EQUIVARIANT MODELS OF SPHERICAL VARIETIES
MIKHAIL BOROVOI
WITH AN APPENDIX BY

arXiv:1710.02471v3 [math.AG] 17 Feb 2018

GIULIANO GAGLIARDI

Abstract. Let G be a connected semisimple group over an algebraically closed field
k of characteristic 0. Let Y = G/H be a spherical homogeneous space of G, and let
Y ′ be a spherical embedding of Y . Let k0 be a subfield of k. Let G0 be a k0 -model
(k0 -form) of G. We show that if G0 is an inner form of a split group and if the subgroup
H of G is spherically closed, then Y admits a G0 -equivariant k0 -model. If we replace
the assumption that H is spherically closed by the stronger assumption that H coincides
with its normalizer in G, then both Y and Y ′ admit G0 -equivariant k0 -models, and these
models are unique.

Contents
0.

Introduction

2

1.

Semi-morphisms of k-schemes

4

2.

Semi-morphisms of G-varieties

9

3.

Quotients

11

4.

Semi-morphisms of homogeneous spaces

12

5.

k-automorphisms of homogeneous spaces

14

6.

Equivariant models of G-varieties

15

7.

Spherical homogeneous spaces and their combinatorial invariants

17

8.

Action of an automorphism of the base field
on the combinatorial invariants of a spherical homogeneous space

21

9.

Equivariant models of automorphism-free spherical homogeneous spaces

24

10.

Equivariant models of spherically closed spherical homogeneous spaces

25

11.

Equivariant models of spherical embeddings
of automorphism-free spherical homogeneous spaces

31

Appendix A. Algebraically closed descent
for spherical homogeneous spaces
Appendix B.

The action of the automorphism group on the colors
of a spherical homogeneous space

References

32
34
36

Date: February 20, 2018.
2010 Mathematics Subject Classification. 14M27, 14M17, 14G27, 20G15.
Key words and phrases. Spherical variety, spherical homogeneous space, spherical embedding, color,
model, form, semi-automorphism.
This research was partially supported by the Hermann Minkowski Center for Geometry and by the
Israel Science Foundation (grant No. 870/16).
1

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MIKHAIL BOROVOI

WITH AN APPENDIX BY

GIULIANO GAGLIARDI

0. Introduction
Let G be a connected semisimple group over an algebraically closed field k of characteristic 0. Let Y be a G-variety, that is, an (irreducible) algebraic variety over k together
with a morphism
θ : G ×k Y → Y
defining an action of G on Y . We say that (Y, θ) is a G-k-variety or just that Y is a
G-k-variety.
Let k0 ⊂ k be a subfield. Let G0 be a k0 -model (k0 -form) of G, that is, an algebraic
group over k0 together with an isomorphism of algebraic k-groups
∼

κG : G0 ×k0 k → G.
By a G0 -equivariant k0 -model of the G-k-variety (Y, θ) we mean a G0 -k0 -variety (Y0 , θ0 )
∼
together with an isomorphism κY : Y0 ×k0 k → Y such that the diagram (26) commutes,
see Section 6 below.
From now on till the end of the Introduction we assume that Y is a spherical homogeneous space of G. This means that Y = G/H (with the natural action of G) for some
algebraic subgroup H ⊂ G and that a Borel subgroup B of G has an open orbit in Y .
Let Y ֒→ Y ′ be a spherical embedding of Y = G/H. This means that Y ′ is a G-kvariety, that Y ′ is a normal variety, and that Y ′ contains Y as an open dense G-orbit.
Then B has an open dense orbit in Y ′ .
Inspired by the works of Akhiezer and Cupit-Foutou [ACF14], [Akh15], [CF15], for a
given k0 -model G0 of G we ask whether there exist a G0 -equivariant k0 -model Y0 of Y and
a G0 -equivariant k0 -model Y0′ of Y ′ .
Since char k = 0, by a result of Alexeev and Brion [AB05, Theorem 3.1], see Knop’s
MathOverflow answer [Kn17b] and Appendix A below, the spherical subgroup H of G
is conjugate to some (spherical) subgroup defined over the algebraic closure of k0 in k.
Therefore, from now on we assume that k is an algebraic closure of k0 . We set Γ =
Gal(k/k0 ) (the Galois group of k over k0 ).
Let T be a maximal torus of G contained in a Borel subgroup B. We consider the
Dynkin diagram Dyn(G) = Dyn(G, T, B). The k0 -model G0 of G defines the so-called
∗-action of Γ = Gal(k/k0 ) on the Dynkin diagram Dyn(G), see Tits [Tits66, Section 2.3,
p. 39]. In other words, we obtain a homomorphism
ε : Γ → Aut Dyn(G).
The k0 -group G0 is called an inner form (of a split group) if the ∗-action is trivial, that
is, if εγ = id for all γ ∈ Γ. For example, if G is a simple group of any of the types
A1 , Bn , Cn , E7 , E8 , F4 , G2 , then any k0 -model G0 of G is an inner form, because in these
cases Dyn(G) has no nontrivial automorphisms. If G is a split k-group, then of course G
is an inner form.
Let D(Y ) denote the set of colors of Y = G/H, that is, the (finite) set of the closures of
B-orbits of codimension one in Y . A spherical subgroup H ⊂ G is called spherically closed
if the automorphism group AutG (Y ) = NG (H)/H acts on D = D(Y ) faithfully, that is, if
the homomorphism
AutG (Y ) → Aut(D)
is injective. Here NG (H) denotes the normalizer of H in G.
Example 0.1. Let k = C, G = PGL2,C , H = T (a maximal torus), Y = G/T . Then
|NG (T )/T | = 2, and the spherical homogeneous space Y of G has exactly two colors, which
are swapped by the non-unit element of NG (T )/T . We see that the subgroup H = T of
G is spherically closed.

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

3

Theorem 0.2. Let G be a connected semisimple group over an algebraically closed field
k of characteristic 0. Let Y = G/H be a spherical homogeneous space of G. Let k0 be a
subfield of k such that k is an algebraic closure of k0 . Let G0 be a k0 -model of G. Assume
that:
(i) G0 is an inner form,
(ii) H is spherically closed.
Then Y admits a G0 -equivariant k0 -model Y0 .
Theorem 0.2 is a special case of the more general Theorem 10.2 below, where instead of
assuming that G0 is an inner form, we assume only that for all γ ∈ Γ the automorphism εγ
of Dyn(G) preserves the combinatorial invariants (Luna-Losev invariants) of the spherical
homogeneous space Y . Note that we have to assume that char k = 0 because in the proof
we use Losev’s uniqueness theorem [Lo09, Theorem 1], which has been proved only in
characteristic 0. Note also that Y = G/H might have no G0 -equivariant k0 -models if H
is not spherically closed, see Example 10.11 below.
Theorem 0.2 was inspired by Theorem 1.1 of Akhiezer [Akh15] and by Corollary 1 of
Cupit-Foutou [CF15, Section 2.5].
Note that the G0 -equivariant k0 -model Y0 in Theorem 0.2 is in general not unique. The
following theorem is a special case of the more general theorem 10.13 below.
Theorem 0.3. In Theorem 0.2 the set of isomorphism classes of G0 -equivariant k0 -models
of Y = G/H is naturally a principal homogeneous space of the abelian group
H 1 (Γ, AutG (Y )) ≃ (Hom(Γ, S2 ))Ω

(2)

.

Here S2 is the symmetric group on two symbols (isomorphic to Z/2Z), Ω(2) = Ω(2) (Y ) is
(2)
the finite set defined in Section 7 below (before Definition 7.3), and ( . )Ω denotes the
group of maps from the set Ω(2) to the group in the parentheses.
In particular, for k0 = R we have Hom(Γ, S2 ) = S2 , and therefore, the number of these
isomorphism classes is 2s , where s = |Ω(2) |. For G and Y as in Example 0.1 we have s = 1,
hence for each of the two R-models of G there are exactly two non-isomorphic equivariant
R-models of Y , see Example 10.15 below.
Corollary 0.4 (Akhiezer’s theorem). In Theorem 0.2 instead of (ii) assume that
(ii′ ) H is self-normalizing, that is, NG (H) = H.
Then Y = G/H admits a G0 -equivariant k0 -model Y0 , and this model is unique up to a
unique isomorphism.
Indeed, since H is self-normalizing, it is spherically closed. By Theorem 0.2 Y admits
a G0 -equivariant k0 -model. The uniqueness assertion is obvious because AutG (Y ) = {1}.
Corollary 0.4 generalizes Theorem 1.1 of Akhiezer [Akh15], where the case k0 = R was
considered.
Theorem 0.5. Under the assumptions of Corollary 0.4, any spherical embedding Y ′ of
Y = G/H admits a G0 -equivariant k0 -model Y0′ . This k0 -model Y0′ is compatible with the
unique G0 -equivariant k0 -model Y0 of Y from Corollary 0.4, and hence is unique up to a
unique isomorphism.
Theorem 0.5 generalizes Theorem 1.2 of Akhiezer [Akh15], who proved in the case
k0 = R that the wonderful embedding of Y admits a unique G0 -equivariant R-model.
Our proof of Theorem 0.5 uses results of Huruguen [Hu11]. Note that in Theorem 0.5 we
do not assume that Y ′ is quasi-projective.

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WITH AN APPENDIX BY

Theorems 0.2, 0.3, and 0.5 seem to be new even in the case k0 = R.
The plan of the rest of the paper is as follows. In Sections 1–6 we consider semilinear
morphisms and models for general G-varieties and homogeneous spaces of G, not necessarily spherical. In Sections 7–8 we consider combinatorial invariants of spherical homogeneous spaces. Following ideas of Akhiezer [Akh15, Theorem 1.1] and Cupit-Foutou [CF15,
Theorem 3(1), Section 2.2], for γ ∈ Γ = Gal(k/k0 ) we give a criterion of isomorphism of
a spherical homogeneous space Y = G/H and the “conjugate” variety γ∗ Y = G/γ(H)
in terms of the action of γ on the combinatorial invariants of G/H. In Sections 9–11 we
prove Corollary 0.4, Theorem 0.2, Theorem 0.3, and Theorem 0.5. In Appendix A for a
connected reductive group G0 defined over an algebraically closed field k0 of characteristic
0 and for an algebraically closed extension k ⊃ k0 , it is proved that any spherical subgroup
H of the base change G = G0 ×k0 k is conjugate to a (spherical) subgroup defined over k0 .
In Appendix B, following Friedrich Knop’s MathOverflow answer [Kn17a] to the author’s
question, Giuliano Gagliardi gives a proof of an unpublished theorem of Ivan Losev that
describes the image of AutG (G/H) = NG (H)/H in the group of permutations of D(G/H).
Our proofs of Theorems 0.2, 0.3, and 0.5 use this result of Losev.
Acknowledgements. The author is very grateful to Friedrich Knop for answering the
author’s numerous MathOverflow questions, especially for the answer [Kn17a], to Giuliano
Gagliardi for writing Appendix B, and to Roman Avdeev for suggesting Example 10.9 and
proving Proposition 10.10. It is a pleasure to thank Michel Brion for very helpful e-mail
correspondence. The author thanks Dmitri Akhiezer, Stéphanie Cupit-Foutou, Cristian
D. González-Avilés, David Harari, Boris Kunyavskiı̆, and Stephan Snigerov for helpful
discussions. This preprint was written during the author’s visits to the University of La
Serena (Chile) and to the Paris-Sud University, and he is grateful to the departments of
mathematics of these universities for support and excellent working conditions.
Notation and assumptions.
k is a field. In Section 2 and everywhere starting Section 4, k is algebraically closed.
k0 is a subfield of the algebraically closed field k such that k is a Galois extension of k0
(except for Appendix A), hence k0 is perfect.
A k-variety is a reduced separated scheme of finite type over k, not necessarily irreducible.
An algebraic k-group is a smooth k-group scheme of finite type over k, not necessarily
connected. All algebraic k-subgroups are assumed to be smooth.
1. Semi-morphisms of k-schemes
Let k be a field and let Spec k denote the spectrum of k. By a k-scheme we mean a pair
(Y, pY ), where Y is a scheme and pY : Y → Spec k is a morphism of schemes. Let (Y, pY )
and (Z, pZ ) be two k-schemes. By a k-morphism
λ : (Y, pY ) → (Z, pZ )
we mean a morphism of schemes λ : Y → Z such that the following diagram commutes:
Y

λ

/Z
pZ

pY



Spec k

id



/ Spec k

Let γ : k → k be an automorphism of k (we write γ ∈ Aut(k)). Let
γ ∗ := Spec γ : Spec k → Spec k
denote the induced automorphism of Spec k, then (γγ ′ )∗ = (γ ′ )∗ ◦ γ ∗ .

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

5

Let (Y, pY ) be a k-scheme. By abuse of notation we write just that Y is a k-scheme. We
define the γ-conjugated k-scheme γ∗ (Y, pY ) = (γ∗ Y, γ∗ pY ) to be the base change of (Y, pY )
from Spec k to Spec k via γ ∗ . By abuse of notation we write just γ∗ Y for γ∗ (Y, pY ).
Lemma 1.1. Let (Y, pY ) be a k-scheme, and let γ ∈ Aut(k). Then the γ-conjugated
k-scheme γ∗ (Y, pY ) is canonically isomorphic to (Y, (γ ∗ )−1 ◦ pY )
Proof. Write (X, pX ) = γ∗ (Y, pY ), then X comes with a canonical morphism λ : X → Y
such that the following diagram commutes:
λ

/Y

γ∗



X

pY

pX



Spec k

/ Spec k

Since (γ −1 )∗ (γ∗ (Y, pY )) is canonically isomorphic to (Y, pY ), one can easily see that λ is
an isomorphism of schemes. From the above diagram we obtain a commutative diagram
λ

X

/Y
pY



Spec k

pX

(γ ∗ )−1



Spec k

id



/ Spec k
∼

which gives a canonical isomorphism of k-schemes (X, pX ) → (Y, (γ ∗ )−1 ◦ pY ).



We define an action of γ : k → k on k-points. Let y be a k-point of Y , that is, a
morphism y : Spec k → Y such that pY ◦ y = idSpec k . We denote
(1)

γ! (y) = y ◦ γ ∗ : Spec k → Spec k → Y,

then an easy calculation shows that γ! (y) a k-point of γ∗ Y . Thus we obtain a bijection
(2)

γ! : Y (k) → (γ∗ Y )(k),

y 7→ γ! (y).

Let G be a k-group scheme. Following Flicker, Scheiderer, and Sujatha [FSS98, (1.2)],
we define the k-group scheme γ∗ G to be the base change of G from Spec k to Spec k via
γ ∗ . Then the map (2)
γ! : G(k) → (γ∗ G)(k)
is an isomorphism of groups (because for any field extension λ : k ֒→ k′ the corresponding
map on rational points
λ! : G(k) → (G ×k k′ )(k′ )
is a homomorphism). If H ⊂ G is a k-group subscheme, then γ∗ H is naturally a k-group
subscheme of γ∗ G (because a base change of a group subscheme is a group subscheme).
From the commutative diagram
H(k)


G(k)

γ!

