$\mathrm{SO}(3)$-homogeneous decomposition of the flag scheme of $\mathrm{SL}_3$ over $\mathbb{Z}\left[1/2\right]$

In this paper, we give $\mathbb{Z}\left[1/2\right]$-forms of $\mathrm{SO}(3,\mathbb{C})$-orbits in the flag variety of $\mathrm{SL}_3(\mathbb{C})$. We also prove that they give a $\mathbb{Z}\left[1/2\right]$-form of the $\mathrm{SO}(3,\mathbb{C})$-orbit decomposition of the flag variety of $\mathrm{SL}_3(\mathbb{C})$.


Introduction
Motivated by applications to special values of automorphic L-functions, Michael Harris, Günter Harder, and Fabian Januszewski started to work on (g, K)-modules over number fields and localization of the rings of their integers in the 2010s ( [9,10,8,19,20,18]).For general theory of (g, K)-modules over commutative rings, see [13,12].Among those, Harris proposed to construct rational models of discrete series representations from the corresponding closed K-orbits in the flag variety and line bundles on them over the field C of complex numbers through the localization.
In [16], we studied descent properties of rings of definition of certain closed K-orbits in the moduli scheme of parabolic subgroups of G for reductive group schemes K ⊂ G in the sense of [2,Définition 2.7].As a consequence, we established real and smaller arithmetic forms of A q (λ)-modules ([16, Section 6.2]).
In this paper, we study rings of definition of the remaining three SO(3, C)-orbits in the complex flag variety of SL 3 .The main result is to establish a Z [1 2]-analog of the SO(3, C)-orbit decomposition of the complex flag variety of SL 3 : Theorem 1.1 (Theorem 3.4, Theorem 3.13, Lemma 3.14, Theorem 4.1).The flag scheme B SL 3 of SL 3 over Z [ 1 2] is decomposed into four affinely imbedded subschemes which are SO(3)-homogeneous in the étale topology in the sense of [1, Proposition et défintion 6.7.3].
1.1.First Perspective: (g, K)-Modules over Commutative Rings.In the representation theory of Lie groups, there are many phenomena which we can understand through real and complex geometry.For instance, there are two geometric realizations of principal series representations of the special linear Lie group SL 3 (R).
For simplicity, we restrict ourselves to (the Harish-Chandra module of) the principal series representation X ps with trivial parameter.We can realize X ps by using the real flag manifold SL 3 (R) B std (R), where B std (R) is the Borel subgroup of SL 3 (R) consisting of upper triangular matrices.The representation X ps can be realized as the space of functions of SL 3 (R) B std (R).According to the Iwasawa decomposition, it can be identified with SO(3, R) B std (R) ∩ SO(3, R), where SO(3, R) is the special orthogonal group.This is a geometric explanation why X ps is induced from the trivial representation of SO(3, R)∩B std (R) as a representation of SO (3, R).We also note that X ps admits a natural real structure by this construction.The other realization is to use the complex flag variety of SL 3 .We define the complex algebraic groups SL 3 (C), B std (C), SO(3, C) in a similar way.Then, we have a unique open SO(3, C)-orbit in the complex flag variety SL 3 (C) B std (C).As a complex SO(3, C)-variety, it is given by SO(3, C) SO(3, C) ∩ B std (C).The representation X ps can be realized as the space of regular functions on this orbit.We can think of these regular functions as global sections of the pushforward of the coordinate ring of this orbit to SL 3 (C) B std (C).These two realizations are related by the analytic continuation.
What happens to other kinds of representations?The Beilinson-Bernstein correspondence tells us that irreducible Harish-Chandra modules of SL 3 (R) with trivial infinitesimal character are obtained by D-modules on the complex flag variety.For example, the fundamental representation A b (0) is attached to the unique closed SO(3, C)-orbit.Can we obtain them from real geometric objects?The transitivity of the SO(3)-action on the real flag manifold of SL 3 tells us that principal series representations should be the only representations which we can obtain from the real flag manifold.
To give another nice answer to the above question, we shall change the point of view.Since there is only one closed SO(3, C)-orbit in SL 3 (C) B std (C), this orbit is stable under formation of the complex conjugation.Hence, it is naturally defined over the real numbers by Galois descent.In fact, we realized this object as the real flag variety of SO (3) in [16,Section 5].More strongly, we gave an equivariant closed immersion of flag schemes over Z [ 1 2].The point is that this real algebraic variety which does exist does not admit real points.Hence, it cannot appear in the formalism of manifolds.It is not an SO(3)-orbit as well by the same reason.However, it is homogeneous in the sense of [1,Proposition et défintion 6.7.3].This appears to be a nice formalism to study "orbit-like" objects over general base fields (rings).This real algebraic subvariety gives rise to a real form of the fundamental series representation A b (0) via localization ([16, Section 6.2]).The existence of the real form of A b (0) was proved algebraically in [19,Theorem 7.3].
The main purpose of this paper is to construct Z [1 2]-forms of the remaining SO(3, C)-orbits explicitly.Philosophically, this result says that all irreducible admissible representations of SL 3 (R) (with regular infinitesimal character) are controlled by real algebro-geometric objects (cf.[16, Definition 3.5.1,Corollary 4.2.2]).We will even control representations over Z[1 2] at the level of orbits.As a consequence of our direct computational approach, we find that the Z[1 2]-forms of the orbits are represented by affine schemes except the closed one.This fact makes the study of the global section modules of the direct and proper direct image twisted D-modules (cf.[14, Appendix A]).1.2.Second Perspective: Combinatorics of Orbit Decomposition.Traditional problems in the theory of combinatorics of orbit decomposition are summarized as follows: Problem 1.2.Let k be a commutative ring, and X be a k-scheme, equipped with an action of a group scheme K over k.Classify the K(F )-orbits in X(F ) for a field F over k.
