Some topics in geometric invariant theory over non algebraically closed fields

Some topics in geometric invariant theory over
non-algebraically closed fields
Dao Phuong Bac∗ and Nguyˆen
˜ Quˆo´c Thˇan
´ g†

Abstract
In this paper, we review some problems related with the study of
geometric and relative orbits for the actions of algebraic groups on
affine varieties defined over non-algebraically closed fields.
Mathematics Subject Classification (AMS 2000) : Primary : 14L24.
Secondary: 14L30, 20G15.

Plan.
I. Introduction.
II. An overview of geometric invariant theory. Observability and related notions.
III. Stability in geometric invariant theory over non-algebraically closed fields.
IV. Topology of relative orbits for actions of algebraic groups over completely
valued fields.

1

Introduction

Let G be a smooth affine algebraic group acting morphically on an affine
variety X, all defined over a field k. Many results of (geometric) invariant
theory related to the orbits of the action of G are obtained in the geometric
∗

Department of Mathematics, VNU of Science, 334 Nguyen Trai, Hanoi, Vietnam.
E-mail : ,

†
Institute of Mathematics, 18-Hoang Quoc Viet, Hanoi, Vietnam.
E-mail :
Support in part by NAFOSTED and VIASM.

1


case, i.e., when k is an algebraically closed field. However, since the very beginning of modern geometric invariant theory, as presented in [Mu], [MFK],
there has been a need to consider the relative case of the theory. For example, Mumford has considered many aspects of the theory already over
sufficiently general base schemes, with arithmetical aim (say, to construct
arithmetic moduli of abelian varieties, as in Chap. 3 of [Mu], [MFK]). Also
some questions or conjectures due to Borel ([Bo1]), Tits ([Mu]) ... ask for
extensions of results obtained (in the case of algebraically closed fields) to
the case of non-algebraically closed fields. As typical examples, we cite the
results by Birkes [Bi], Kempf [Ke], Raghunathan [Ra], which gave solutions
to some of the above mentioned questions or conjectures.
This article has its aim to overview some of the recent results in this
direction, while trying to put them in a coherent form. In fact, since the
results in the field are quite diverse, to give a reasonable account of (the
majority of) all existing results would require a whole book. Therefore, the
reades will find that we are concerned mostly with some basic topics such as
the geometry and the topology of the orbits.
Throughout, we consider only smooth affine (i.e. linear) algebraic groups
defined over some field k, which are called also shortly as k-groups. For basic theory of smooth affine (linear) algebraic groups over non-algebraically
closed field we refer to [Bo2], and for a k-group G, the notion of a rational
k-module V for G is as in [Gr1], [Gr2]. However, in few places we give some
account of the recent development, which is directly related to our discussion
here.

2
2.1

An overview of geometric invariant theory.
Observability and related notions
Some basic definitions and facts

To keep things simple, we describe the basic notions in their simplest form
(thus not in most general form), with the hope that the readers will either
be able themselves either to extend to a more general setting later on, or find
the corresponding ones in the literature.

2


Action and Orbit. Let k be a field, k¯ an algebraic closure of k. Let
G be an affine algebraic group, V an affine variety all are defined over k. For
¯ Assume that there exists
simplicity, we identify G, V with their points in k.
a regular k-morphism ϕ : G × V → V , (g, v) → g.v such that the following
holds:
1) g.(h.v) = (gh).v, ∀ g, h ∈ G, v ∈ V ;
2) e.v = v, ∀ v ∈ V , where e denotes the identity element of G.
Then one says that we are given an action of G on V defined over k, or
that G acts k-regularly on V. For a fixed v ∈ V , the image of G × {v} via ϕ
is called the orbit of v under G and denoted by G.v.
Consider a smooth affine algebraic group G acting morphically on an
affine variety V , all are defined over a field k. One of the basic subjects in
our study is the orbit G.v, v ∈ V , under the action of G. From the geometric
point of view, the most important objects are the closed orbits and open orbits

(with respect to Zariski topology). The first natural question is the following:
Is there any closed (open) orbit in V ?
One of the first, though elementary but very basic, results is the following well-known statement, which assures the existence of closed orbits. On
should also note that the open orbits may not exist in general.
2.1.1. Theorem. (Cf. [Bo1], [Hum]) With above assumption,
a) Each orbit G.v is a smooth locally closed subvariety of V;
b) Each orbit G.v contains an open dense subset of its closure;
c) The boundary of G.v is the union of orbits of lower dimension;
d) There exists G-orbits which are closed (in Zariski topology) in V.
Linear action. Linearization. Among those affine varieties which are
the most important, one can single out the class of varieties V with linear
action of G, i.e., V are vector spaces and G acts on V via a rational (i.e.
linear) representation ρ : G → GL(V ). We call such V also G-modules (or
k-G-modules if they are defined over k).
It is no doubt that the study of representation theory of G by using
(or from the point of view of) geometric invariant theory should play some
definite role.
3