γ!

/ (γ∗ H)(k)

/ (γ∗ G)(k)

we see that
(3)

(γ∗ H)(k) = γ! (H(k)) ⊂ (γ∗ G)(k).

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Let (Y, θ) be a G-k-scheme (a G-scheme over k), where
θ : G ×k Y → Y,
is an action of G on Y . By abuse of notation we write just that Y is a G-k-scheme. Again
we define the γ∗ G-k-scheme γ∗ (Y, θ) = (γ∗ Y, γ∗ θ) to be the base change of (Y, θ) from
Spec k to Spec k via γ ∗ .
Definition 1.2. Let (Y, pY ) and (Z, pZ ) be two k-schemes. A semilinear morphism
(γ, ν) : (Y, pY ) → (Z, pZ )
is a pair (γ, ν) where γ : k → k is an automorphism of k, and ν : Y → Z is a morphism of
schemes such that the following diagram commutes:
Y

ν

/Z



(γ ∗ )−1



pZ

pY

Spec k

/ Spec k

We shorten “semilinear morphism” to “semi-morphism”. We write “ν : (Y, pY ) → (Z, pZ )
is a γ-semi-morphism” if (γ, ν) : (Y, pY ) → (Z, pZ ) is a semi-morphism. Then by abuse of
notation we write just that ν : Y → Z is a γ-semi-morphism.
Note that if we take γ = idk , then a idk -semi-morphism (Y, pY ) → (Z, pZ ) is just a
morphism of k-schemes.
Lemma 1.3. If (γ, ν) : (Y, pY ) → (Z, pZ ) is a semi-morphism of nonempty k-schemes,
then the morphism of schemes ν : Y → Z uniquely determines γ.
Proof. We may and shall assume that Y and Z are affine, Y = Spec RY , Z = Spec RZ .
Then we have a commutative diagram
(4)

RO Y o

ν∗

RO Z

γ −1

ko
k
Since k is a field, the vertical arrows are injective, and therefore, the homomorphism of
rings ν ∗ uniquely determines the automorphism γ −1 .

We define an action of a semi-morphism (γ, ν) : (Y, pY ) → (Z, pZ ) on k-points. If
y : Spec k → Y is a k-point of (Y, pY ), we set
(5)

(γ, ν)(y) = ν ◦ y ◦ γ ∗ : Spec k → Z,

which is a k-point of (Z, pZ ). This formula is compatible with the usual formula for
the action of a k-morphism on k-points. By abuse of notation we write ν(y) instead of
(γ, ν)(y).
Definition 1.4. By a γ-semi-isomorphism ν : (Y, pY ) → (Z, pZ ) we mean a γ-semimorphism ν : (Y, pY ) → (Z, pZ ) for which the morphisms of schemes ν : Y → Z is an
isomorphism. By a γ-semi-automorphism of a k-scheme (Y, pY ) we mean a γ-semiisomorphism µ : (Y, pY ) → (Y, pY ).
Let us fix γ ∈ Aut(k). The commutative diagram
(6)

ν

/Z
Y ▲▲
▲▲▲
▲▲γ▲∗ pY
pZ
pY
▲▲
▲▲▲

&
∗
−1

(γ )
/ Spec k
Spec k

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

7

shows that
(7)

(γ, ν) : (Y, pY ) → (Z, pZ )

is a semi-morphism (that is, ν : (Y, pY ) → (Z, pZ ) is a γ-semi-morphism) if and only if
(8)

(idk , ν) : γ∗ (Y, pY ) → (Z, pZ )

is a semi-morphism (that is, ν : γ∗ (Y, pY ) → (Z, pZ ) is a k-morphism). For brevity we
write
ν ♮ : γ∗ Y → Z

(9)

for the k-morphism (8), then the k-morphism ν♮ acts on k-points as follows:
(10)

(y ′ : Spec k → γ∗ Y ) 7−→ (ν ◦ y ′ : Spec k → Z).

Example 1.5. Let (Y, pY ) be a k-scheme. Recall that γ∗ (Y, pY ) = (Y, (γ ∗ )−1 ◦ pY ). The
commutative diagram
idY

Y

/Y
pY



pY

Spec k
(γ ∗ )−1



Spec k

(γ ∗ )−1



/ Spec k

shows that (γ, idY ) : Y → γ∗ Y is a γ-semi-isomorphism. We denote this γ-semi-isomorphism
by
γ! : Y → γ∗ Y.
Comparing formulas (1) and (5), we see that the γ-semi-isomorphism γ! : Y → γ∗ Y acts
on k-points as the bijective map γ! : Y (k) → (γ∗ Y )(k) defined by formula (1).
Now let ν : Y → Z be a γ-semi-morphism. The commutative diagram (6) shows that
(11)

γ

ν♮

ν = ν♮ ◦ γ! = (idk , ν) ◦ ((γ ∗ )−1 , idY ) : Y −−!→ γ∗ Y −−→ Z,

where γ! is a γ-semi-isomorphism and ν♮ is a k-morphism (an idk -semi-morphism). It
follows that
(12)

ν(y) = ν♮ (γ! (y)) for y ∈ Y (k)

(this follows also from comparing formulas (1), (5), and (10)).
Example 1.6. Let Y0 be a k0 -scheme, where k0 is a subfield of k. Let γ ∈ Aut(k/k0 ),
that is, γ is an automorphism of k that fixes all elements of k0 . Consider
Y := Y0 ×k0 k = Y0 ×Spec k0 Spec k
and
µγ = idY0 × (γ ∗ )−1 : Y → Y.
It follows from the construction of µγ that the following diagram commutes:
Y

µγ

/Y
pY

pY



Spec k

(γ ∗ )−1



/ Spec k

We see that µγ is a γ-semi-automorphism of Y .
Let Y be an affine k-variety, Y = Spec RY , then RY is the ring of regular functions on
Y . If f ∈ RY , then for any y ∈ Y (k) the value f (y) ∈ k is defined.

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Lemma 1.7. Let ν : (Y, pY ) → (Z, pZ ) be a γ-semi-isomorphism of affine k-varieties,
where γ : k → k is an automorphism of k. Let Y = Spec RY , Z = Spec RZ , and let
ν ∗ : RZ → RY denote the morphism of rings corresponding to ν. Let fZ ∈ RZ . Then
fZ (ν(y)) = γ((ν ∗fZ )(y))

(13)

for all y ∈ Y (k).

Proof. The assumption that ν : Y → Z is a γ-semi-morphism means that the diagram (4)
commutes. A k-point y ∈ Y (k) corresponds to a homomorphism of k-algebras ϕy : RY →
k, and the following diagram commutes:
RY o

ν∗

RZ

γ −1



ϕν(y)

ϕy



ko
hence ϕν(y) = γ ◦ ϕy
that

◦ ν ∗.

We set fY =

ν ∗f

Z

k

∈ RY , then fY (y) = ϕy (fY ), and (13) means

(γ ◦ ϕy ◦ ν ∗ )(fZ ) = γ(ϕy (ν ∗ fZ )),
which is obvious.



Now let ν : (Y, pY ) → (Z, pZ ) be a γ-semi-isomorphism of irreducible k-varieties, where
γ : k → k is an automorphism of k. Then the isomorphism of schemes ν : Y → Z induces
an isomorphism of the fields of rational functions
ν∗ : K(Y ) → K(Z),

f 7→ ν∗ f.

For f ∈ K(Y ) and y ∈ Y (k), the value f (y) ∈ k ∪ {∞} of f at y is defined, where we
write f (y) = ∞ if f is not regular at y.
Corollary 1.8. With the above notation and assumptions we have
(ν∗ fY )(z) = γ(fY (ν −1 (z)))

for all fY ∈ K(Y ), z ∈ Z(k).

Proof. We consider the isomorphism
ν ∗ = ν∗−1 : K(Z) → K(Y ),

fZ 7→ ν ∗fZ ,

where fZ ∈ K(Z).

Set fZ = ν∗ fY , then fY = ν ∗ fZ . We must prove that (13) holds. We may and shall
assume that Y and Z are affine varieties, Y = Spec RY , Z = Spec RZ , fZ ∈ RZ , and
that the morphism ν corresponds to a homomorphism of rings ν ∗ : RZ → RY . Now the
corollary follows from Lemma 1.7.

Remark 1.9. (Classical language.) In this remark we describe the variety γ∗ Y and the
map γ! : Y (k) → (γ∗ Y )(k) in the language of the classical algebraic geometry. First,
consider the affine space Ank , then Ank (k) = kn . Let k0 be the prime subfield of k, that
is, the subfield generated by 1, then Ank = Ank0 ×k0 k. Let γ ∈ Aut(k) = Aut(k/k0 ), then
γ induces a γ-semi-automorphism µγ : Ank → Ank , see Example 1.6. For i = 1, . . . , n, let
fi denote the i-th coordinate function on Ank , which is a regular function. Since fi comes
from a regular function on Ank0 , we have (µγ )∗ fi = fi , and by Lemma 1.7 we have
fi (µγ (x)) = γ(fi (x)) for x ∈ Ank (k) = kn .
If we write x = (xi )ni=1 ∈ kn , where xi = fi (x) ∈ k, then
µγ (x) = γ(xi )ni=1 .
Now let Y ⊂ Ank be an affine variety. Let ι : Y ֒→ Ank denote the inclusion morphism,
then γ induces a k-morphism
γ∗ ι : γ∗ Y → γ∗ Ank .
We have a k-isomorphism
(µγ )♮ : γ∗ Ank → Ank ,

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

9

and we obtain a k-morphism
(γ∗ ι)′ = (µγ )♮ ◦ γ∗ ι : γ∗ Y → Ank .
From the commutative diagram
Y (k)

γ!

/ (γ∗ Y )(k)
γ∗ ι

ι



Ank (k)

γ!



/ (γ∗ An )(k)
k

we see that
(γ∗ ι)(γ! (y)) = γ! (ι(y))

for y ∈ Y (k),

hence
(γ∗ ι)′ (y) = (µγ )♮ (γ! (ι(y))) = µγ (ι(y)).
Now we assume that k is algebraically closed. As usual in the classical algebraic geometry, we identify Y with the algebraic set
Y (k) ⊂ Ank (k) = kn .
Furthermore, we identify γ∗ Y with the algebraic set
(γ∗ ι)′ (Y (k)) = µγ (Y (k)) ⊂ kn .
We see that
γ∗ Y = µγ (Y ) = {γ(yi )ni=1 | (yi )ni=1 ∈ Y },
and that the map
γ! : Y (k) → (γ∗ Y )(k)
sends a point y with coordinates (yi )ni=1 to the point with coordinates γ(yi )ni=1 . If Y ⊂ kn
is defined by a family of polynomials (Pα )α∈A , then γ∗ Y ⊂ kn is defined by the family
(γ(Pα ))α∈A , where γ(Pα ) is the polynomial obtained from Pα by acting by γ on the
coefficients.
2. Semi-morphisms of G-varieties
2.1. In this section k is an algebraically closed field, and Y is a k-variety, that is, a reduced
separated scheme of finite type over k.
Let G be an algebraic group over k (we write also “an algebraic k-group”), that is, a
smooth group scheme of finite type over k. Let (Y, θ) be a G-k-variety, that is, a k-variety
Y together with an action
θ : G ×k Y → Y
of G on Y . If g ∈ G(k) and y ∈ Y (k), we write just g ∗ y or g · y for θ(g, y) ∈ Y (k).
Y

Definition 2.2 (cf. [FSS98, (1.2)]). Let γ ∈ Aut(k). A γ-semi-automorphism of an
algebraic k-group G is a γ-semi-automorphism of k-schemes τ : G → G such that the
corresponding isomorphism of k-varieties τ♮ : γ∗ G → G, see (9), is an isomorphism of algebraic k-groups. This condition is the same as to require that certain diagrams containing
τ commute, see [Brv93, 1.2].
Let H ⊂ G be an algebraic k-subgroup. By Definition 2.2 we have τ (H) = τ♮ (γ∗ H)),
hence τ (H) is a k-subgroup of G. We have (τ (H))(k) = τ (H(k)).
We shall always assume that G = G0 ×k0 k, where G0 is an algebraic group defined over
a subfield k0 of k, and that γ ∈ Aut(k/k0 ), that is, γ is an automorphism of k fixing all
elements of k0 . Then we have a γ-semi-automorphism
σγ = idG0 × (γ ∗ )−1 : G → G,

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compare Example 1.6. If α is any k-automorphism of G, then
τ := α ◦ σγ : G → G
is a γ-semi-automorphism of G, and all γ-semi-automorphisms of G (for given γ) can be
obtained in this way.
Definition 2.3. Let G be an algebraic k-group, and let (Y, θY ) and (Z, θZ ) be two Gk-varieties. Let γ ∈ Aut(k), and let τ : G → G be a γ-semi-automorphism of G. A
τ -equivariant γ-semi-morphism
ν : (Y, θY ) → (Z, θZ )
is a γ-semi-morphism ν : Y → Z such that the following diagram commutes:
G ×k Y

(14)

θY

/Y
ν

τ ×ν



G ×k Z

θZ



/Z

where we write τ × ν for the product of τ and ν over the automorphism (γ ∗ )−1 of Spec k.
Since k is algebraically closed, G is smooth (reduced), and Y and Z are reduced, we see
that the diagram (14) commutes if and only if
ν(g · y) = τ (g) · ν(y) for all g ∈ G(k), y ∈ Y (k).
Construction 2.4. Let G be an algebraic k-group, and let (Y, θY ) be a G-k-variety. The
group γ∗ G naturally acts on γ∗ Y : the action θ : G ×k Y → Y gives an action
γ∗ θ : γ∗ G ×k γ∗ Y → γ∗ Y.

(15)

By definition, a γ-semi-automorphism τ of G defines an isomorphism of algebraic k-groups
τ♮ : γ∗ G → G.
We identify G and γ∗ G via τ♮ and obtain from (15) an action
τ ∗ γ∗ θ : G ×k γ∗ Y → γ∗ Y,

(g, y ′ ) 7→ (γ∗ θ)(τ♮−1 (g), y ′ )

for g ∈ G(k), y ′ ∈ (γ∗ Y )(k).

By abuse of notation we write τ ∗ θ for τ ∗ γ∗ θ and we write γ∗ Y for the G-k-variety
(γ∗ Y, τ ∗ γ∗ θ). We write
g ∗τ y ′

for (τ ∗ θ)(g, y ′ ),

where g ∈ G(k), y ′ ∈ (γ∗ Y )(k).