The ring k in Problem 1.2 should be usually a certain localization of the ring of integers or the field of rational numbers.The description of the classification depends on F in most cases.For instance, the number of orbits may differ by F .On the other hand, one can find by experience that some parts of the classifications are independent of F .In fact, we (possibly implicitly) happen to solve equations on the course of classification by nature of the algebro-geometric formulation.Among those equations, some may be defined over k.As far as such equations are concerned, there are two factors why the dependence happens: 1. Existence of solutions; 2. Dependence on choice of solutions (Galois symmetry).Hence, the larger rings k ′ we replace k by, the more uniform classification we obtain.
In this paper, we suggest the three things.
1. We quit solving equations whose solutions essentially depend on fields F . 2. Attach a Galois extension k → k ′ with Galois group Γ to an independent equation.Then, we find Γ-invariant parts of a decomposition of X(F ′ ) for fields F ′ over k ′ .3. We do these things at the level of schemes to get a decomposition of X into K-invariant subspaces Z λ as a set.Decomposition into subspaces Z λ may not give a complete answer to Problem 1.2 but instead that we obtain a uniform decomposition of X(F ) in F into K(F )invariant subsets Z λ (F ).In fact, Z λ (F ) can have multiple K(F )-orbits.We explain below that Z λ (F ) can be also empty.
Problem 1.3.Decompose X into smaller pieces of K-invariant subspaces represented by k-schemes.
We would like to suggest a basic strategy to get Z λ , which consists of four phases.
1. Take a Galois extension k Study the Galois orbit of the set of K ⊗ k k ′ -orbits to get subspaces over k by the Galois descent.The last phase says that we prove that a K ⊗ k k ′ -orbit admits a k-form by showing that it admits a Galois action; otherwise, we get a k-scheme by joining K ⊗ k k ′orbits.For example, see [16,Example 5.2.22].We can regard that [16, Proposition 5.1.1]is the case that the Galois orbit has two elements . This is the reason why Z λ (k) may be empty.This observation tells us that even if the Galois orbit is a singleton, the expected "orbits" (subspaces) Z λ may not have a base point.The author believes that the key ingredients to achieve each phase of our program lie in the combinatorial study of Problem 1.2.More specifically, we shall think of the K C -orbit decomposition of flag varieties.For this, recall the Matsuki classification ( [22], see also [25] for similar results over algebraically closed fields of characteristic ≠ 2).Let G be a connected real reductive algebraic group, and g 0 be its Lie algebra.Let K be a maximal compact subgroup of the group G(R) of real points of G, and K C be its complexification.Let θ be the Cartan involution relative to K.For a θ-stable Cartan subalgebra h 0 ⊂ g 0 , set where N and Z denote the normalizer and the centralizer respectively.Let B G be the flag variety of G.There exists a bijection where h 0 runs through fixed representatives of K-conjugacy classes of θ-stable Cartan subalgebras.
Example 1.4.The special orthogonal group SO(2, R) acts on the real projective line P 1 (R) transitively.On the other hand, there are three SO(2, C)-orbits in P 1 (C).This difference suggests us to decompose P 1 into the SO(2)-orbit containing real points and the others which are closed.Since the complex conjugation switches the two closed orbits { √ −1} and {− √ −1} in C ∪ {∞} ≅ P 1 (C), the two SO(2, C)-orbits are not defined over R but their union is.Technically, observe that the Galois action Example 1.5.Put G = SL 3 .The compact Lie group SO(3, R) acts transitively on B SL 3 (R), and there are four SO(3, C)-orbits in B SL 3 (C).One should separate the SO(3)-orbit containing real points and the others.We can see that the unique closed SO(3, C)-orbit is defined over the real numbers from [16,Proposition 5.5.2].The key idea for its proof is that a corresponding Borel subgroup of SL 3 is determined by a regular cocharacter to SO(3, C), and that its conjugation is expressed by the action of w 0 = diag(1, −1, −1) ∈ SO(3, R).That is, w 0 plays the role of the Galois action.This should happen since w 0 = −1 as an element of the Weyl group of SO(3, C).The main idea of this paper is as follows: w 0 also plays the role of the Galois action to certain Borel subgroups corresponding to the other two orbits since the Weyl group of SL 3 (C) does not contain −1.For example, we can explain it at the Lie algebra level as follows: set and h fun,0 ⊂ sl(3, R) be the fundamental Cartan subalgebra containing t fun,0 .Let h fun,0 = t fun,0 ⊕ a fun,0 denote the Cartan decomposition.Then, w 0 acts on t fun,0 by −1.Since w 0 ≠ −1 and w 2 0 = 1 as an element of the Weyl group of SL 3 , and dim a fun,0 = 1, w 0 acts on a fun,0 by 1.Since an element of h fun determining each of the above two Borel subgroups belongs to √ −1t fun,0 ⊕ a fun,0 , the action of w 0 on this element coincides with the conjugate action.In this paper, we improve this idea to work over Z [ 1 2].
Like the last part of Example 1.5, we will have to analyze combinatorial results carefully in general to get hints from them.The author is working in progress on a Z [1 2]-analog of the K C -orbit decomposition of the flag varieties for higher rank classical groups by proactive use of [25].There is also a more general formalism: Problem 1.6.Let G be a reductive group scheme over k, and K be a closed subgroup scheme of G. Decompose the moduli scheme of parabolic subgroups of G into smaller pieces of K-invariant subschemes.
Note that symmetric subgroups in the sense of [15, Example 3.1.2]will be typical examples of K.

