/H is the space of quadratic forms x_{1}ξ^{2} + 2x_{2}ξη + x_{3}η^{2} with discriminant x_{2}^{2} − x_{1}x_{3} = −1. Therefore X(ℝ) is a 2-sheeted hyperboloid. Its closure in P^{3} is the smooth quadric x_{2}^{2} − x_{1}x_{3} = −x_{0}^{2} with signature (3, 1). Thus X_{st}(ℝ) ≅ S^{2}, the 2-sphere. The boundary Y (ℝ) is given by x_{0} = 0 which defines a great circle S^{1} ⊆ S^{2} which separates X_{st}(ℝ) in two parts X_{±}. Of course, X_{st}(ℝ) can be identified with the Riemann sphere with X_{±} being the upper and lower halfplane. When we cut X_{st}(ℝ) in Y (ℝ) one gets a disjoint union of two closed disks. Thus, G(ℝ) has two open and two closed orbits in X.

ii) H = SO(1, 1). In this case, G/H is the space of quadratic forms x_{1}ξ^{2} + 2x_{2}ξη + x_{3}η^{2} with discriminant x_{2}^{2} − x_{1}x_{3} = +1. Therefore X(ℝ) is a 1-sheeted hyperboloid. Its closure in P^{3} is the smooth quadric x_{2}^{2} − x_{1}x_{3} = x_{0}^{2} with signature (2, 2). Thus X_{st}(ℝ) ≅ P^{1}(ℝ) × P^{1}(ℝ) ≅ S^{1} × S^{1}, a 2-torus. The boundary Y (ℝ) corresponds to the diagonal in S^{1} × S^{1} which does not separate X_{st}(ℝ). Cutting along Y (ℝ) results in a cylinder S^{1} × [0, 1]. Thus G(ℝ) has one open and two closed orbits in X. Observe that X is not the closure of an open G(ℝ)-orbit in X_{st}(ℝ). This example also shows that connected components of X can have more than one closed G(ℝ)^{0}-orbit.

iii) H_{1} = N_{G}(SO(2)) or H_{2} = N_{G}(SO(1, 1)). These two subgroups lead to the same ℝvariety. In fact, X is the space similitude classes of quadratic forms x_{1}ξ^{2} + 2x_{2}ξη + x_{3}η^{2} with discriminant x_{2}^{2} − x_{1}x_{3} ≠ 0. This shows that X_{st}(ℝ) is the projective space P^{2}(ℝ). The boundary is a smooth conic which separates P^{2}(ℝ) in two parts: one, ≅ G/H_{1}, is diffeomorphic to an open disk, the other, ≅ G/H_{2}, is diffeomorphic to an open Möbius strip. Cutting along Y (ℝ) results in disjoint union of a closed disk (compactification of G/H_{1}) and a closed Möbius strip (compactification of G/H_{2}). Thus, G(ℝ) has two open and two closed orbits in X.

iv) In the last example let H = N be a maximal unipotent subgroup. It is horospherical, hence the compactification contains two closed orbits Y_{1} and Y_{2}. More precisely, if we take for F the two halflines in Z_{ℝ}(X) = ℝ (there is no other possibility) then X(F) equals the projective space P^{2} blown up at the origin. Thus, X′ is the non-orientable S^{1}-bundle over S^{1}, i.e., a Klein bottle. The two closed orbits form two sections of this bundle. Since the normal bundle is non-trivial, cutting along each of them does not result in two circles. Instead one gets a single circle which projects as a two-fold cover to S^{1}. Thus X consists of one open orbit and two closed ones. Topologically, X is a closed annulus. Note that the two closed orbits are no longer real points of algebraic varieties. Indeed, the isotropy group is the connected component P (ℝ)^{0} which is a non-algebraic subgroup of SL(2, ℝ).

References

[Ber57] Marcel Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177.

[Bor91] Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991.

[BJ06] Armand Borel and Lizhen Ji, Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.

[BS64] Armand Borel and Jean-Pierre Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164.

[BT65] Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150.

[Bou68] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.

[BP15] Paolo Bravi and Guido Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), 105–123, arxiv:1109.6777.

[Bri87] Michel Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), 189–208.

[Bri90] Michel Brion, Vers une généralisation des espaces symétriques, J. Algebra 134 (1990), 115– 143.