By formula (12) we have
τ (g) = τ♮ (γ! (g)),
hence
(16)

g ∗τ y ′ = (γ∗ θ)(τ♮−1 (g), y ′ ) = γ! (θ(γ!−1 (τ♮−1 (g)), γ!−1 (y ′ )))
= γ! (θ(τ −1 (g), γ!−1 (y ′ )) = γ! (τ −1 (g) ∗ γ!−1 (y ′ )).
Y

Lemma 2.5. Let G be an algebraic k-group, and let (Y, θ) be a G-k-variety. Let γ ∈
Aut(k), and let τ : G → G be a γ-semi-automorphism of G. Let y (0) ∈ Y (k) be a k-point,
and write H = StabG (y (0) ). Consider the action
τ ∗ θ : G ×k γ∗ Y → γ∗ Y.
Then the stabilizer in G(k) of the point γ! (y (0) ) ∈ (γ∗ Y )(k) under the action τ ∗ θ is
τ (H(k)) = (τ (H))(k).

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

11

Proof. By formula (16) we have
g ∗τ γ! (y (0) ) = γ! (τ −1 (g) · y (0) ).
Since the stabilizer in G(k) of y (0) ∈ Y (k) is H(k), the lemma follows.



Note that ν in Definition 2.3 defines a k-morphism
ν♮ : γ∗ Y → Z,
see (9).
Lemma 2.6. Let γ ∈ Aut(k) and let τ : G → G be a γ-semi-automorphism of G. Let
(Y, θY ) and (Z, θZ ) be two G-k-varieties. A morphism of schemes ν : Y → Z is a τ equivariant γ-semi-morphism if and only if ν♮ : γ∗ Y → Z is a G-equivariant morphism of
k-varieties.
Proof. By (6) the morphism of schemes ν is a γ-semi-morphism Y → Z if and only if it is
a k-morphism γ∗ Y → Z.
Let g ∈ G(k), y ′ ∈ (γ∗ Y )(k). Using formula (16) we obtain
ν♮ (g ∗τ y ′ ) = ν(γ!−1 (g ∗τ y ′ )) = ν(τ −1 (g) ∗ γ!−1 (y ′ )).
Y

We have also
g ∗ ν(γ!−1 (y ′ )) = g ∗ ν♮ (y ′ ).
Z

Z

If ν is τ -equivariant, then
ν(τ −1 (g) ∗ γ!−1 (y ′ )) = g ∗ ν(γ!−1 (y ′ )),
Z

Y

and we obtain that
ν♮ (g ∗τ y ′ ) = g ∗ ν♮ (y ′ ) for all g ∈ G(k), y ′ ∈ (γ∗ Y )(k),
Z

hence ν♮ is G-equivariant.
Conversely, if ν♮ is G-equivariant, we obtain from the above calculations that
ν(τ −1 (g) ∗ γ!−1 (y ′ )) = ν♮ (g ∗τ y ′ ) = g ∗ ν♮ (y ′ ) = g ∗ ν(γ!−1 (y ′ )).
Z

Y

Set y =

γ!−1 (y ′ ),

Z

g′ = τ −1 (g), then we obtain that
ν(g ′ ∗ y) = τ (g′ ) ∗ ν(y) for all g′ ∈ G(k), y ∈ Y (k).
Y

Z

Thus ν is τ -equivariant.



Corollary 2.7. Let γ be an automorphism of k, and let τ : G → G be a γ-semi-automorphism
of G. Let (Y, θ) be a G-k-variety. There exists a τ -equivariant γ-semi-automorphism
µ : Y → Y if and only if the G-k-variety (γ∗ Y, τ ∗ θ) is isomorphic to (Y, θ).
Proof. We take Z = Y in Lemma 2.6.



3. Quotients
Let k be a field (not necessarily algebraically closed). By an algebraic scheme over k
we mean a scheme of finite type over k. By an algebraic group scheme over k we mean a
group scheme over k whose underlying scheme is of finite type over k.
Let H be an algebraic group subscheme of an algebraic group k-scheme G. A quotient
of G by H is an algebraic scheme Y over k equipped with an an action θ : G ×k Y → Y
and a point y (0) ∈ Y (k) fixed by H satisfying certain properties (a) and (b), see Milne
[Mi18, Definition 5.20].

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By [Mi18, Theorem 5.28] there exists a quotient of G by H. By [Mi18, Proposition 5.22]
this quotient (Y, θ, y (0) ) has the following universal property:
(U) Let Z be a k-scheme on which G acts, and let z (0) ∈ Z(k) be a point fixed by H. Then
there exists a unique G-equivariant map Y → Z making the following diagram commute:
g7→g·y (0)

/Y
◆◆◆
◆◆◆
◆◆
g7→g·z (0) ◆◆◆ 
&

G ◆◆

Z

Clearly the universal property (U) uniquely determines the quotient up to a unique
isomorphism, so we may take (U) as a definition of the quotient.
We return to our settings: k is an algebraically closed field, G is a linear algebraic kgroup (a smooth affine group k-scheme) and H is a smooth algebraic k-subgroup of G.
Since G is smooth, so is the quotient Y , see [Mi18, Corollary 5.26]. Since G is smooth and
affine, the quotient Y is a separated algebraic scheme, see [Mi18, Theorem 7.18]. Thus Y
is a k-variety, and therefore, in the universal property U defining Y we may assume that Z
is a k-variety. Since k is algebraically closed and H is smooth, the condition “fixed by H”
is equivalent to “fixed by H(k)”. Thus we arrive to the following definition of Springer:
Definition 3.1 (cf. Springer [Sp98, Section 5.5]). Let k be an algebraically closed field,
and let G be a linear algebraic k-group. Let H ⊂ G be a smooth k-subgroup. A quotient
of G by H is a pointed G-k-variety (Y, θ : G ×k Y → Y, y (0) ∈ Y (k) ) such that H(k) fixes
y0 , with the following universal property:
(U′ ) For any pointed G-k-variety (Z, θZ , z (0) ) such that the k-point z (0) ∈ Z(k) is fixed by
H(k), there exists a unique morphism of pointed G-k-varieties (Y, θ, y (0) ) → (Z, θZ , z (0) ).
For G and H as in Definition 3.1, let (Y, θ, y (0) ) be a quotient of G by H. The action
of G on Y induces a G-k-morphism
(17)

G → Y,

g 7→ g · y (0) ,

where G acts on itself by left translations.
As usual, we write G/H for Y and g · H or gH for g · y (0) , where g ∈ G(k). In
particular, we write 1 · H for y (0) . The G-equivariant morphism G/H = Y → Z of (U′ )
sends 1·H ∈ (G/H)(k) to z (0) , hence for any g ∈ G(k) it sends the k-point gH ∈ (G/H)(k)
to g · z (0) ∈ Z(k). Thus the quotient G/H has the following universal property:
(U′′ ) For any pointed G-k-variety (Z, θZ , z (0) ) such that the k-point z (0) is fixed by H(k),
there exists a unique G-k-morphism G/H → Z sending gH to g · z (0) for any g ∈ G(k).
By [Mi18, Definition 5.20(a)] the morphism (17) induces an injective map G(k)/H(k) →
(G/H)(k) sending g · H(k) to gH. By [Mi18, Proposition 5.25] the morphism (17) is
faithfully flat, and therefore, since k is algebraically closed, we see that the induced map
G(k)/H(k) → (G/H)(k) is surjective. We conclude that this map is bijective. Thus any
k-point of G/H is of the form gH, where g ∈ G(k).
4. Semi-morphisms of homogeneous spaces
Let k be an algebraically closed field.
Lemma 4.1 (well-known). Let G be a linear algebraic k-group over an algebraically closed
field k, and let H1 , H2 be two k-subgroups. Then Y1 = G/H1 and Y2 = G/H2 are isomorphic as G-k-varieties if and only if the subgroups H1 and H2 are conjugate. To be more
precise, for a ∈ G(k) the following two assertions are equivalent:

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

13

(i) There exists an isomorphism of G-k-varieties φa : G/H1 → G/H2 taking g · H1 to
ga−1 · H2 for g ∈ G(k);
(ii) H1 = a−1 H2 a.
Proof. (i)⇒(ii). Clearly StabG(k) (1 · H1 ) = H1 (k) and StabG(k) (a−1 · H2 ) = a−1 · H2 (k) · a.
Since φa (1 · H1 ) = a−1 · H2 , these stabilizers coincide, whence (ii).
(0)

(0)

(ii)⇒(i). Set Y2 = G/H2 , y2 = 1 · H2 ∈ Y2 (k), y ′ = a−1 · y2 ∈ Y2 (k), then
StabG(k) (y ′ ) = a−1 H2 (k)a = H1 (k), so by the property (U′′ ) of the quotient G/H1 there
exists a unique morphism of G-varieties φa : G/H1 → G/H2 such that
φa (g · H1 ) = g · a−1 · H2

for g ∈ G(k).

Similarly, since the stabilizer in G(k) of a · H1 ∈ (G/H1 )(k)
unique morphism of G-varieties ψa : G/H2 → G/H1 such that
ψa (g · H2 ) = g · a · H1

is H2 (k), there exists a

for g ∈ G(k).

Clearly these two morphisms are mutually inverse, hence both φa and ψa are isomorphisms.

4.2. Let k be an algebraically closed field. Let G be a linear algebraic group over k. Let
γ ∈ Aut(k). Let τ : G → G be a γ-semi-automorphism of G.
Let H ⊂ G be a smooth k-subgroup. Set Y = G/H, then we have a morphism
θ : G ×k Y → Y defining the action of G on Y . Furthermore, the variety Y has a k-point
y (0) = 1 · H such that StabG(k) (y (0) ) = H(k), and the group of k-points G(k) acts on Y (k)
transitively.
Consider the variety γ∗ Y , the action γ∗ θ : γ∗ G ×k γ∗ Y → γ∗ Y of γ∗ G on γ∗ Y , and the
k-point γ! (y (0) ) ∈ (γ∗ Y )(k). As in Construction 2.4 we obtain an action
τ ∗ θ : G ×k γ∗ Y → γ∗ Y.
Lemma 4.3. Let k, G, γ, τ , H be as in Subsection 4.2. Set Y = G/H and let θ : G×k Y →
Y denote the canonical action. Consider the map on k-points
(18)

(G/H)(k) → (G/τ (H))(k),

g · H 7→ τ (g) · τ (H)

for g ∈ G(k).

Then the following assertions hold:
(i) The pointed G-k-variety (γ∗ Y, τ ∗ θ, γ! (y (0) )) is isomorphic to G/τ (H);
(ii) the map (18) is induced by some γ-semi-isomorphism ν : G/H → G/τ (H).
Proof. Let (Z, θZ , z (0) ) be a pointed G-k-variety, and assume that τ (H(k)) fixes z (0) . Consider the pointed G-k-variety
((γ −1 )∗ Z, (τ −1 )∗ θZ , γ!−1 (z (0) )),
where the action
(19)

(τ −1 )∗ θZ : G ×k (γ −1 )∗ Z → (γ −1 )∗ Z

is defined as in Construction 2.4, but for the pair (γ −1 , τ −1 ) instead of (γ, τ ). By Lemma
2.5, H(k) fixes γ!−1 (z (0) ) ∈ (γ −1 )∗ Z. For any morphism of pointed G-k-varieties
(20)

κ : (Y, θ, y (0) ) → ((γ −1 )∗ Z, (τ −1 )∗ θZ , γ!−1 (z (0) ))

we obtain a morphism of pointed G-k-varieties
(21)

γ∗ κ : (γ∗ Y, τ ∗ θ, γ! (y (0) )) → (Z, θZ , z (0) ).

We see that the map κ 7→ γ∗ κ is a bijection between the set of morphisms as in (20)
and the set of morphisms as in (21). Since Y = G/H and H(k) fixes γ!−1 (z (0) ) under the
action (19), we conclude by the universal property (U′′ ) for the quotient Y = G/H that

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the former set contains exactly one element. It follows that the latter set contains exactly
one element, that is, the triple (γ∗ Y, τ ∗ θ, γ! (y (0) )) has the universal property (U′′ ). This
means that (γ∗ Y, τ ∗ θ, γ! (y (0) )) is a quotient of G by τ (H), which proves (i). It follows
that there exists an isomorphism of G-k-varieties
λ : γ∗ Y → G/τ (H) such that g ∗τ γ! (y (0) ) 7−→ g · τ (H).

(22)
We set

γ

λ

ν = λ ◦ γ! : G/H = Y −−!→ γ∗ Y −−→ G/τ (H),

(23)

where γ! : Y → γ∗ Y is the γ-semi-morphism of Example 1.5. Then we have ν♮ = λ. Since ν♮
is an isomorphism of G-k-varieties, by Lemma 2.6 ν is a τ -equivariant γ-semi-isomorphism.
Since
ν(1 · H) = ν(y (0) ) = λ(γ! (y (0) )) = 1 · τ (H)
by (22), we have
ν(g · H) = ν(g · (1 · H)) = τ (g) · ν(1 · H) = τ (g) · ν(H),
which proves (ii).



Corollary 4.4. Let G be a linear algebraic k-group and H ⊂ G be an algebraic k-subgroup.
Set Y = G/H. Let γ ∈ Aut(k) and let τ : G → G be a γ-semi-automorphism of G. The
following three conditions are equivalent:
(i) There exists a τ -equivariant γ-semi-automorphism µ : Y → Y ;
(ii) The G-k-variety G/τ (H) is isomorphic to G/H;
(iii) The algebraic subgroup τ (H) ⊂ G is conjugate to H.
Proof. By Corollary 2.7 there exists µ : Y → Y as in (i) if and only if the G-k-variety
(γ∗ Y, τ ∗ θ) is isomorphic to (Y, θ). By construction (Y, θ) = G/H, and by Lemma 4.3
(γ∗ Y, τ ∗ θ) ∼

= G/τ (H). Thus (i)⇔(ii). By Lemma 4.1 (ii)⇔(iii).
4.5. With the assumptions of Subsection 4.2, assume also that G is connected, then
the homogeneous spaces Y1 = G/H and Y2 = G/τ (H) of G are irreducible k-varieties.
We consider the fields of rational functions K(Y1 ) and K(Y2 ). The γ-semi-isomorphism
ν : Y1 → Y2 of (23) induces an isomorphism of fields
∼

ν∗ : K(Y1 ) → K(Y2 ),

f1 7→ f2 = ν∗ f1 .

and by Lemma 1.8 we have
f2 (y2 ) = γ! (f1 (ν −1 (y2 )))

(24)

for y2 ∈ Y2 (k).

5. k-automorphisms of homogeneous spaces
Let G be a linear algebraic group over an algebraically closed field k. Let Y be a G-kvariety. We denote by AutG (Y ) the group of G-equivariant k-automorphisms of Y , that
is, of k-automorphisms ψ : Y → Y such that
ψ(g · y) = g · ψ(y)

for g ∈ G, y ∈ Y.

We assume that Y is a homogeneous space of G, that is, Y = G/H, where H is a
k-subgroup of G. Set N = NG (H), the normalizer of H in G.
Lemma 5.1 (well-known). For n ∈ N (k) we define a map on k-points
n∗ : G/H → G/H, gH 7→ gn−1 H

for g ∈ G(k).