Organization of this Paper.
In Section 2, we collect some general results on decompositions of schemes to verify ideas of Section 1.2.In Appendix A, we collect some general results on descent techniques in abstract algebraic geometry.They will be helpful when we try to find forms of orbit decompositions of schemes, based on our general program of Section 1.2.In Section 3, we use them to construct Z [1 2]-forms of the SO(3, C)-orbits in B SL 3 (C).We also give their moduli descriptions.We cost many pages to this section for confirming the moduli descriptions (particularly Theorem 3.4 and Lemma 3.9) because we perform the Gram-Schmidt process and its versions explicitly and independently to the general matrices of size 3 × 3.As a result, we find explicit formulas of the defining relations of the SO(3)-homogeneous subschemes of B SL3 .We also cost pages to the proof of the isomorphism SO( 3 in Proposition 3.6, based on the sheaf-theoretic definition of SO(3) SO (2).In fact, we construct the inverse of the canonical map from left to right étale locally.On these courses and the formulations to these results, we meet many matrices of size 3 × 3. Section 4 is devoted to the conclusion.In Appendix B, we use ideas of this paper to give a reasonable realization of the flag scheme of SO(3) over Z [ 1 2].We also establish a Z [1 2]-form of the SO(3, C)-orbit decomposition of a proper complex partial flag variety of SL 3 .We again meet matrices of size 3 × 3 which take large space.Totally, the many pages are needed for the case-by-case studies of the orbits through computations of large matrices.
1.4.Notation.We follow [15] for the notations and conventions.In the below, we list additional notations.
To save space, we denote vertical vectors in R 3 as (a 1 a 2 a 3 ) T for a commutative ring R.
For a field F , we denote its algebraic closure by F .
Let k be a commutative ring.Let CAlg red k denote the full subcategory of CAlg k consisting of reduced k-algebras.A sheaf on the big affine étale site over k will be called an étale k-sheaf, which will be identified with a copresheaf on CAlg k in this paper.If necessary, see [24,Section 2] for the general formalism of sheaves on sites.We will regard k-schemes as étale k-sheaves by the restricted Yoneda functor (see [24,Theorem 4.1.2]).For a copresheaf F on CAlg k and a k-algebra R, we will sometimes identify an element x ∈ F (R) with a natural transformation Spec R → F of copresheaves by the Yoneda lemma.It is evident by definitions that for any homomorphism k → k ′ of commutative rings, the base change − ⊗ k k ′ sends étale k-sheaves to étale k ′ -sheaves.
For the formalism of quotient by group schemes, we adopt the quotient in the étale topology.That is, let k be a commutative ring, G be a group k-scheme, and H ⊂ G be a subsegroup k-scheme.Then, G H is the étale sheafification of the k-space defined by R ↦ G(R) H(R).See [1] for the general formalism.Although G H is not represented by a k-scheme in general, we will see that the quotients appearing in this paper are representable.
For a reductive group scheme G over a scheme S in the sense of [2, Définition 2.7], the moduli scheme of Borel subgroups of G will be denoted by B G (see [3,Corollaire 5.8