[BLV86] Michel Brion, Domingo Luna, and Thierry Vust, Espaces homogènes sphériques, Invent. Math. 84 (1986), 617–632.

[DCP83] Corrado De Concini and Claudio Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982), Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 1–44.

[Die44] Jean Dieudonné, Sur les fonctions continues p-adiques, Bull. Sci. Math. (2) 68 (1944), 79–95.

[Jän68] Klaus Jänich, On the classification of O(n)-manifolds, Math. Ann. 176 (1968), 53–76.

[Joy12] Dominic Joyce, On manifolds with corners, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 225–258, arxiv:0910.3518v2.

[Kap50] Irving Kaplansky, The Weierstrass theorem in fields with valuations, Proc. Amer. Math. Soc. 1 (1950), 356–357.

[Kem78] George Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), 299–316.

[KKMSD73] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973.

[Kno93] Friedrich Knop, Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind, Math. Ann. 295 (1993), 333–363.

[Kno94] Friedrich Knop, The asymptotic behavior of invariant collective motion, Invent. Math. 116 (1994), 309–328.

[Kno96] Friedrich Knop, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc. 9 (1996), 153–174.

[KKS15a] Friedrich Knop, Bernhard Krötz, and Henrik Schlichtkrull, The local structure theorem for real spherical varieties, Compos. Math. 151 (2015), 2145–2159, arxiv:1310.6390.

[KKS15b] Friedrich Knop, Bernhard Krötz, and Henrik Schlichtkrull, The tempered spectrum of a real spherical space (2015), 52 p., arxiv:1509.03429.

[KKSS14] Friedrich Knop, Bernhard Krötz, Eitan Sayag, and Henrik Schlichtkrull, Volume growth, temperedness and integrability of matrix coefficients on a real spherical space (2014), 20 p.,

           arxiv:1407.8006.

[KKSS15] Friedrich Knop, Bernhard Krötz, Eitan Sayag, and Henrik Schlichtkrull, Simple compactifications and polar decomposition of homogeneous real spherical spaces, Selecta Math. (N.S.) 21 (2015), 1071–1097, arxiv:1402.3467.

[Krä79] Manfred Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen,

Compositio Math. 38 (1979), 129–153.

[Los09] Ivan Losev, Demazure embeddings are smooth, Int. Math. Res. Not. IMRN 14 (2009), 2588– 2596.

[Lun73] Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81–105. Bull. Soc. Math. France, Paris, Mémoire 33.

[LV83] Domingo Luna and Thierry Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), 186–245.

[Mik86] Igor V. Mikityuk, Integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Mat. Sb. (N.S.) 129(171) (1986), 514–534, 591.

[Oda88] Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 15, Springer-Verlag, Berlin, 1988.

[Ros56] Maxwell Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443.

[Ros61] Maxwell Rosenlicht, On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer. Math. Soc. 101 (1961), 211–223.

[SV12] Yiannis Sakellaridis and Akshay Venkatesh, Periods and harmonic analysis on spherical varieties (2012), 291p., arxiv:1203.0039.

[Sat60] Ichirô Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. (2) 71 (1960), 77–110.

[Sat63] Ichirô Satake, On the theory of reductive algebraic groups over a perfect field, J. Math. Soc. Japan 15 (1963), 210–235.

[Spr98] Tonny A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998.

[Sum75] Hideyasu Sumihiro, Equivariant completion. II, J. Math. Kyoto Univ. 15 (1975), 573–605.

[Tit59] Jacques Tits, Sur la classification des groupes algébriques semi-simples, C. R. Acad. Sci. Paris 249 (1959), 1438–1440.

[Tit66] Jacques Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62.

[VS13] Bart Van Steirteghem, Various interpretations of the root system(s) of a spherical variety, Mini-workshop: Spherical Varieties and Automorphic Representations, May 12–18, 2013, Oberwolfach Rep., Vol. 10, 2013, pp. 1464–1467.

[VK78] Èrnest B. Vinberg and Boris N. Kimel′fel′d, Homogeneous domains on flag manifolds and spherical subsets of semisimple Lie groups, Funktsional. Anal. i Prilozhen. 12 (1978), 12–19, 96.

[Vus74] Thierry Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France 102 (1974), 317–333 (French).

[Was96] Benjamin Wasserman, Wonderful varieties of rank two, Transform. Groups 1 (1996), no. 4, 375–403.