Then
(i) The map n∗ is induced by some automorphism φn ∈ AutG (G/H);

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

15

(ii) The map
φ : N (k) → AutG (G/H),

(25)

n 7→ φn

is a homomorphism inducing an isomorphism
∼

N (k)/H(k) → AutG (G/H).
Proof. By assumption n−1 Hn = H, and by Lemma 4.1 there exists an isomorphism
φn : G/H → G/H such that φn (g · H) = gn−1 · H, which proves (i).
Clearly the map φ of (25) is a homomorphism with kernel H(k). To prove (ii) it remains
to show that φ is surjective.
Let ψ ∈ AutG (G/H). Write ψ(1 · H) = a · H with a ∈ G(k). Since ψ is an isomorphism
of G-k-varieties, we have StabG(k) (a · H) = StabG(k) (1 · H) = H(k). On the other hand,
StabG(k) (a · H) = a · H(k) · a−1 . Thus a · H · a−1 = H, hence a ∈ N (k). We have
ψ(g · H) = ga · H. Write a = n−1 , then n ∈ N (k) and ψ = φn . Thus the homomorphism
φ is surjective, as required.

Corollary 5.2. If NG (H) = H, then AutG (G/H) = {1}.



6. Equivariant models of G-varieties
Let k be an algebraically closed field, and k0 ⊂ k be a subfield such that k is a Galois
extension of k0 , that is, k0 is a perfect field and k is an algebraic closure of k0 . We write
Γ = Gal(k/k0 ) := Aut(k/k0 ).
Let Y be a k-variety. Let γ, γ ′ ∈ Γ. If µ is a γ-semi-automorphism of Y , and µ′ is a
of Y , then µ ◦ µ′ is a γ ◦ γ ′ -semi-automorphism of Y and µ−1 is a
−1
γ -semi-automorphism of Y .

γ ′ -semi-automorphism

We denote by SAutk/k0 (Y ) or just by SAut(Y ) the group of all γ-semi-automorphisms
µ of Y where γ runs over Γ = Gal(k/k0 ).
A k0 -model of Y is a k0 -variety Y0 together with an isomorphism of k-varieties
∼

κY : Y0 ×k0 k → Y.
Note that γ ∈ Γ defines a γ-semi-automorphism of Y0 ×k0 k
idY0 × (γ ∗ )−1 : Y0 ×Spec k0 Spec k → Y0 ×Spec k0 Spec k
and thus, via κY , a γ-semi-automorphism µγ of Y . We obtain a homomorphism
Γ → SAut(Y ), γ 7→ µγ .
Conversely:
Lemma 6.1 (Borel and Serre [BS64, Lemma 2.12]). Let k, k0 , Γ, Y be as above. Assume
that for any γ ∈ Γ we have a γ-semi-automorphism µγ of Y such that
(i) the map Γ → SAutk/k0 (Y ), γ 7→ µγ , is a homomorphism,
(ii) the restriction of this map to Gal(k/k1 ) for some finite Galois extension k1 /k0 in
k comes from a k1 -model Y1 of Y ,
(iii) Y is quasi-projective.
Then there exists a k0 -model Y0 of Y that defines this homomorphism γ 7→ µγ .



6.2. Let G be a linear algebraic group over k. We assume that we are given a k0 -model of
G, that is, a linear algebraic group G0 over k0 together with an isomorphism of algebraic
∼
k-groups over κG : G0 ×k0 k → G. For γ ∈ Γ the automorphism (γ ∗ )−1 of Spec k induces a
γ-semi-automorphism idG0 × (γ ∗ )−1 of G0 ×Spec k0 Spec k. We identify G with G0 ×Spec k0

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Spec k via κG , then for any γ ∈ Γ we obtain a γ-semi-automorphism σγ : G → G. The
map
Γ → SAut(G), γ 7→ σγ
is a homomorphism. We identify γ∗ G with G using (σγ )♮ : γ∗ G → G.
Let (Y, θ) be a G-k-variety. By a G0 -equivariant k0 -model of the G-k-variety Y we mean
∼
a G0 -k0 -variety (Y0 , θ0 ) together with an isomorphism κY : Y0 ×k0 k → Y such that the
following diagram commutes:
(26)

G0,k ×k Y0,k

θ0,k

/ Y0,k
κY

κG ×κY



G ×k Y



θ

/Y

where G0,k := G0 ×k0 k and Y0,k := Y0 ×k0 k. For a given k0 -model G0 of G we ask whether
there exists a G0 -equivariant k0 -model Y0 of Y .
Let (Y, θ) be a G-k-variety. We write g · y for θ(g, y). Recall (Definition 2.3) that a
γ-semi-automorphism µ of Y is σγ -equivariant if the following diagram commutes:
G ×k Y

θ

/Y

θ



µ

σγ ×µ



G ×k Y

/Y

Since k is algebraically closed, G is smooth (reduced) and Y is reduced, this is the same
as to require that
µ(g · y) = σγ (g) · µ(y) for all g ∈ G(k), y ∈ Y (k).
We ask whether there exists such µ.
A G0 -equivariant k0 -model of Y defines a homomorphism
Γ → SAutk/k0 (Y ),

γ 7→ µγ ,

where for any γ ∈ Γ, the γ-semi-automorphism µγ of Y is σγ -equivariant. Conversely:
Lemma 6.3. Let k, k0 , Γ = Gal(k/k0 ), G, (Y, θ) be as above and let G0 be a k0 -model
of G. Assume that for any γ ∈ Γ we have a γ-semi-automorphism µγ of Y such that the
following conditions are satisfied:
(i) the map Γ → SAutk/k0 (Y ), γ 7→ µγ is a homomorphism,
(ii) the restriction of this map to Gal(k/k1 ) for some finite Galois extension k1 /k0 in
k comes from a G1 -equivariant k1 -model Y1 of Y , where G1 = G0 ×k0 k1 ,
(iii) Y is quasi-projective,
(iv) for any γ ∈ Γ, the γ-semi-automorphism µγ is σγ -equivariant.
Then there exists a G0 -equivariant k0 -model Y0 of Y that defines this homomorphism
γ 7→ µγ .
Proof. By Lemma 6.1 the homomorphism
Γ → SAut(Y ),

γ 7→ µγ

defines a k0 -model Y0 of Y . Using Galois descent for morphisms (see e.g. Jahnel [Ja00,
Proposition 2.8]) we obtain from condition (iv) that θ comes from some morphism θ0 : G0 ×k0
Y0 → Y0 , and the k0 -model (Y0 , θ0 ) of (Y, θ) is G0 -equivariant.


EQUIVARIANT MODELS OF SPHERICAL VARIETIES

17

Remark 6.4. If in Lemma 6.3 we do not assume that Y is quasi-projective, then we obtain
a k0 -model Y0 in the category of algebraic k0 -spaces (see Wedhorn [We15, Proposition
8.1]), but not necessarily in the category of k0 -schemes (even when k = C and k0 = R,
see Huruguen [Hu11, Theorem 2.35]).
Lemma 6.5. Let k, k0 , Γ = Gal(k/k0 ), G, (Y, θ) be as above and let G0 be a k0 -model
of G. Assume that AutG (Y ) = {1}. Assume that for any γ ∈ Γ there exists a γ-semiautomorphism µγ of Y satisfying condition (iv) of Lemma 6.3. Then such µγ is unique,
and the map γ 7→ µγ satisfies conditions (i) and (ii) of Lemma 6.3.
G
′
Proof. If µ′γ another such γ-semi-automorphism, then µ−1
γ µγ ∈ Aut (Y ) = {1}, hence
G
µ′γ = µγ . If γ, δ ∈ Γ, then µ−1
γδ µγ µδ ∈ Aut (Y ) = {1}, hence µγδ = µγ µδ , hence the map
γ 7→ µγ is a homomorphism, that is, condition (i) holds.

Since G and Y are of finite type over k, there exists a finite Galois extension k1 /k in k
and a G1 -equivariant k1 -model (Y1 , θ1 ) of (Y, θ), where G1 := G0 ×k0 k1 . This k1 -model
defines a homomorphism
(27)

γ 7→ µ′γ : Gal(k/k1 ) → SAut(Y )

such that µ′γ is σγ -equivariant for all γ ∈ Gal(k/k1 ). Since a σγ -equivariant γ-semiautomorphism is unique, we see that for all γ ∈ Gal(k/k1 ) we have µγ = µ′γ , and hence,
the restriction of the map
(28)

γ 7→ µγ : Γ → SAut(Y )

to Gal(k/k1 ) comes from the k1 -model (Y1 , θ1 ) of (Y, θ), that is, condition (ii) of Lemma
6.3 is satisfied.

7. Spherical homogeneous spaces and their combinatorial invariants
Let G be a connected reductive group over an algebraically closed field k. We describe
combinatorial invariants (invariants of Luna and Losev) of a spherical homogeneous space
Y = G/H of G.
We start with combinatorial invariants of G. We fix T ⊂ B ⊂ G, where B is a Borel
subgroup and T is a maximal torus. Let BRD(G) = BRD(G, T, B) denote the based root
datum of G. We have
BRD(G, T, B) = (X, X ∨ , R, R∨ , S, S ∨ )
where
X = X∗ (T ) := Hom(T, Gm,k ) is the character group of T ;
X ∨ = X∗ (T ) := Hom(Gm,k , T ) is the cocharacter group of T ;
R = R(G, T ) ⊂ X is the root system;
R∨ ⊂ X ∨ is the coroot system;
S = S(G, T, B) ⊂ R is the system of simple roots (the basis of R) defined by B;
S ∨ ⊂ R∨ is the system of simple coroots.
There is a canonical pairing X × X ∨ → Z, (χ, x) 7→ hχ, xi, and a canonical bijection
α 7→ α∨ : R → R∨ such that S ∨ = {α∨ | α ∈ S}. See Springer [Sp79, Sections 1 and 2] for
details.
We consider also the Dynkin diagram Dyn(G) = Dyn(G, T, B), which is a graph with
the set of vertices S. The edge between two simple roots α, β ∈ S is described in terms of
the integers hα, β ∨ i and hβ, α∨ i.
We call a pair (T, B) as above a Borel pair. If (T ′ , B ′ ) is another Borel pair, then by
Theorem 11.1 and Theorem 10.6(4) in Borel’s book [B91], there exists g ∈ G(k) such that
(29)

g · T · g −1 = T ′ ,

g · B · g −1 = B ′ .

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This element g induces an isomorphism
∼

g∗ : BRD(G, T ′ , B ′ ) → BRD(G, T, B).
If g′ ∈ G(k) another element as in (29), then g = gt for some t ∈ T (k), and therefore, the
isomorphism
∼
(g′ )∗ : BRD(G, T ′ , B ′ ) → BRD(G, T, B)
coincides with g∗ . Thus we can canonically identify BRD(G, T ′ , B ′ ) with BRD(G, T, B)
and write BRD(G) for BRD(G, T, B). We say that BRD(G) is the canonical root datum
of G. We see that the based root datum BRD(G) is an invariant of G. In particular,
the character lattice X = X∗ (T ) with the subset S ⊂ X is an invariant, and the Dynkin
diagram Dyn(G) is an invariant.
Now we describe combinatorial invariants of a homogeneous spherical G-variety Y =
G/H. Let K(Y ) denote the field of rational functions of Y . The group G(k) acts on K(Y )
by
(g · f )(y) = f (g−1 · y) for f ∈ K(Y ), g ∈ G(k), and y ∈ Y (k).
(B)

For χ ∈ X∗ (B) let K(Y )χ denote the space of χ-eigenfunctions in K(Y ), that is, the
k-space of rational functions f ∈ K(X) such that
b · f = χf (b) · f

for all b ∈ B(k).
(B)

Since B has an open dense orbit in Y , the k-dimension of K(Y )χ is ≤ 1. Let X =
(B)
X (Y ) ⊂ X∗ (B) denote the set of characters χ of B such that K(Y )χ 6= 0, which is a
subgroup of X∗ (B) called the weight lattice of Y . We set
V = V (Y ) = HomZ (X , Q).
Let Val(K(Y )) denote the set of Q-valued valuations of the field K(Y ) that are trivial
on k. The group G(k) naturally acts on K(Y ) and on Val(K(Y )). We will consider the
set ValB (K(Y )) of B(k)-invariant valuations, and the set ValG (K(Y )) of G(k)-invariant
valuations. We have a canonical map
ρ : ValB (K(Y )) → V, v 7→ (χ 7→ v(fχ )),
(B)

where v ∈ ValB (K(Y )), χ ∈ X , fχ ∈ K(Y )χ , fχ 6= 0. It is known, see Knop [Kn89,
Corollary 1.8], that the restriction of ρ to ValG (K(Y )) is injective. We denote by
V = V(Y ) := ρ(ValG (K(Y ))) ⊂ V
the image of ValG (K(Y )) in V . It is a cone in V called the valuation cone of Y .
Let D = D(Y ) denote the set of colors of Y , that is, the set of closures of B-orbits of
codimension 1 in Y ; it is finite. Each D ∈ D is a B-invariant divisor, which defines a
B-invariant valuation of K(Y ) that we denote by val(D). Thus we obtain a map
val : D → ValB (K(Y )).
By abuse of notation we denote ρ(val(D)) ∈ V by ρ(D). Thus we obtain a map
ρ : D → V,
which in general is not injective (for example, it is not injective for G and Y as in Example
0.1).
For D ∈ D, let StabG (D) denote the stabilizer of D ⊂ Y in G. Clearly StabG (D) ⊃
B, hence StabG (D) is a parabolic subgroup of G. For α ∈ S, let Pα ⊃ B denote the
corresponding minimal parabolic subgroup of G containing B. Let ς(D) denote the set of
α ∈ S for which Pα is not contained in StabG (D). We obtain a map
ς : D → P(S),
where P(S) denotes the set of all subsets of S.

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19

Lemma 7.1. Any fiber of the map ς has ≤ 2 elements.
Proof. Let D ∈ D. Since Y is a homogeneous G-variety, clearly StabG (D) 6= G, hence
ς(D) 6= ∅. We see that there exists α ∈ ς(D). Consider the set D(α) consisting of those
D ∈ D for which α ∈ ς(D). By Proposition B.2 in Appendix B below we have |D(α)| ≤ 2,
and the lemma follows.

Consider the map
ρ × ς : D → V × P(S).
Corollary 7.2. Any fiber of the map ρ × ς has ≤ 2 elements.



Consider the subset Ω := im(ρ × ς) ⊂ V × P(S). Let Ω(2) (resp. Ω(1) ) denote the subset
of Ω consisting of the elements with two preimages (resp. with one preimage) in D. We
obtain two subsets Ω(1) , Ω(2) ⊂ V × P(S), and by Corollary 7.2 we have Ω = Ω(1) ∪ Ω(2)
(disjoint union).
Definition 7.3. Let G be a connected reductive group over an algebraically closed field
k. Let Y = G/H be a spherical homogeneous space of G. By the combinatorial invariants
of Y we mean
X ⊂ X∗ (B),

V ⊂ V := HomZ (X , Q), and Ω(1) , Ω(2) ⊂ V × P(S).