.3 (i)]).
Let X be a scheme over k.Let X denote the underlying set of X.We will use a similar notation for morphisms of schemes.For a point x ∈ X , let κ(x) denote the residue field at x.If we are given a field F over k and an element x ∈ X(F ), we will also denote the residue field of X at the image of the point of Spec F along the map x ∶ Spec F → X by the same symbol κ(x).For a point x ∈ X , we denote the geometric point Spec κ(x) → Spec κ(x) → X by x.
Let B std denote the Borel subgroup of SL 3 over Z [1 2] consisting of upper triangular matrices.

Remark on Set-Theoretic Decomposition of Schemes
Fix k as a commutative ground ring.Let X be a k-scheme, and {i λ ∶ Z λ → X} λ∈Λ be a small set of monomorphisms of k-schemes.We say that {Z λ } exhibits a settheoretic decomposition of X if the canonical map ∐ λ∈Λ Z λ → X is a bijection.The goal of this paper is to decompose the flag scheme B SL3 into SO(3)-invariant subschemes as a set.In this section, we note some general results on set-theoretic decompositions of schemes to relate them with the results of combinatorics.
Theorem 2.1.The following conditions are equivalent: The canonical map ∐ λ∈Λ Z λ (F ) → X(F ) be a bijection for every field F over k.(c) The canonical map ∐ λ∈Λ Z λ (F ) → X(F ) be a bijection for every algebraically closed field F over k.
Proof.It is clear that (b) implies (c).Suppose that {Z λ } satisfies (c).We prove {Z λ } exhibits a set-theoretic decomposition of X.Let x ∈ X .Since the map is a bijection, there exist an index λ and a unique element . Hence it will suffice to show that the images of i λ are disjoint in X .Let λ, µ ∈ Λ be distinct indices, z λ ∈ Z λ , and x Then, we can find an algebraically closed field This shows that x Spec F is in the images of both i λ and i µ .It contradicts to the assumption that {Z λ } exhibits a fieldwise decomposition.This proves the implication (c)⇒(a).Finally, suppose that {Z λ } satisfies (a).We wish to show that {Z λ } satisfies (b).Fix a field F over k.To see that the map (i λ ) ∶ ∐ λ∈Λ Z λ (F ) → X(F ) is injective, it will suffice to show that the images of Z λ (F ) in X(F ) are disjoint since i λ are monomorphisms.Let λ, µ ∈ Λ be distinct indices, z λ ∈ Z λ (F ), and z µ ∈ Z µ (F ).Let us wirte the corresponding elements in Z λ and Z µ by the same symbols z λ and z µ respectively.Suppose i λ (z λ ) = i µ (z µ ).Then, we have i λ (z λ ) = i µ (z µ ), which contradicts to the condition (a) since λ ≠ µ.
The proof is completed by showing that Then, the corresponding element x ∈ X can be expressed as i λ (z λ ) for some index λ and an element z λ ∈ Z λ .Let Z λ,x be the fiber of i λ at x ∈ X .