Fortunately, the general action of affine groups on affine varieties is not
far from this linear action, as the following statement shows.
2.1.2. Theorem. (Cf. [Bi], [Bre], [Ke], [KSS]) Let G be an affine algebraic group acting regularly on an affine variety V, all defined over a field k.
Then there exists a closed embedding ψ : V → W and a rational representation ρ : G → GL(W ), all defined over k, such that ρ(g)(ψ(x)) = ψ(g.x) for
all x ∈ V .
One should also note that a closely related and somewhat more general
result also holds. Namely we have
2.1.3. Theorem. (Cf. [Su1, Theorem 1], [Su2, Theorem 2.5]) Let G be a
connected affine algebraic group acting regularly on a normal quasi-projective
variety V . Then there exists a projective embedding ψ : V → W := Pn

and a projective group representation ρ : G → PGLn = Aut(W ) such that
ρ(g)(ψ(x)) = ψ(g.x) for all x ∈ V .
Therefore it is no harm to consider any affine (resp. quasi-projective) Gvariety V just as a G-stable closed k-subvariety of some k-G-module (resp.
some projective G-space) W . The proofs of all these facts give also very explicit way to construct such linear (resp. projective spaces), where the action
is linearized. The setting in [Su2] is scheme-theoretic, thus it may be applied
to a more general situation.
Quotients. To study the actions of algebraic groups on algebraic varieties, it
is convenient to consider the set of orbits (the quotient sets) for such actions.
However, such sets usually do not give much information and the first question comes to mind, when we are considering the action of algebraic groups
on algebraic varieties, is how to recognize if there is any other structure on
such sets.
Naturally, if an algebraic group G acts on a variety V , all defined over
a field k, it also acts on the affine algebra k[V ]. To use the correspondence
Geometry ↔ Algebra, the quotient set V /G should have the algebra
of functions which, being considered as functions on V , are constant on the
G-orbits, thus the G-invariant functions. In order that the quotient of V
under the action of G exist, it is necessary that the subalgebra k[V ]G of
all G-invariant functions of k[V ] be finitely generated. Assuming this, we
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say that a variety W , together with a morphism p : V → W is an algebraic (or categorical) quotient of V under the action of G, if the comorphism
p∗ : k[W ]
k[V ]G is a k-isomorphism of k-algebras from k[W ] onto the
subalgebra of all G-invariant functions in k[G], and we denote W = V //G.
Further, a categorical quotient is called a geometric quotient, if all the fibers
of p are closed, in other words, all G-orbits are closed. If it is the case, we
denote V /G the corresponding geometric quotient. One should mention also
that in order that the geometric quotient exist, it is necessary that all the
G-orbits be closed. Meanwhile, the following holds

2.1.4. Theorem. (Rosenlicht) (Cf. [Do, Thm. 6.2], [Sp, Satz 2.2]) Given
any action of an affine algebraic group G on an affine variety V, there exists
an open G-stable subvariety U → V such that with the induced action of G
on U, the geometric quotient U/G exists.
One should mention the following notions. An affine algebraic group G
is called linearly reductive if any linear representation ρ : G → GL(V ) is
completely reducible. Equivalently, G is linearly reductive, if for any such
ρ and non-zero G-invariant vector v ∈ V , there exists a linear G-invariant
form F on V such that F (v) = 0. G is called geometrically reductive if for
any linear representation ρ : G → GL(V ) and non-zero G-invariant vector
v ∈ V , there exists a homogeneous G-invariant polynomial F on V such that
F (v) = 0. It is tautollogical that linearly reductive ⇒ geometrically reductive. The converse is also true in characteristic 0 (and this is also equivalent
to the property of being reductive (the solvable radical is a torus)) but fails
in general in characteristic p > 0. Finally, G is called reductive if its unipotent radical Ru (G) is trivial. Then we have the following criteria for linearly
reductivity due to Kemper [K].
2.1.5. Theorem. [K] Let k be a field, G a linear algebraic group defined
over k. Then G is linearly reductive if and only if the following conditions
hold:
a) For every affine G-scheme X, the invariant ring k[X]G is finitely generated
over k,
b) For every G-module V, the invariant ring k[V ]G is a Cohen-Macaulay ring.