7.4. Let G be a connected reductive k-group. Let H1 ⊂ G be a spherical subgroup, then
we set Y1 = G/H1 . We consider the set of colors D(Y1 ) and the canonical maps
ρ1 : D(Y1 ) → V (Y1 ),

ς1 : D(Y1 ) → P(S).

If H2 ⊂ G is another spherical subgroup, then we set Y2 = G/H2 and consider the set of
colors D(Y2 ) and the canonical maps
ρ2 : D(Y2 ) → V (Y2 ),

ς2 : D(Y2 ) → P(S).

Now assume that there exists a ∈ G(k) such that H2 = aH1 a−1 . Then we have an
isomorphism of G-varieties of Lemma 4.1
φa : Y1 → Y2 ,

g · H1 7→ ga−1 · H2 .

It follows that X (Y1 ) = X (Y2 ), V (Y1 ) = V (Y2 ), V(Y1 ) = V(Y2 ). Moreover, the Gequivariant map φa : Y1 → Y2 induces a bijection
(φa )∗ : D(Y1 ) → D(Y2 )
satisfying
(30)

ρ2 ◦ (φa )∗ = ρ1 ,

ς2 ◦ (φa )∗ = ς1 .

It follows that
Ω(1) (Y1 ) = Ω(1) (Y2 ) and

Ω(2) (Y1 ) = Ω(2) (Y2 ).

Conversely:
Proposition 7.5 (Losev’s Uniqueness Theorem [Lo09, Theorem 1]). Let G be a connected
reductive group over an algebraically closed field k of characteristic 0. Let H1 , H2 ⊂ G
be two spherical subgroups, and let Y1 = G/H1 and Y2 = G/H2 be the corresponding
spherical homogeneous spaces. If X (Y1 ) = X (Y2 ), V(Y1 ) = V(Y2 ), Ω(1) (Y1 ) = Ω(1) (Y2 ),
and Ω(2) (Y1 ) = Ω(2) (Y2 ), then there exists a ∈ G(k) such that H2 = aH1 a−1 .
7.6. Consider the group AutG (Y ) = NG (H)/H, this group acts on D. We consider the
surjective map
ζ = ρ × ς : D → Ω.
By Corollary 7.2 each of the fibers of ζ has either one or two elements. We denote by
AutΩ (D) the group of permutations π : D → D such that ζ ◦ π = ζ. It is clear that

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the group AutG (Y ), when acting on D, acts on the fibers of the map ζ, so we obtain a
homomorphism
AutG (Y ) → AutΩ (D).
Theorem 7.7 (Losev, unpublished). Let G be a connected reductive group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space
of G. Then, with the above notation, the homomorphism
AutG (Y ) → AutΩ (D).

(31)
is surjective.

This theorem will be proved in Appendix B, see Theorem B.4.
Corollary 7.8 (Strong version of Losev’s Uniqueness Theorem). Let G, H1 , H2 , Y1 =
G/H1 , Y2 = G/H2 be as in Proposition 7.5, in particular, X (Y1 ) = X (Y2 ), V(Y1 ) = V(Y2 ),
Ω(1) (Y1 ) = Ω(1) (Y2 ), and Ω(2) (Y1 ) = Ω(2) (Y2 ). Let ϕ : D(Y1 ) → D(Y2 ) be any bijection
satisfying
ρ2 ◦ ϕ = ρ1 , ς2 ◦ ϕ = ς1
(such a bijection exists because Ω(1) (Y1 ) = Ω(1) (Y2 ) and Ω(2) (Y1 ) = Ω(2) (Y2 )). Then there
exists a′ ∈ G(k) such that H2 = a′ H1 (a′ )−1 and
(φa′ )∗ = ϕ : D(Y1 ) → D(Y2 ).
Proof. By Proposition 7.5 there exists a ∈ G(k) such that H2 = aH1 a−1 . This element a
defines a map
(φa )∗ : D(Y1 ) → D(Y2 )
satisfying (30). Set
ψ = (φa )−1
∗ ◦ ϕ : D(Y1 ) → D(Y1 ),
then ψ satisfies
ρ1 ◦ ψ = ρ1 ,

ς1 ◦ ψ = ς1 ,

hence ψ ∈ AutΩ D(Y1 ). By Theorem 7.7 there exists n ∈ NG (H1 ) such that
(φn )∗ = ψ : D(Y1 ) → D(Y1 ).
We set a′ = an, then a′ H1 (a′ )−1 = H2 , φa′ = φa ◦ φn , and
(φa′ )∗ = (φa )∗ ◦ (φn )∗ = (φa )∗ ◦ ψ = ϕ.



Corollary 7.9. If in Theorem 7.7 H is spherically closed, then the homomorphism (31)
is an isomorphism.
Proof. Indeed, since H is spherically closed, the homomorphism (31) is injective, and by
Theorem 7.7 it is surjective, hence it is bijective, as required.

Note that
AutΩ (D) =

Y

Aut(ζ −1 (ω)),

ω∈Ω

Aut(ζ −1 (ω))

where
is the group of permutations of the set ζ −1 (ω). It is clear that for any
ω ∈ Ω, the restriction homomorphism
(32)

AutΩ (D) → Aut(ζ −1 (ω))

is surjective.
Corollary 7.10. If in Theorem 7.7 NG (H) = H, then the surjective map ζ is bijective,
hence D injects into V × P(S).

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

21

Proof. It follows from Theorem 7.7 and the surjectivity of the homomorphism (32), that
the group AutG (Y ) = NG (H)/H acts transitively on the fiber ζ −1 (ω) for any ω ∈ Ω.
Since by assumption NG (H)/H = {1}, we conclude that each fiber of ζ has exactly one
element, hence ζ is bijective, as required.

8. Action of an automorphism of the base field
on the combinatorial invariants of a spherical homogeneous space
8.1. Let k is an algebraically closed field, G be a connected reductive group over k, H1 ⊂ G
be a spherical subgroup, and Y1 = G/H1 be the corresponding spherical homogeneous
space.
Let k0 ⊂ k be a subfield and let γ ∈ Aut(k/k0 ). We assume that “G is defined over k0 ”,
that is, we are given a k0 -model G0 of G. Then we have a γ-semi-automorphism σγ of G,
see Subsection 6.2. Set H2 = σγ (H1 ) ⊂ G and denote by Y2 := G/H2 the corresponding
spherical homogeneous space.
We wish to know whether the spherical homogeneous spaces Y1 and Y2 are isomorphic
as G-varieties. For this end we compare their combinatorial invariants.
We fix a Borel pair (T, B), then T ⊂ B ⊂ G. Consider
σγ (T ) ⊂ σγ (B) ⊂ G.
Then (σγ (T ), σγ (B)) is again a Borel pair, hence there exists gγ ∈ G(k) such that
gγ · σγ (T ) · gγ−1 = T,

gγ · σγ (B) · gγ−1 = B.

Set τ = inn(gγ ) ◦ σγ : G → G, then τ is a γ-semi-automorphism of G, and
(33)
Set H2′
and Y2′

τ (B) = B,
Y2′

τ (T ) = T.

G/H2′ .

= τ (H1 ) ⊂ G and
=
We have H2′ = gγ · H2 · gγ−1 , so by Lemma 4.1 Y2
are isomorphic, and we wish to know whether Y1 and Y2′ are isomorphic.

By (33), τ acts on the characters of T and B; we denote the corresponding automorphism
by εγ . By definition
(34)

εγ (χ)(b) = γ(χ(τ −1 (b)))

for χ ∈ X∗ (B), b ∈ B(k),

and the same for the characters of T (recall that X∗ (B) = X∗ (T )). Since τ (B) = B, we see
that εγ , when acting on X∗ (T ), preserves S = S(G, T, B) ⊂ X∗ (T ). It is well known (see
e.g. [BKLR14, 3.2 and Proposition 3.1(a)]) that the automorphism εγ does not depend
on the choice of gγ and that the map
ε : Aut(k/k0 ) → Aut(X∗ (T ), S),
is a homomorphism. Since εγ acts on
εγ (Ω(1) (Y1 )), εγ (Ω(2) (Y1 )).

X∗ (B)

γ 7→ εγ

and on S, one can define εγ (X (Y1 )), εγ (V(Y1 )),

Following Akhiezer [Akh15], we compute the combinatorial invariants of the spherical
homogeneous space Y2′ . We define a map
(35)

Y1 (k) → Y2′ (k),

g · H1 7→ τ (g) · H2′ , where g ∈ G(k).

By Lemma 4.3 the map (35) is induced by some τ -equivariant γ-semi-isomorphism
ν : Y1 → Y2′ ,
and so we obtain an isomorphism of the function fields
ν∗ : K(Y1 ) → K(Y2′ ).
(B)

(B)

Lemma 8.2. Let χ1 ∈ X∗ (B) and assume that f1 ∈ K(Y1 )χ1 . Then ν∗ f1 ∈ K(Y2′ )χ2 ,
where χ2 = εγ (χ1 ).

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Proof. By assumption
f1 (b−1 y1 ) = χ1 (b) · f1 (y1 ) for all y1 ∈ Y1 (k), b ∈ B(k).
We write f2′ = ν∗ f1 ∈ K(Y2′ ). Since ν : Y1 → Y2′ is a γ-semi-isomorphism, by Corollary 1.8
we have
f2′ (y2′ ) = γ(f1 (ν −1 (y2′ ))) for y2′ ∈ Y2′ (k).
Note that τ −1 : G → G is a γ −1 -semi-automorphism of G, and ν −1 : Y2′ → Y1 is a τ −1 equivariant γ −1 -semi-isomorphism. Moreover, τ −1 (T ) = T and τ −1 (B) = B. We compute:
f2′ (b−1 · y2′ ) = γ(f1 (ν −1 (b−1 · y2′ ))) = γ(f1 ((τ −1 (b))−1 · ν −1 (y2′ )))
= γ(χ1 (τ −1 (b))) · γ(f1 (ν −1 (y2′ ))) = εγ (χ1 )(b) · f2′ (y2′ ).
(B)

Thus f2′ ∈ K(Y2′ )χ2 , where χ2 = εγ (χ1 ), as required.



Corollary 8.3. X (Y2′ ) = εγ (X (Y1 )).
Proof. By Lemma 8.2 we have εγ (X (Y1 )) ⊂ X (Y2′ ). Applying Lemma 8.2 to the triple
(γ −1 , τ −1 , ν −1 ), we obtain that εγ −1 (X (Y2′ )) ⊂ X (Y1 ), hence X (Y2′ ) ⊂ εγ (X (Y1 )). Thus
X (Y2′ ) = εγ (X (Y1 )), as required.

Let v1 ∈ ValB (K(Y1 )). We define ν∗ v1 ∈ ValB (K(Y2′ )) by
(ν∗ v1 )(f2′ ) = v1 (ν∗−1 (f2′ ))

for f2′ ∈ K(Y2′ ).

We consider the maps
ρ1 : ValB (K(Y1 )) → V (Y1 ) and

ρ′2 : ValB (K(Y2′ )) → V (Y2′ ).

Lemma 8.4. For any v1 ∈ ValB (K(Y1 )) we have
ρ′2 (ν∗ v1 ) = εγ (ρ1 (v1 )).
Proof. See Huruguen [Hu11, Proposition 2.18].



Corollary 8.5. V(Y2′ ) = εγ (V(Y1 )).



Let D1 ∈ D(Y1 ) be a color, that is, D1 is the closure of a B-orbit of codimension one in
Y1 . We set D2′ := ν(D1 ) ⊂ Y2′ , then D2′ ∈ D(Y2′ ). We also write ν∗ D1 for ν(D1 ).
Let D1 ∈ D(Y1 ) and D2′ ∈ D(Y2′ ), then we denote by val1 (D1 ) ∈ ValB (K(Y1 )) and
val′2 (D2′ ) ∈ ValB (K(Y2′ )) the corresponding B-invariant valuations.
Lemma 8.6. For any D1 ∈ D(Y1 ) we have
val′2 (ν∗ D1 ) = ν∗ (val1 (D1 )).
Proof. See Huruguen [Hu11, Proposition 2.19].



Remark 8.7. Propositions 2.18 and 2.19 of Huruguen [Hu11, Section 2.2] are proved in
his paper under certain additional assumptions. Namely, Huruguen assumes that that
k/k0 is a Galois extension, that the triple (G, Y, θ) has a k0 -model (G0 , Y0 , θ0 ), and that
Y0 has a k0 -point y (0) . Those assumptions are not used in his proof.
By abuse of notation, if D1 ∈ D(Y1 ) and D2′ ∈ D(Y2′ ), we write ρ1 (D1 ) for ρ1 (val1 (D1 )) ∈
V (Y1 ) and ρ′2 (D2′ ) for ρ′2 (val′2 (D2′ )) ∈ V (Y2′ ).
Corollary 8.8 (from Lemma 8.4 and Lemma 8.6). For any D1 ∈ D(Y1 ) we have
ρ′2 (ν∗ D1 ) = εγ (ρ1 (D1 )).

Lemma 8.9. For any D1 ∈ D(Y1 ) we have:

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

23

(i) StabG (ν∗ D1 ) = τ (StabG (D1 ));
(ii) ς(ν∗ D1 ) = εγ (ς(D1 )).
Proof. (i) follows from the fact that the map ν : Y1 (k) → Y2′ (k) is τ -equivariant, and (ii)
follows from (i).

Corollary 8.10 (from Corollary 8.8 and Lemma 8.9).
Ω(1) (Y2′ ) = εγ (Ω(1) (Y1 ))

and

Ω(2) (Y2′ ) = εγ (Ω(2) (Y1 )).


Proposition 8.11.
X (Y2 ) = εγ (X (Y1 )),

V(Y2 ) = εγ (V(Y1 )),
Ω(1) (Y2 ) = εγ (Ω(1) (Y1 )),

Ω(2) (Y2 ) = εγ (Ω(2) (Y1 )).

Proof. Since H2′ and H2 are conjugate, by Lemma 4.1 the G-varieties Y2′ and Y2 are
isomorphic, hence they have the same combinatorial invariants, and the proposition follows
from Corollaries 8.3, 8.5, and 8.10.

Note that Proposition 8.11 generalizes Propositions 5.2, 5.3, and 5.4 of Akhiezer [Akh15].
Namely, in the case when γ 2 = 1, our Proposition 8.11 is equivalent to those results of
Akhiezer. Our proofs are similar to his.
Proposition 8.12. With the notation and assumptions of Subsection 8.1, if the subgroups
H1 and H2 = σγ (H1 ) are conjugate, then εγ preserves the combinatorial invariants of Y1 ,
that is
(36)

εγ (X (Y1 )) = X (Y1 ),

εγ (V(Y1 )) = V(Y1 ),
εγ (Ω(1) (Y1 )) = Ω(1) (Y1 ),

εγ (Ω(2) (Y1 )) = Ω(2) (Y1 ).