Consider the commutative diagram
The left vertical arrow in this diagram is an isomorphism by [6, Remarque 8.11.5.1] since Z λ,x is nonempty.Hence the morphism i λ sends the element of Z λ (F ) given by Spec Proof.The first part is clear from the definition of base changes in terms of copresheaves.Let k ′ be a faithfully flat k-algebra.Let F be an algebraically closed field over k.Then, k ′ ⊗ k F is nonzero by the hypothesis on k ′ .Choose an algebraically closed field where the bottom arrow is a bijection since {Z λ ⊗ k k ′ } exhibits a fieldwise decomposition of X ⊗ k k ′ .The left upper vertical arrow is injective since the embedding F → F ′ is faithfully flat.Therefore, the upper horizontal arrow is injective.Suppose that we are given an element x ∈ X(F ).Then, there exist an index λ ∈ Λ and Let K be a group scheme over k, X be a scheme over k, equipped with an action of K. Let {i λ ∶ Z λ → X} be a set-theoretic decomposition of X. Suppose that for each index λ, the following conditions are satisfied: (i) The action of K on X restricts to Z λ .
(ii) Every geometric fiber of Z λ is nonempty and locally of finite type.
(iii) For every algebraically closed field F over k, K(F ) acts transitively on Z λ (F ).
Such a set-theoretic decomposition is minimal in the following sense: Corollary 2.3.For each λ, suppose that we are given a set-theoretic decomposition {Z ′ λµ ↪ Z λ }.Then, each set {Z ′ λµ ↪ Z λ } is a singleton if the following conditions are satisfied for every pair (λ, µ): (i) The action of K on X restricts to Z ′ λµ .(ii) Every geometric fiber of Z ′ λµ is nonempty and locally of finite type.Proof.This is an immediate consequence of Hilbert's Nullstellensatz.[16] by the Galois descent.In this section, we construct Z [1 2]-forms of the remaining three SO(3, C)-orbits.We also give their moduli descriptions.To achieve them, remark that if we are given a Borel subgroup of SL 3 over a Z [1 2]-algebra R, the stabilizer of the action of SO(3) at B ∈ B SL 3 (R) is SO(3) ∩ B since the normalizer of B coincides with itself ([3, Corollaire 5.3.12 and Proposition 5.1.3]).
By definitions, we have The equivalences are now obvious.

Open Orbit
]), and i op ∶ U ↪ B SL 3 be the corresponding embedding.
Lemma 3.2.The group scheme SO(3) ∩ B std is a finite étale diagonalizable group scheme.We next show that U is the locus in B SL 3 where we can apply the Gram-Schmidt process in order to prove that i op is an affine open immersion.In other words, U can be identified with the moduli scheme of flags where the Gram-Schmidt process works.

Proof. Consider the embedding Spec
Property 3.3.Let F be an algebraically closed field of characteristic different from 2. We say that a full flag We remark that a nonzero vector v ∈ V such that (v 1 , v) = 0 is unique up to nonzero scalar since dim V 2 = 2. Theorem 3.4.
(1) Let R be an arbitrary Z [1 2]-algebra.For a Borel subgroup B ∈ B SL 3 (R), the following conditions are equivalent: k), and v 3 (k) are orthogonal to each other.Since det is alternating multilinear, we have det k = det g = 1.Therefore, we obtain As a consequence, k belongs to SO(3, R).Since belongs to the image of i op .This shows (1).For (3), we show that for every test affine scheme Spec R over Z [1 2] and a morphism Spec R → B SL 3 , the base change Spec R × BSL 3 U → Spec R is an affine open immersion.Let B be the Borel subgroup corresponding to Spec R → B SL3 .Since the assertion is étale local in Spec R, we may again assume B = gB std g −1 for some g ∈ SL 3 (R).For a ring homomorphism f ∶ R → S, the following conditions are equivalent: Therefore, U × BSL 3 Spec R is isomorphic to Spec R c1c2 .This completes the proof.
. Then, SO(3) acts on Spec A by the restriction of the canonical action of SO(3) on Spec Z 1 2, Proposition 3.6.
Proof.We only prove the assertions for B 1 .The other is proved in a similar way.For (1), we may prove the equality (1)

SO(3)-HOMOGENEOUS DECOMPOSITION OF THE FLAG SCHEME OF SL3 OVER Z [1 2]11
by passing to the conjugate by g −1 1 .Let R be an arbitrary Z 1 2, √ −1 -algebra.Then, the computation of µ 2 in [15, Section 3.2] implies We next prove (2).It is easy to show that the stabilizer subgroup of SO( 3) at We thus obtain a monomorphism To see that it is an isomorphism, it will suffice to show that the identity map of A is étale locally expressed as g(0 0 1) T for some g ∈ SO(3).Since the affine schemes Spec A 1 x 2 + y 2 , Spec A 1 y 2 + z 2 , and Spec A 1 √ z 2 + x 2 form an étale cover of Spec A. Set Then, we have g xy (0 0 1) T = (x y z) T on this étale locus.One can find similar matrices sending (0 0 1) T to (x y z) T on the other étale loci.This shows (2).For (3), observe that the automorphism σ on A naturally extends to A 1 x 2 + y 2 by σ( x 2 + y 2 ) = x 2 + y 2 .By construction of i 1 , the Borel subgroups of SL 3 corresponding to i 1 on Spec A 1 . Since σ(g xy g 1 ) = g xy w 0 σ(g 1 ) = g xy g 1 w 0 for w 0 ∶= diag(1, −1, −1), i 1 is Γ-invariant on this étale locus.Similar arguments work on the other loci.This completes the proof.
Remark 3.7.The argument of (2) clearly works if we replace A by

Put an action of SO(3) on Spec
).In view of Theorem A.3 (3) and Proposition A.4, we obtain two SO(3)-equivariant monomorphisms Z j ∶= Spec A Γ ↪ B SL 3 , which we denote by the same symbol i j .For a digression, we describe A Γ : Proposition 3.8.Define an action of SO(3) on in a similar way to that on Spec A. Then, there is an The resulting morphism is clearly an SO(3)-equivariant isomorphism.Since the action of SO(3) on We demonstrate similar computations to the proof of Theorem 3.4 (b)⇒(a) to give moduli descriptions of Z 1 and Z 2 , and to prove that i 1 and i 2 are affine immersions: Lemma 3.9.Let R be a Z [1 2]-algebra, and g ∈ SL 3 (R).Set B = gB std g −1 .
(1) The Borel subgroup B belongs to the image of i 1 if and only if c 1 (g) = 0 and c 3 (g) ∈ R × .(2) The Borel subgroup B belongs to the image of i 2 if and only if c 1 (g) ∈ R × and c 2 (g) = 0.
Proof.We remark that all conditions are local in the étale topology by Lemma A.5 and Lemma A.1.Hence we may replace R by R √ −1 to assume that R is a .

One can easily check
Suppose that B belongs to the image of i 1 .To prove that c 1 (g) = 0 and c 3 (g) ∈ R × , we may assume that there exists k ∈ SO(3, R) such that B = kB 1 k −1 .Then, [3, Corollaire 5.3.12 and Proposition 5.