Conversely, if equalities (36) hold and char k = 0, then H1 and H2 are conjugate.
Proposition 8.12 generalizes Theorem 3(1) of Cupit-Foutou [CF15], where the case k0 =
R was considered.
Proof. If H1 and H2 are conjugate, then by Lemma 4.1 the homogeneous spaces Y1 = G/H1
and Y2 = G/H2 are isomorphic as G-varieties, hence then they have the same combinatorial
invariants, that is,
(37)

X (Y2 ) = X (Y1 ),

V(Y2 ) = V(Y1 ),
Ω(1) (Y2 ) = Ω(1) (Y1 ),

Ω(2) (Y2 ) = Ω(2) (Y1 ),

and (36) follows from Proposition 8.11. Conversely, if equalities (36) hold and char k =
0, then by Proposition 8.11 the equalities (37) hold, and by Proposition 7.5 (Losev’s
Uniqueness Theorem) the subgroups H1 and H2 are conjugate.

Corollary 8.13. With the notation and assumptions of Subsection 8.1, if there exists
a σγ -equivariant γ-semi-automorphism µ : Y1 → Y1 , then εγ preserves the combinatorial
invariants of Y1 , that is, equalities (36) hold. Conversely, if equalities (36) hold and
char k = 0, then there exists a σγ -equivariant γ-semi-automorphism µ : Y1 → Y1 .
Proof. By Corollary 4.4 there exists a σγ -equivariant γ-semi-automorphism µ : Y1 → Y1 if
and only if the subgroup σγ (H1 ) of G is conjugate to H1 . Now the corollary follows from
Proposition 8.12.


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9. Equivariant models of automorphism-free spherical homogeneous spaces
9.1. Let k0 be a perfect field and let k be a fixed algebraic closure of k0 with Galois group
Γ = Gal(k/k0 ).
Let G be a connected reductive group over k. Let T ⊂ B ⊂ G be as in Section 7. We
consider the based root datum BRD(G) = BRD(G, T, B).
Let G0 be a k0 -model of G. For any γ ∈ Γ, this model defines a γ-semi-automorphism
σγ : G → G,
which induces an automorphism εγ ∈ AutBRD(G), see Section 8. We obtain a homomorphism
ε : Γ → AutBRD(G), γ 7→ εγ .
Let Y = G/H be a spherical homogeneous space of G. We consider the combinatorial
invariants of Y :
X = X (Y ) ⊂ X∗ (T ),

V = V(Y ) ⊂ HomZ (X , Q),
Ω(1) = Ω(1) (Y ),

Ω(2) = Ω(2) (Y ) ⊂ HomZ (X , Q) × P(S),

see Section 7. Since εγ acts on BRD(G), we can define
εγ (X ), εγ (V), εγ (Ω(1) ), εγ (Ω(2) ).
Recall that Y = G/H. By Lemma 4.3(i) we have γ∗ Y ∼
= G/σγ (H).
Proposition 9.2. If Y = G/H admits a G0 -equivariant k0 -model Y0 , then for all γ ∈ Γ,
εγ preserves the combinatorial invariants of Y , that is
(38)

εγ (X ) = X , εγ (V) = V, εγ (Ω(1) ) = Ω(1) , εγ (Ω(2) ) = Ω(2) .

Proposition 9.2 follows from formulas of Huruguen [Hu11, Section 2.2]. For the reader’s
convenience we prove it here.
Proof. A G0 -equivariant k0 -model Y0 of Y defines, for any γ ∈ Γ, a σγ -equivariant γ-semiautomorphism µγ of Y , hence an isomorphism of G-k-varieties (µγ )♮ : G/σγ (H) = γ∗ Y →
Y . We see that the G-varieties G/H and G/σγ (H) are isomorphic, hence they have the
same combinatorial invariants. By Proposition 8.11 the combinatorial invariants of the
spherical homogeneous space G/σγ (H) are
εγ (X ), εγ (V), εγ (Ω(1) ), εγ (Ω(2) ),
and (38) follows.



The next theorem is a partial converse of Proposition 9.2.
Theorem 9.3. Let k, k0 , Γ, G, H, G0 be as in 9.1. Assume that:
(i) For all γ ∈ Γ, εγ preserves the combinatorial invariants of Y , that is, equalities
(38) hold;
(ii) NG (H) = H;
(iii) char k = 0.
Then Y = G/H admits a G0 -equivariant k0 -model Y0 . This k0 -model is unique up to a
unique isomorphism.
Proof. Let γ ∈ Γ. Since char k = 0 and εγ preserves the combinatorial invariants of Y , by
Corollary 8.13 there exists a σγ -equivariant γ-semi-automorphism
µγ : Y → Y.

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

25

Thus condition (i) of Lemma 6.3 is satisfied.
Since NG (H) = H, by Corollary 5.2 AutG (Y ) = {1}, and by Lemma 6.5 conditions (ii)
and (iii) of Lemma 6.3 are satisfied.
The variety Y = G/H is quasi-projective, hence condition (iv) of Lemma 6.3 is satisfied.
By Lemma 6.3 there exists a G0 -equivariant k0 -model Y0 of Y . Since AutG (Y ) = {1},
for any given γ ∈ Γ the γ-semi-automorphism µγ is unique, hence the model Y0 is unique
up to a unique isomorphism.

Recall that a k0 -model G0 of a connected reductive k-group G is called an inner form
(of a split group) if εγ = 1 for all γ ∈ Γ = Gal(k/k0 ).
Lemma 9.4. Let k, k0 , Γ, G, H, G0 be as in 9.1. Then each of the conditions below
imply condition (i) of Theorem 9.3.
(i) G0 is an inner form;
(ii) G0 is absolutely simple (that is, G is simple) of one of the types A1 , Bn , Cn , E7 ,
E8 , F4 , G2 ;
(iii) G0 is split.
Proof. (i) If G0 is an inner form, then εγ = 1 for any γ ∈ Γ, hence condition (i) of Theorem
9.3 is clearly satisfied.
(ii) In these cases Dyn(G) has no nontrivial automorphisms, hence Γ acts trivially on
Dyn(G). We see that (ii) implies (i).
(iii) In this case clearly εγ = 1 for all γ ∈ Γ.



Corollary 9.5. If char k = 0, NG (H) = H, and at least one of the conditions (i–iii) of
Lemma 9.4 is satisfied, then Y admits a G0 -equivariant k0 -model, and this k0 -model is
unique.

Remark 9.6. Assume that k = R and NG (H) = H. The assertion that if condition (iii)
of Lemma 9.4 is satisfied, then Y has a unique G0 -equivariant R-model Y0 , is Theorem
4.12 of Akhiezer and Cupit-Foutou [ACF14]. The similar assertion when only condition
(i) of Lemma 9.4 is satisfied, is Theorem 1.1 of Akhiezer [Akh15]. Our paper is inspired
by this result of Akhiezer, and our proof of Theorem 9.3 is similar to his proof.
10. Equivariant models of spherically closed spherical homogeneous spaces
In this section we do not assume that NG (H) = H.
Let k, G, H, Y = G/H, T ⊂ B ⊂ G be as in Section 7, in particular, k is algebraically
closed, and we assume that char k = 0. The group AutG (Y ) = NG (H)/H acts on Y and
on the set D of colors of Y .
Definition 10.1. A spherical homogeneous space Y = G/H is called spherically closed if
NG (H)/H acts on D faithfully, that is, if the homomorphism
AutG (Y ) → Aut(D)
is injective. (Here Aut(D) denotes the group of permutations of the finite set D.)
Let k0 ⊂ k be a subfield such that k is an algebraic closure of k0 . Let G0 be a k0 -model
of G, and for γ ∈ Γ := Gal(k/k0 ) let σγ : G → G be the γ-semi-automorphism defined by
G0 . Let εγ : X∗ (T ) → X∗ (T ) be as in (34).
Theorem 10.2. Let k be an algebraically closed field. Let G, H, Y = G/H be as in
Section 7. Let G0 be a k0 -model of G, where k0 ⊂ k is a subfield such that k is an
algebraic closure of k0 . Assume that

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(i) εγ preserves the combinatorial invariants of Y for all γ ∈ Γ,
(ii) Y is spherically closed,
(iii) char k = 0.
Then Y admits a G0 -equivariant k0 -model.
Theorem 10.2 generalizes the existence assertion of Theorem 9.3. It was inspired by
Corollary 1 of Cupit-Foutou [CF15, Section 2.5], where the case k0 = R was considered.
In order to prove the theorem we need a few lemmas.
Lemma 10.3. Let ζ : D → Ω be a mapping of nonempty finite sets. Let Γ be a group
acting on Ω by a homomorphism
s : Γ → Aut(Ω),

γ 7→ sγ .

Assume that for any γ ∈ Γ there exists a permutation mγ : D → D covering sγ , that is,
such that the following diagram commutes:
D

mγ

/D

ζ

ζ



Ω

sγ



/Ω

Then there exists a homomorphism m′ : Γ → Aut(D) such that:
(i) for any γ ∈ Γ the permutation m′γ covers sγ ;
(ii) for any γ ∈ Γ we have m′γ = aγ ◦ mγ , where aγ ∈ AutΩ (D);
(iii) m′γ = idD for all γ ∈ ker s.
Proof. We may and shall assume that Γ acts transitively on Ω. Let ω, ω ′ ∈ Ω, then there
exists γ ∈ Γ such that sγ (ω) = ω ′ . By hypotheses there exists mγ ∈ Aut(D) covering
sγ , then mγ induces a bijection ζ −1 (ω) → ζ −1 (ω ′ ), hence the cardinalities of ζ −1 (ω) and
ζ −1 (ω ′ ) are equal. We see that ω 7→ |ζ −1 (ω)| is a constant function on Ω; we denote its
value by n. For each ω ∈ Ω we fix some bijection between ζ −1 (ω) and the set {1, . . . , n};
(i)
we denote the element of ζ −1 (ω) ⊂ D corresponding to i ∈ {1, . . . , n} by dω . Then we
define m′γ ∈ Aut(D) by
(i)

m′γ (d(i)
ω ) = dsγ (ω) .
Since s : γ 7→ sγ is a homomorphism, we see that m′ : γ 7→ m′γ is a homomorphism, and
clearly m′γ covers sγ , which proves (i). Set aγ = m′γ ◦ m−1
γ , then clearly (ii) holds, and the
assertion (iii) holds by construction.

10.4. Write
ζ = ρ × ς : D → V × P(S),

Ω = im ζ,

sγ = εγ |Ω : Ω → Ω,

then the map
Γ → Aut(Ω),

γ 7→ sγ

is a homomorphism. Assume that for all γ ∈ Γ there exists a σγ -equivariant γ-semiautomorphism µ : Y → Y , that is, a γ-semi-automorphism satisfying
(39)

µ(g · y) = σγ (g) · µ(y) for all g ∈ G(k), y ∈ Y (k).

The following lemma is obvious:
Lemma 10.5. If γ, δ ∈ Aut(k/k0 ), µ is a σγ -equivariant γ-semi-automorphism, and ν is a
σδ -equivariant δ-semi-automorphism, then µν is a σγδ -equivariant γδ-semi-automorphism
and µ−1 is a σγ −1 -equivariant γ −1 -semi-automorphism.


EQUIVARIANT MODELS OF SPHERICAL VARIETIES

27

10.6. Consider σγ (T ) ⊂ σγ (B) ⊂ G. There exists gγ ∈ G(k) such that if we set σγ′ =
inn(gσ ) ◦ σγ , then
σγ′ (T ) = T

(40)

and σγ′ (B) = B.

For µ : Y → Y as in (39), we define a γ-semi-automorphism
µ′ = gγ ◦ µ : Y → Y,

y 7→ gγ · µ(y)

for y ∈ Y (k).

Then for g ∈ G(k), y ∈ Y (k) we have
(41)

µ′ (g · y) = gγ · µ(g · y) = gγ · σγ (g) · µ(y)
= (gγ · σγ (g) · gγ−1 ) · (gγ · µ(y)) = σγ′ (g) · µ′ (y).

Let D ∈ D = D(Y ) be a color, this means that D is the closure of a codimension one
B-orbit in Y . Since σγ′ (B) = B, it follows from (41) that the divisor µ′ (D) in Y is the
closure of a codimension one B-orbit, that is, a color. We obtain a permutation
mµ : D → D,

(42)

D 7→ µ′ (D),

covering sγ . Since gγ for which (40) holds is defined uniquely up to multiplication on the
left by an element t ∈ T (k) ⊂ B(k), we see that mµ depends only on µ and does not
depend on the choice of gγ .
Lemma 10.7. The map µ 7→ mµ is a homomorphism: for γ, µ, δ, ν as in Lemma 10.5,
we have mµ◦ν = mµ ◦ mν .
Proof. Let (γ, µ) and (δ, ν) be as in Lemma 10.5. Choose gγ , gδ ∈ G(k) such that
(43)

gγ · σγ (T, B) · gγ−1 = (T, B),

gδ · σδ (T, B) · gδ−1 = (T, B).

Set
µ′ = gγ ◦ µ : Y → Y,

ν ′ = gδ ◦ ν : Y → Y.

Then for y ∈ Y (k) we have
(44)

(µ′ ν ′ )(y) = µ′ (ν ′ (y)) = gγ · µ(gδ · ν(y)) = gγ · σγ (gδ ) · (µν)(y).

On the other hand, from (43) we obtain
gγ · σγ (gδ · σδ (T, B) · gδ−1 ) · gγ−1 = (T, B),
hence
gγ σγ (gδ ) · σγδ (T, B) · (gγ σγ (gδ ))−1 = (T, B).
Thus we may set
(µν)′ = (gγ σγ (gδ )) ◦ µν ,
that is,
(µν)′ (y) = gγ σγ (gδ ) · (µν)(y)

for y ∈ Y (k).

Comparing with (44), we see that with this (µν)′ we have
(µν)′ = µ′ ◦ ν ′ .
hence
mµν = mµ ◦ mν ,
as required.