SO(3)-HOMOGENEOUS DECOMPOSITION OF THE FLAG SCHEME OF SL3 OVER Z [1 2]13
By a similar argument to Theorem 3.4, k belongs to SO(3, R).In view of Lemma 3.1, we may replace g by k −1 g to assume v Then, we have gb = g ′ 1 since det g = 1.The assertion now follows from We next prove (2).The "only if" direction follows by a similar argument to (1).Suppose that c 1 (g) ∈ R × and c 2 (g) = 0. We may pass to the étale cover Spec R √ r 1 , and multiply to g from the right side to assume c 1 = 1 and (v 1 , v 2 ) = 0. Since c 2 = 0, we have We then replace g by It is clear that the equalities Then, k belongs to SO(3, R).Replace g by k −1 g to assume Then, we have The assertion now follows from This completes the proof.
The first part of (2) follows since these three subschemes do not admit Z [1 2]points.The latter assertion of ( 2) is evident by the constructions of these three subschemes.
(2) Put an action of Γ on R ⊗ k k ′ by the base change.Then, we have a canonical Γ-equivariant isomorphism R ⊗ k k ′ ≅ S which we denote by (j, g).In particular, the canonical homomorphism R → R ⊗ k k ′ is a Galois extension of Galois group Γ for this action.
(3) Let α ∈ X(S) Γ ⊂ X(S) = (X ⊗ k k ′ )(S).Let ᾱ ∈ X(R) be an element satisfying X(j)(ᾱ) = α which uniquely exists by Galois descent.We denote the corresponding morphisms α ∶ Spec S → X ⊗ k k ′ and Spec R → X by the same symbols α and ᾱ respectively.Then, the composite map coincides with the base change of ᾱ.
Proof.Part ( 1) and ( 2) are proved in a similar way to [4, Theorems 14.86 and 14.85].For (3), it will suffice to compare the images of id Then, the image along ᾱ ⊗ k k ′ is computed as Hence the two images coincide.This completes the proof.
For the relation of descent of spaces and actions of groups, the following result is useful: Proposition A.4.Let k → k ′ be a faithfully flat ane étale homomorphism of commutative rings.Let i ∶ X → Y be a monomorphism of étale k-sheaves, and K be a group étale k-sheaf.Suppose that K acts on G.If the induced action of K ⊗ k k ′ on Y ⊗ k k ′ restricts to X ⊗ k k ′ , the action of K on Y restricts to X.Moreover, the base change of the resulting action on X coincides with the given action on X ⊗ k k ′ .
Proof.We denote the action map K ×Y → Y (resp. be the canonical homomorphism onto the jth factor.Take g ∈ K(R) and x ∈ X(R).We check the descent condition for φ ′ ○ (K × X)(l)(g, x) along l.Observe that for j ∈ {1, 2}, we have

We used the hypothesis that
Since i is monic, we have X(ι 1 )○φ ′ ○(K ×X)(l)(g, x) = X(ι 2 )○φ ′ ○(K ×X)(l)(g, x).Since X is an étale sheaf, there is a unique element φ(g, x) ∈ X(R) such that X(l)(φ(g, x)) = φ ′ ○ (K × X)(l)(g, x).To see that this gives the restriction of the action of K on Y to X, notice that Since l is étale and faithfully flat, Y (l) is injective.Therefore, we get the equality i(φ(g, x)) = ψ(g, i(x)) as desired.In particular, φ determines the restriction of the action of ψ (use the hypothesis that i is monic for the naturality of φ).
The proof is completed by showing φ Since i is monic, we have φ ′ (g, x) = φ(g, x).
In Section 3.3, we consider orbit sheaves in the scheme B SL 3 .To compute their moduli description, the following observation is useful since many objects are local in the étale topology: Lemma A.5.Let k be a commutative ring, and f ∶ R → R ′ be a faithfully flat étale homomorphism of k-algebras.Let i ∶ F → G be a monomorphism of étale k-sheaves, and y ∈ G(R).Then, y belongs to the image of i if and only if G(f )(y) is so.
Proof.The "only if" direction is clear.To see the "if" direction, suppose that we are given an element x ′ ∈ F (R ′ ) such that i(x ′ ) = G(f )(y).For j ∈ {1, 2}, let ι j denote the caonical homomorphism R ′ → R ′ ⊗ R R ′ onto the jth factor.Then, we have In [3], the moduli space B SO(3) of Borel subgroups of SO( 3) is proved to be represented by a projective scheme over Z [ 1 2].The key result for its proof is that the moduli space of subgroups of SO(3) of type (R) is represented by a quasi-projective scheme.In this appendix, we realize B SO(3) as a moduli subspace of P 2 = Proj Z [1 2, x, y, z] by using ideas in this paper.To be precise, set To achieve them, let us recall a moduli description of P 2 : for a commutative Z [1 2]-algebra R, P 2 (R) is naturally bijective to the set of equivalence classes of line bundles L on Spec R with generators (a 1 , a 2 , a 3 ).If L is the structure sheaf O Spec R of Spec R, we will denote it by [a 1 a 2 a 3 ] T .