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Proof of Theorem 10.2. Let γ ∈ Γ. Since char k = 0 and εγ preserves the combinatorial invariants of Y = G/H, by Corollary 8.13 there exists a σγ -equivariant γ-semi-automorphism
µγ : Y → Y . Set
mγ = mµγ ∈ Aut(D),
see Subsection 10.6, then mγ covers sγ , where sγ ∈ Aut(Ω) is the restriction of εγ to Ω.
By Lemma 10.3 there exists a homomorphism
m′ : Γ → Aut(D),

γ 7→ m′γ

such that for any γ ∈ Γ the permutation m′γ ∈ Aut(D) covers sγ (property (i)) and
we have m′γ = aγ ◦ mγ , where aγ ∈ AutΩ (D) (property (ii)). By Theorem 7.7 there
exists an automorphism e
aγ ∈ AutG (Y ) inducing aγ on D. We set µ′γ = e
aγ ◦ µγ , then
′
′
µγ is a σγ -equivariant γ-semi-automorphism of Y , and by Lemma 10.7, µγ acts on D by
aγ ◦ mγ = m′γ .
Let γ, δ ∈ Γ, then µ′γ µ′δ (µ′γδ )−1 ∈ AutG (Y ) and by Lemma 10.7 it acts on D by
m′γ m′δ (m′γδ )−1 = idD . Since Y is spherically closed, we conclude that µ′γ µ′δ (µ′γδ )−1 = idY ,
hence µγδ = µγ ◦ µδ . Thus the map γ 7→ µ′γ is a homomorphism.
It is easy to see that Y admits a Gk2 -equivariant k2 -model Y2 over some finite Galois
extension k2 /k0 in k. Let Γ2 = Gal(k/k2 ), and for γ ∈ Γ2 let µ′′γ denote the γ-semiautomorphism of Y defined by the k2 -model Y2 . After passing to a finite extension, we
may assume that for γ ∈ Γ2 we have sγ = idΩ , and by property (iii) of Lemma 10.3 we
have m′γ = idD . Moreover, we may assume that for γ ∈ Γ2 the semi-automorphism µ′′γ
acts trivially on D. It follows that (µ′′γ )−1 µ′γ acts trivially on D, and clearly (µ′′γ )−1 µ′γ ∈
AutG (Y ). Since Y is spherically closed, we conclude that (µ′′γ )−1 µ′γ = idY , and hence,
µ′γ = µ′′γ for γ ∈ Γ2 . We see that the homomorphism γ 7→ µ′γ satisfies condition (iii) of
Lemma 6.3. Note that Y = G/H is quasi-projective, that is, condition (iv) of Lemma 6.3
is satisfied as well. By Lemma 6.3 the homomorphism γ 7→ µ′γ defines a G0 -equivariant
k0 -model Y0 of Y , which completes the proof of the theorem.

10.8. In Example 0.1 we considered a spherically closed spherical variety Y = G/H,
where G = SL2,k and H = T , a maximal torus in G. In this case it is obvious that for
any k0 -model G0 of G there exists a G0 -equivariant k0 -model Y0 of Y = G/H. Indeed,
there exists a maximal torus T0 ⊂ G0 defined over k0 , and it is clear that Y0 := G0 /T0 is a
G0 -equivariant k0 -model of Y = G/T . In the following example we consider a spherically
closed spherical subgroup that is not conjugate to a subgroup defined over k0 .
Example 10.9. Let k = C, k0 = R. Following a suggestion of Roman Avdeev, we take
G = SO2n+1,C , where n ≥ 2, and we take for H a Borel subgroup of SO2n,C , where
SO2n,C ⊂ SO2n+1,C = G. By Proposition 10.10 below, H is a spherically closed spherical
subgroup of G and NG (H) 6= H. Take G0 = SO2n+1,R , then G0 is an anisotropic (compact)
R-model of G. Since the Dynkin diagram of G has no nontrivial automorphisms, G0 is an
inner form. We wish to show that Y = G/H admits a G0 -equivariant R-model. Clearly
H is not conjugate to any subgroup of G0 defined over R because H is not reductive,
and therefore, we cannot argue as in Subsection 10.8. Since NG (H) 6= H, we cannot
apply Theorem 9.3 either. However, since H is spherically closed, by Theorem 10.2 the
homogeneous variety Y = G/H does admit a G0 -equivariant R-model Y0 .
Proposition 10.10 (Roman Avdeev, private communication). Let G = SO2n+1,C , where
n ≥ 2. Let H be a Borel subgroup of SO2n,C , where SO2n,C ⊂ SO2n+1,C = G. Then H is
a spherically closed spherical subgroup of G, and NG (H) 6= H.
Proof. Set g = Lie(G). Choose a Borel subgroup B ⊂ G and a maximal torus T ⊂ B. Let
X = X∗ (T ) denote the character lattice of T and let R = R(G, T ) ⊂ X be the root system.
The Borel subgroup B defines a set of positive roots R+ ⊂ R and the corresponding set of

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

29

simple roots S ⊂ R+ ⊂ R. Let U denote the unipotent radical of B and put u = Lie(U ).
We have
M
M
g = Lie(T ) ⊕
gβ ,
u=
gβ ,
β∈R

β∈R+

where gβ is the root subspace corresponding to a root β.

Let Rl ⊂ R denote the root subsystem consisting of the long roots. Observe that R is
a root system of type Bn , and Rl is a root system of type Dn . Set Rl+ = R+ ∩ Rl . We set
M
M
gl = Lie(T ) ⊕
gβ ,
ul =
gβ .
β∈R+
l

β∈Rl

Let Gl (resp., Ul ) be the connected algebraic subgroup of G with Lie algebra gl (resp., ul ).
Set H = T Ul . Then Gl ≃ SO2n,C and H is a Borel subgroup of Gl .
It is well known that H is a spherical subgroup of G. For example, this fact follows from
Theorem 1 of Avdeev [Av11] (to apply this theorem one needs to check that all the short
positive roots in R are linearly independent). By [Av15, Proposition 5.25] H is spherically
closed.
We consider the Weyl group W = W (G, T ) = W (R). Let r ∈ W = NG (T )/T denote
the reflection with respect to the short simple root, and let ρ be a representative of r
in NG (T ). Since r preserves Rl+ , we see that ρ ∈ NG (H). Since ρ ∈ NG (T ) r T and
NG (T ) ∩ B = T , we see that ρ ∈
/ B. By construction H ⊂ B, and we conclude that ρ ∈
/ H,
hence NG (H) 6= H. In fact, NG (H) = H ∪ ρH by [Av13, Theorem 3].

The following example shows that G/H might have no G0 -equivariant k0 -model when
H is not spherically closed.
Example 10.11. Let k = C, k0 = R. Let n ≥ 1, G = Sp2n,C ×C Sp2n,C , Y = Sp2n,C ,
the group G acts on Y by
(g1 , g2 ) ∗ y = g1 y g2−1 ,

g1 , g2 , y ∈ Sp2n (C).

Let H denote the stabilizer in G of 1 ∈ Sp2n (C) = Y (C), then H = Sp2n,C embedded
diagonally in G. We have Y = G/H, and Y is a spherical homogeneous space of G. We
have NG (H) = Z(G)·H, where Z(G) denotes the center of G. It follows that NG (H)/H ≃
{±1} =
6 {1}. Clearly NG (H)/H acts trivially on D(G/H), so H is not spherically closed.
Consider the following real model of G:
G0 = Sp2n,R ×R Sp(n),
where Sp(n) is the compact real form of Sp2n . We show that Y cannot have a G0 equivariant real model, although G0 is an inner form of a split group.
Indeed, assume for the sake of contradiction that such a real model Y0 of Y exists.
We have Y = Sp2n,C , and Y0 is simultaneously a principal homogeneous space of Sp2n,R
and of Sp(n). Since H 1 (R, Sp2n,R ) = 1, we see that Y0 (R) is not empty. It follows that
the topological space Y0 (R) is simultaneously a principal homogeneous space of Sp(2n, R)
and of Sp(n). Thus Y0 (R) is simultaneously homeomorphic to the noncompact Lie group
Sp(2n, R) and to the compact Lie group Sp(n), which is clearly impossible. Thus, there
is no G0 -equivariant real model Y0 of Y .
10.12. Let k, k0 , Γ, G, H, G0 be as in Subsection 9.1, in particular, Γ = Gal(k/k0 ).
We do not assume that char k = 0. We assume that H is spherically closed and that
Y = G/H admits a G0 -equivariant k0 -model Y0 . Then by Corollary 8.13 εγ preserves
the combinatorial invariants of Y for all γ ∈ Γ, in particular, Γ acts on the finite set
Ω(2) = Ω(2) (Y ). Let U1 , U2 , . . . , Ur be the orbits of Γ in Ω(2) . For each i = 1, 2, . . . , r, let
us choose a point ui ∈ Ui . Set Γi = StabΓ (ui ).

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Theorem 10.13. With the notation and assumptions of 10.12 we have:
(i) The set of isomorphism classes of G0 -equivariant k0 -models of Y is canonically a
1
principal homogeneous
Qr space of the abelian group H (Γ, AutΩ (D));
1
(ii) H (Γ, AutΩ (D)) ≃ i=1 Hom(Γi , S2 ), where S2 is the symmetric group on two
symbols.
Note that S2 ∼
= {±1} ∼
= Z/2Z.
Corollary 10.14. In Theorem 10.13 assume that |Γ| = 2. Then the number of isomorphism classes of G0 -equivariant k0 -models of Y = G/H is 2s , where s is the number of
fixed points of Γ in Ω(2) .
Proof. Let Ui be an orbit of Γ in Ω(2) . If |Ui | = 2, then Γi = {1}, hence |Hom(Γi , S2 )| = 1.
If |Ui | = 1, then Γi = Γ, and hence, |Hom(Γi , S2 )| = 2. Now the corollary follows from the
theorem.

Proof of Theorem 10.13. Since Y is quasi-projective, the set of the isomorphism classes
in the theorem is in a canonical bijection with the pointed set H 1 (Γ, AutG (Y )); see Serre
[Se97], Proposition 5 in Section III.1.3. By Theorem 2 of Losev [Lo09] (see also Theorem
B.1 below), the group AutG (Y ) is abelian, hence H 1 (Γ, AutG (Y )) is an abelian group,
and the set of isomorphism classes in the theorem is canonically a principal homogeneous
space of this abelian group. By Corollary 7.9 there is a canonical isomorphism of abelian
∼
groups AutG (Y ) → AutΩ (D), and (i) follows.
We compute H 1 (Γ, AutΩ (D)). Recall that we have a surjective map ζ : D → Ω. Set
D (2) = ζ −1 (Ω(2) ), then clearly


r
Y
Y
Y

AutΩ (D) = AutΩ(2) (D (2) ) =
S2 =
S2  ,
ω∈Ω(2)

hence

1

H (Γ, AutΩ (D)) =

r
Y

H 1 (Γ,

i=1

i=1

Y

ω∈Ui

S2 ).

ω∈Ui

Since Γ acts on Ui transitively, by the lemma of Faddeev and Shapiro, see Serre [Se97,
I.2.5, Proposition 10], we have
Y
H 1 (Γ,
S2 ) ≃ H 1 (Γi , S2 ) = Hom(Γi , S2 ).
ω∈Ui

Thus
H 1 (Γ, AutΩ (D)) ≃

r
Y

Hom(Γi , S2 ),

i=1

which proves (ii).



Example 10.15. Let G = SO3,C ≃ PGL2,C . Let T ⊂ G be a maximal torus. We take
H = T and consider Y = G/H = G/T , which is a spherical homogeneous space of G. We
have
AutG (G/T ) = NG (T )/T ≃ {±1}.
Let G0 be an R-form of G, and let T0 be a maximal torus of G0 (defined over R), then
clearly G0 /T0 is a G0 -equivariant R-model of Y = G/T (so we do not have to refer to
Theorem 10.2 in order to see that Y admits a G0 -equivariant real model). Since
H 1 (Γ, AutG (G/T )) = Hom(Γ, {±1}) ≃ {±1},

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

31

Y has exactly two G0 -equivariant R-models. We describe such models for each R-model
of G = SO3,C .
Consider the indefinite real quadratic form in three variables
F2,1 (x1 , x2 , x3 ) = x21 + x22 − x23 ,

xi ∈ R.

Set G0 = SO(F2,1 ) = SO2,1 , which is a noncompact (split) R-model of G. We consider
+
the affine quadric Y2,1
⊂ A3R given by the equation F2,1 (x) = +1, and the affine quadric
−
+
−
3
Y2,1 ⊂ AR given by the equation F2,1 (x) = −1. Then Y2,1
and Y2,1
are SO2,1 -equivariant
+
R-models of Y = G/T . It is well known that Y2,1 (R) is a hyperboloid of one sheet, hence
−
it is connected, while Y2,1
(R) is a hyperboloid of two sheets, hence it is not connected.
+
−
It follows that the R-varieties Y2,1
and Y2,1
are two non-isomorphic SO2,1 -equivariant
R-models of Y = G/T .
Now consider the positive definite real quadratic form in three variables
F3 (x1 , x2 , x3 ) = x21 + x22 + x23 ,

xi ∈ R.

Set G0 = SO(F3 ) = SO3,R , which is a compact (anisotropic) R-model of G. We consider
the affine quadric Y3+ ⊂ A3R given by the equation F3 (x) = +1, and the affine quadric
Y3− ⊂ A3R given by the equation F3 (x) = −1. Then Y3+ and Y3− are SO3,R -equivariant
R-models of Y = G/T . Clearly, Y3+ (R) is the unit sphere in R3 , hence it is nonempty,
while Y3− (R) is empty. It follows that the R-varieties Y3+ and Y3− are two non-isomorphic
SO3,R -equivariant R-models of Y = G/T .
11. Equivariant models of spherical embeddings
of automorphism-free spherical homogeneous spaces
In this section we assume that NG (H) = H.
Theorem 11.1. Let k, G, H, Y = G/H, k0 , Γ, G0 be as in Section 9.1 . We assume
that
(i) G0 is an inner form of a split group,
(ii) NG (H) = H,
(iii) char k = 0.
Let Y ֒→ Y ′ be an arbitrary spherical embedding of Y . Then Y ′ admits a G0 -equivariant
k0 -model Y0′ . This model is compatible with the k0 -model Y0 of Y from Theorem 9.3 and
is unique up to a canonical isomorphism.
This theorem generalizes Theorem 1.2 of Akhiezer [Akh15], who considered the case
k0 = R. Note that Akhiezer considered only the wonderful embedding of Y , while we
consider an arbitrary spherical embedding, so our result is new even in the case k0 = R.
Proof. We show that a k0 -model of Y ′ , if exists, is unique. Indeed, let Y0′ be such a
k0 -model. For γ ∈ Γ := Gal(k/k0 ), let µ′γ : Y ′ → Y ′ be the corresponding γ-semiautomorphism of Y ′ . Since Y is the only open G-orbit in Y ′ , it is stable under µ′γ for all
γ ∈ Γ. Since k0 is a perfect field, this defines a G0 -equivariant k0 -model Y0 of Y , which is
unique because NG (H) = H and hence AutG (Y ) = {1}. Since Y is Zariski-dense in Y ′ , we
conclude that the model Y0′ of Y ′ is unique. (This argument does not use the assumption
that char k = 0.)
We prove the existence. By Theorem 9.3 Y admits a unique G0 -equivariant k0 -model
Y0 . The model Y0 defines an action of Γ on the finite set D, see e.g. Huruguen [Hu11,
2.2.5]. Namely, for any γ ∈ Γ we have a σγ -equivariant γ-semi-automorphism µγ , which
induces an automorphism mγ : D → D covering sγ : Ω → Ω, see (42). We show that this

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action of Γ on D is trivial. Indeed, since NG (H) = H, by Corollary 7.10 the surjective
map
ζ: D → Ω
is bijective. Since by assumption G0 is an inner form, for all γ ∈ Γ we have εγ = 1, hence
sγ = 1. Thus Γ acts trivially on Ω and on D.
Let CF(Y ′ ) denote the colored fan of Y ′ (see Knop [Kn89] or Perrin [Pe14, Definition
3.1.9]) which is a set of colored cones (C, F) ∈ CF(Y ′ ), where C ⊂ V and F ⊂ D. We
know that Γ acts trivially on V = HomZ (X , Q) and on D. It follows that for any γ ∈ Γ
and for any colored cone (C, F) ∈ CF(Y ′ ), we have
(45)

γ∗ (C) = C,

γ∗ (F) = F.

It follows that the colored fan CF(Y ′ ) is Γ-stable. Moreover, it follows from (45) that the
hypothesis of Theorem 2.26 of Huruguen [Hu11] is satisfied, that is, Y ′ has a covering by
G-stable and Γ-stable open quasi-projective subvarieties. By this theorem Y ′ admits a
G0 -equivariant k0 -model compatible with Y0 .

Remark 11.2. Huruguen [Hu11] assumes that Y0 has a k0 -point, but he does not use
that assumption.
Remark 11.3. In Theorem 11.1 we do not assume that Y ′ is quasi-projective.
Appendix A. Algebraically closed descent
for spherical homogeneous spaces
The proofs in this appendix were communicated to the author by experts.
Theorem A.1. Let G0 be a connected reductive group defined over an algebraically
closed field k0 of characteristic 0. Let k ⊃ k0 be a larger algebraically closed field. We set
G = G0 ×k0 k, the base change of G0 from k0 to k. Let H ⊂ G be a spherical subgroup of
G (defined over k). Then H is conjugate to a (spherical) subgroup defined over k0 .
Proof. The theorem will be proved in five steps.
1) Let X0 be a variety equipped with an action of G0 . Then X0 is the disjoint union
of locally closed G0 -stable subvarieties X0m consisting of all orbits of a fixed dimension m.
The orbits of maximal (resp. minimal) dimension form an open (resp. closed) subvariety.
To see this, consider the product G0 × X0 and the subvariety Y0 consisting of the pairs
(g, x) such that gx = x, equipped with the projection to X0 . By a theorem of Chevalley, see
EGA [Gr66, 13.1.3], the dimension of fibers of this projection is an upper semicontinuous
function on Y0 . Restricting this function to X0 (viewed as the closed subvariety of Y0 on
which g = e), it follows that the dimension of the G0 -orbit is an upper semicontinuous
function on X0 .
2) Take for X0 the variety of Lie subalgebras of g0 = Lie G0 of a fixed codimension, say
r, and let X be the k-variety obtained from X0 by scalar extension. Then X is the variety
of Lie subalgebras of codimension r in g = Lie G. Moreover, the stabilizer of a k-point h
in X is the normalizer of h (viewed as a Lie algebra) in G. So the dimension of the orbit
of h is
dim(G) − dim NG (h) = dim(G) − dim(h) − (dim NG (h) − dim(h)) = r − dim ng (h)/h,
where ng (h) denotes the normalizer of h in g. Thus, if there exists h such that h = ng (h),
then the Lie subalgebras h satisfying this property are the k-points of the open subset of
orbits of maximal dimension. Note that ng (h) is the Lie algebra of NG (h). So if h = ng (h),
then h is an algebraic Lie algebra.

EQUIVARIANT MODELS OF SPHERICAL VARIETIES

33

3) Let H be a spherical subgroup of G with Lie algebra h such that ng (h) = h. We
claim that the orbit G · h in X is open.
To prove this, recall that G · h has maximal dimension among G-orbits in X. Since h is
a spherical Lie subalgebra, all Lie subalgebras h′ in an open neighborhood U of h in X are
spherical and their orbits are of the same (maximal) dimension; thus, NG (h′ ) is a spherical
subgroup of G, with Lie algebra h′ by Step 2. By Theorem 3.1 of Alexeev and Brion [AB05],
only finitely many conjugacy classes of spherical subgroups of the form NG (h′ ) for h′ ∈ U
are obtained in this way; let H1 , . . . , Hr be representatives of the conjugacy classes. We
write hi = Lie Hi , then we see that any the spherical Lie algebra h′ ∈ U is conjugate to
one of h1 , . . . , hr ; in particular, U intersects nontrivially with finitely many G-orbits in X,
and all these orbits are of the same dimension. It follows that all these orbits are open, in
particular, the orbit G · h in X is open.
4) By Step 3, the Lie algebras h of spherical subgroups H of G such that ng (h) = h
form finitely many G-orbits, and the closures of these orbits are irreducible components
of the variety X, which is defined over k0 . It follows that every such orbit is defined over
k0 . Since k0 is algebraically closed, every such G-orbit has a k0 -point, which proves the
theorem for spherical subgroups such that ng (h) = h. Also
NG (h) = NG (H 0 ) = NG (H),
where the latter equality follows from Corollary 5.2 of Brion and Pauer [BP87]. Thus the
condition that ng (h) = h is equivalent to the condition that NG (H)/H is finite.
5) To handle the case of an arbitrary spherical k-subgroup H of G, consider the spherical
closure of H, that is, the algebraic subgroup H of NG (H) containing H such that
H/H = ker [NG (H)/H → Aut D(G/H)] .
By Corollary A.3 below, the spherical closure H is spherically closed, that is, NG (H)/H
acts faithfully on the finite set of colors of G/H, hence the group NG (H)/H is finite, and
therefore, ng (Lie H) = Lie H. By Step 4 we may assume that H is defined over k0 . Now H
is an intersection of kernels of characters of H (since the quotient H/H is diagonalizable)
and every such character is defined over k0 . Thus H is defined over k0 , as required.
An alternative proof, also based on Theorem 3.1 of Alexeev and Brion [AB05], is
sketched in Knop’s MathOverflow answer [Kn17b].

From now on till the end of this appendix we follow Avdeev [Av15]. Let G be a
connected reductive group over an algebraically closed field k of characteristic 0. Fix a
e → G such that G
e is a direct product of a torus with a simply
finite covering group G
e
connected semisimple group. For every simple G-module
V , the corresponding projective
space P(V ) has the natural structure of a G-variety. Every G-variety arising in this way
is said to be a simple projective G-space.
Proposition A.2 (Bravi and Luna [BL11, Lemma 2.4.2], see also Avdeev [Av15, Corollary
3.24]). For any spherical subgroup H of a connected reductive group G over an algebraically
closed field k of characteristic 0, the spherical closure H of H is the common stabilizer in
G of all H-fixed points in all simple projective G-spaces.
Corollary A.3 (well known). Let H be a spherical subgroup of a connected reductive
group G defined over an algebraically closed field k of characteristic 0. Let H ′ denote the
spherical closure of H, and let H ′′ denote the spherical closure of H ′ . Then H ′′ = H ′ ,
that is, H ′ is spherically closed.
This result was stated without proof in Section 6.1 of Luna [Lu01] (see also Avdeev
[Av15, Corollary 3.25]).

34

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WITH AN APPENDIX BY

GIULIANO GAGLIARDI

Deduction of Corollary A.3 from Proposition A.2. Let P(V ) be a simple projective G-space.
Let P(V )H denote the set of fixed points of H in P(V ). Since H ⊂ H ′ , we have
′
′
P(V )H ⊂ P (V )H . By Proposition A.2 applied to H, we have P(V )H ⊃ P (V )H . Thus
′
P(V )H = P(V )H .
By Proposition A.2 applied to H ′ , the group H ′′ (k) is the set of g ∈ G(k) that fix
′
P(V )H for all simple projective G-spaces P(V ). By Proposition A.2 applied to H, the
group H ′ (k) is the set of g ∈ G(k) that fix P(V )H for all simple projective G-spaces P(V ).
′

Since P(V )H = P(V )H , we have H ′′ = H ′ , as required.
Appendix B. The action of the automorphism group on the colors
of a spherical homogeneous space
By Giuliano Gagliardi
In this appendix we prove Theorem 7.7, which we restate below as Theorem B.4. Our
proof is based on Friedrich Knop’s MathOverflow answer [Kn17a] to Borovoi’s question.
Knop writes that Theorem B.4 was communicated to him by Ivan Losev.
Let G be a connected reductive group over an algebraically closed field k of characteristic 0. Let Y = G/H be a spherical homogeneous space.
We present results of Knop [Kn96] and Losev [Lo09] describing AutG (Y ), using the
notation of Section 7. Consider the uniquely determined set Σ ⊂ X of linearly independent
primitive elements in the lattice X such that
\
{v ∈ V : hγ, vi ≤ 0}.
V=
γ∈Σ

The elements of Σ are called the spherical roots of Y . For the description of AutG (Y ) a
related subset Σ ⊂ X is used, see Losev [Lo09, Introduction, before Theorem 2]. Losev
has shown how Σ can be computed from Σ and D.
Theorem B.1 (Knop [Kn96, Theorem 5.5]). For every φ ∈ AutG (Y ) and every χ ∈ X
there exists aφ,χ ∈ k× such that
φ|K(Y )(B) = aφ,χ id.
χ

The resulting homomorphism
AutG (Y ) → Hom(X , k× )
is injective and its image is {ψ ∈ Hom(X , k× ) : ψ(Σ) = {1}}.
For α ∈ S, let D(α) denote the set of colors D ∈ D such that the parabolic subgroup
Pα moves D, that is, α ∈ ς(D). We need the following results of Luna [Lu97], [Lu01]:
Proposition B.2. Let α ∈ S.
(1) We have |D(α)| ≤ 2. Moreover, |D(α)| = 2 if and only if α ∈ Σ ∩ S.
(2) Assume |D(α)| = 2 and write D(α) = {Dα+ , Dα− }. If ρ(Dα+ ) = ρ(Dα− ), then:
(i) hρ(Dα+ ), χi = hρ(Dα− ), χi = 21 hα∨ , χi for all χ ∈ X , where α∨ ∈ X∗ (T ) is the
corresponding simple coroot;
(ii) we have ς(Dα+ ) = ς(Dα− ) = {α}.
Proof. For (1), see Luna [Lu97, Sections 2.6 and 2.7] or Timashev [Tim11, 30.10]. For
(2), we use that [Lu01, Theorem 2] or [Tim11, Theorem 30.22] implies that the invariants
of a spherical homogeneous space satisfy the axioms of a homogeneous spherical datum.
These axioms are stated in [Lu01, Sections 2.1 and 2.2] and [Tim11, Definition 30.21]. In
particular, we have ρ(Dα+ ) + ρ(Dα− ) = α∨ |X and for every β ∈ ς(Dα± ) we have β ∈ X and
hρ(Dα± ), βi = 1. With the assumption ρ(Dα+ ) = ρ(Dα− ), we obtain (i) and then (ii).


EQUIVARIANT MODELS OF SPHERICAL VARIETIES

35

We need the following result of Losev:
Proposition B.3. The set Σ has the following properties:
(1) The set Σ can be obtained from Σ by replacing some elements γ ∈ Σ by 2γ and
leaving the other elements γ ∈ Σ unchanged.
(2) If α ∈ Σ ∩ S and hρ(Dα+ ), χi = hρ(Dα− ), χi = 21 hα∨ , χi for all χ ∈ X , then 2α ∈ Σ
(hence α ∈
/ Σ).
Proof. See Losev [Lo09, Theorem 2 and Definition 4.1.1(1)].



The following theorem is the main result of this appendix:
Theorem B.4 (Losev, unpublished). The homomorphism
(46)

AutG (Y ) → AutΩ (D)

is surjective.
Proof of Theorem B.4. Let A denote the set of simple roots α ∈ S such that |D(α)| = 2
and ρ(Dα+ ) = ρ(Dα− ). By Proposition B.2, for every α ∈ A we have ς(Dα+ ) = ς(Dα− ) = {α},
hence the map α 7→ {Dα+ , Dα− } is a bijection between A and the set of unordered pairs
{Dα+ , Dα− } such that (ρ × ς)(Dα+ ) = (ρ × ς)(Dα− ). Note that there is a canonical bijection
A → Ω(2) ,

α 7→ (ρ × ς)(Dα+ ).

By Proposition B.2(1), for every α ∈ A we have α ∈ S ∩ Σ ⊂ X , hence there exists
(B)
fα ∈ K(Y )α , fα 6= 0. Moreover, from Propositions B.2 and B.3 we obtain 2α ∈ Σ (and
α∈
/ Σ).
We want to show that for any α ∈ A there exists φα ∈ AutG (Y ) such that φα swaps
and Dα− , but fixes all Dβ+ and Dβ− for β ∈ A, β 6= α.

Dα+

Let α ∈ A. Since the field k is algebraically closed and the set Σ ⊂ X is linearly
independent, we can construct a homomorphism ψα : X → k× with ψα (α) = −1 and
ψα (γ) = 1 for every γ ∈ Σ r {α}, and such that ψα is of finite order in the group
Hom(X , k× ). Then we have ψα (Σ) = {1}. By Theorem B.1 there exists an automorphism
of finite order φα ∈ AutG (Y ) with
(
−fβ for β = α,
φα (fβ ) =
(47)
fβ
for β ∈ A r {α}.
e ⊂ NG (H) denote the subgroup containing H such that
Let H

e
H/H
= hφα i ⊂ NG (H)/H = AutG (Y ),

e We use the
where hφα i denotes the finite subgroup generated by φα . We set Ye = G/H.
same notation for the combinatorial objects associated to the spherical homogeneous space
Ye as for Y , but with a tilde above the respective symbol. The morphism of G-varieties
Y → Ye induces an embedding K(Ye ) ֒→ K(Y ), and K(Ye ) is the fixed subfield of φα . Since
K(Ye ) is a G-invariant subfield of K(Y ), we have K(Ye )(B) ⊂ K(Y )(B) and Xe ⊂ X .
By (47) we have φα (fα ) = −fα 6= fα . We see that fα ∈
/ K(Ye ), hence α ∈ X r Xe, in
e
e
particular α ∈
/ Σ. By Proposition B.2(1) we have |D(α)| ≤ 1, hence the two colors in D(α)
are mapped to one color by the map Y → Ye , that is, φα swaps Dα+ and Dα− .
On the other hand, for any β ∈ Ar{α}, by (47) we have φα (fβ ) = fβ , hence fβ ∈ K(Ye )
and β ∈ Xe. Since β is a primitive element of X , it is a primitive element of Xe ⊂ X . The
e (see Knop
natural map V → Ve induced by Y 7→ Ye is bijective and identifies V and V
e = −V.
[Kn89, Section 4]). Since β ∈ Σ is dual to a wall of −V, it is dual to a wall of −V

36

MIKHAIL BOROVOI

WITH AN APPENDIX BY

GIULIANO GAGLIARDI

e hence |D(β)|
e
It follows that β ∈ S ∩ Σ,
= 2, and the two colors in D(β) are mapped to
e

distinct colors under Y → Y , that is, φα fixes Dβ+ and Dβ− .
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Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University,
6997801 Tel Aviv, Israel
E-mail address: borovoi@post.tau.ac.il