SO(3)-HOMOGENEOUS DECOMPOSITION OF THE FLAG SCHEME OF SL3 OVER Z [1 2]19
Property B.1.Let F be an algebraically closed field of characteristic different from 2. We say that a one dimensional subspace Notice that for a field F (of characteristic ≠ 2), F -points of P 2 are identified with one dimensional subspaces of F 3 by the correspondence The open subscheme U ′ is the moduli space of one dimensional subspaces Since the condition (O)' is stable under the action of SO(3), U ′ is an SO( 3)invariant open subscheme of P 2 .
Proof.Take the SO(3)-orbit attached to [0 0 1] T to get a monomorphism To see that this morphism is epic, take an arbitrary Z [1 2]-algebra R and an Rpoint V ∈ U ′ (R).We wish to show that V in the image of the above morphism.We may identify V with a pair of a line bundle L on Spec R and its global sections (a 1 , a 2 , a 3 ) of L which (locally) generate L by [11,Theorem 7.1].
Since the assertion is Zariski local by Lemma A.5, we may assume L is the coordinate ring of Spec R. In particular, a 1 , a 2 , a 3 are generators of R as an Rmodule with a 2 1 + a 2 2 + a 2 3 ∈ R × .We may replace R by R a 2 1 + a 2 2 + a 2 3 to assume a 2 1 + a 2 2 + a 2 3 = 1.Then, use the matrices g xy , g yz , g zx in the proof of Proposition 3.6 to see that there étale locally exists g ′ ∈ SO(3, R) such that g ′ [0 0 1] T = [a 1 a 2 a 3 ] T .This completes the proof.
We can define an SO(3)-equivariant morphism where the map SL 3 B std → SL 3 P is the quotient map attached to B std ⊂ P .
To prove this, we realize i ′ as composition of morphisms between quotient spaces over Z 1 2, √ −1 .Set are smooth surjective (see [21,4 Example 3.37] for the smoothness of Z ′ ).Since the reducedness is local in the smooth topology, B SO(3) and Z ′ are reduced.In view of the uniqueness of reduced structure on the underlying set of a closed subscheme, it will suffice to show that i ′ is surjective onto Z ′ .Let F be an algebraically closed field over Z 1 2, √ −1 .Notice that Since SO(3, F ) acts transitively on B SO(3) (F ), i ′ F factors through Z ′ (F ).Let F v ∈ Z ′ (F ).Then, we can choose a vector u ∈ F 3 such that (v, u) ≠ 0. By the proof of Lemma 3.9 (1), there is a matrix k ∈ SO(3, F ) such that F k(1 − √ −1 0) T = F v. This completes the proof.

1 →
(a) B belongs to the image of i op .(b) The flags corresponding to all geometric fibers of B satisfy Property (O).(2) The sheaf U is represented by an affine Z [1 2]-scheme.(3) The morphism i op is an affine open immersion.Proof.Part (2) follows from (1).In fact, U is a sheaf in the fpqc topology by (1) since the condition (b) is local in the fpqc topology (use Lemma A.1 if necessary).In particular, U is the fpqc quotient of SO(3) by SO(3)∩B std .Part (2) then follows from Lemma 3.2 and [5, Corollaire 5.6] (or [17, I.5.6 (6)]).Let R be a Z [1 2]-algebra, and B ∈ B SL 3 (R).Suppose that B satisfies (a).Since (b) is local in the étale topology, we may pass to an étale cover to assume that there exists an element k ∈ SO(3, R) such that B = kB std k −1 by [17, I 5.4 (4), 5.5, 5.6 (2)].Since B std satisfies (b), B does so by k ∈ SO(3, R).Conversely, suppose that B satisfies (b).Since (a) is local in the étale topology by Lemma A.5, we may assume that B = gB std g −1 for some matrix g ∈ SL 3 (R) by [3, Proposition 5.1.3and Corollaire 5.3.12].Since c 1 = c 1 (g) is nonzero at every (geometric) fiber, c 1 is a unit of R. Let us pass to the étale cover Spec R √ c Spec R. Since B = gbB std b −1 g −1 for every element b ∈ B std (R), one can replace g by (a) f belongs to (U × BSL 3 Spec R)(S) ⊂ (Spec R)(S); (b) f (c 1 ) and f (c 2 ) are nonzero in each residue field of Spec S; (c) f (c 1 ) and f (c 2 ) are units of S. (d) The homomorphism f descends to a map R c1c2 → S.

Remark 3 . 5 .
The formula c 1 c 2 appear more directly by the pull back of this open subscheme along the projection SL 3 → SL 3 B std ≅ B SL 3 .That is, the open subscheme of SL 3 obtained by this base change is defined by c 1 c 2 .Similar results hold in the forms of the other orbits below.3.3.Middle Subschemes Z 1 and Z 2 .We next construct by Galois descent Z [1 2]-forms of the two orbits which are neither open or closed.Let Γ = Z 2Z, and σ denote its nontrivial element.