Quite recently, Alper (et al.) in a series of papers [Al1], [Al2], [AE] have
5


extended several classical results in the case of smooth affine groups over
fields to group schemes. Let S be a scheme, G → S a f ppf group scheme,
BG the classifying stack for G. G is called S-linearly reductive if the morphism BG → S is cohomological affine, i.e., if f is quasi-compact and it
induces an exact functor f∗ : QCoh(BG) → QCoh(S) between the categories of quasi-coherent modules over BG and that over S. Then it was

proved in [Al1] the following characterization of the linear reductivitiy.
2.1.6. Proposition. ([Al1, Prop. 12.6]) Let k be a field, G a separated
k-group scheme of finite type. Then the following statements are equivalent.
i) G is k-linearly reductive (in newly introduced sense);
ii) The functor V → V G from the category Rep(G) of G − representations
to the category V ec of vector spaces is exact;
iii) The functor V → V G from the category Repf in (G) of finite dimensional
G − representations to the category V ec of vector spaces is exact;
iv) Every G-representation is completely reducible.
v) Every finite dimensional G-representation is completely reducible.
vi) For every finite dimensional G-representation V and 0 = v ∈ V G , there
exists a G-invariant functional F ∈ (V ∗ )G such that F (v) = 0.
Regarding the geometric reductivity, we have the following well-known result, basically due to Mumford and Haboush
2.1.7. Theorem. (Cf. [Gros2, Thm. A, p.3], [MFK, Thm. A 1.0]) Let
G be an affine algebraic group defined over a field k. The the following are
equivalent.
1) G is reductive;
2) G is geometrically reductive;
3) For any affine k-algebra A with a G-action, the subalgebra of G-invariants
AG is also an affine k-algebra.
2.1.8. Corollary. If G is a reductive group acting on an affine variety
V , then the categorical quotient V //G always exists.
Remarks. 1) Besides the original proof of the Mumford conjecture (that
in 2.1.7, (1) ⇔ (2), given by Haboush, recently there was given a second
proof by Sastry and Sesahdri [SaSe].
2) Theorem 2.1.7 about the geometric reductivity can be stated in a more
6


general setting of schemes over very general bases, and we refer to [Se] as

a standard reference for this. For yet in a more general stack framework,
we mention the works by Alper (et al.) [Al1], [Al2], [AE], where there were
given a fragments of a program of extending the major results of Geometric
Invariant Theory to this new setting, by employing systematically the stack
approach.
3) There is a vast literature on the Hilbert’s 14th Problem. We just indicate
a few sources and refer the readers to these and the reference there: [Do],
[Gr1], [Kr], [MFK]. Below we consider some examples related with the finiteness of the rings of invariants. Other examples, in particular those due to
Nagata and Roberts, are given, say, in [Gr2, Sec. 8]).
Examples. 1) ([G-P]) Let k be any commutative ring with 1, the dual
number (i.e., 2 = 0, X a variable, and A := k[ , X]. Let Ga act on A
via λ.f := f + λ(∂f /∂X). Then one checks that the ring of invariants
AGa = k[ X, X 2 , ...], which is not finitely generated over k.
2) (Weitzenb¨ock) Let Ga act regularly on a vector space V over a field of
characteristic 0. Then k[V ]Ga is finitely generated.

2.2

Observable and other related subgroups

Isotropy subgroups (stabilizers). For v ∈ V , we denote by Gv the
isotropy (or the same stabilizer) subgroup of v in G. If V is an affine (resp.
quasi-projective) variety, then by 2.1.2 (resp. 2.1.3), the stabilizers can be
realized also as stabilizers arising in a linear (resp. projective) representations.
Due to the fact that there exists a natural bijection G/Gv G.v, which
in fact, is an isomorphism of varieties in many cases, the study of the isotropy
subgroups of a given group G also plays an important role. This lead to the
study of the so-called observable subgroups.
Let G be an affine algebraic group defined over an algebraically closed
field k. Then G acts naturally on its regular function ring k[G] by right

translation (rg .f )(x) = f (x.g), for all x, g ∈ G, f ∈ k[G]. For H a closed
k-subgroup of G, we consider k[G]H := {f ∈ k[G] : rh .f = f, for all h ∈ H},
the k-subalgebra of H-invariant functions of k[G].
By convention, we identify the (smooth) affine algebraic groups considered
7


with their points in a fixed algebraically closed field. For a k-subalgebra R
of k[G], we put R = {g ∈ G : rg .f = f, for all f ∈ R}. Then for any closed
subgroup H ⊂ G we have
H ⊆ (k[G]H ) ⊆ G.
With a motivation from representation theory, Bialynicki-Birula, Hochschild
and Mostow (see [BHM, p. 134]) introduced the concept of ”observable
subgroup”. A closed subgroup H of G is called an observable subgroup of
G if any finite dimensional rational representation of H can be extended
to a finite dimensional rational representation on the whole group G (or,
equivalently, if every finite dimensional rational H-module is a H-submodule
of a finite dimensional rational G-module). In loc. cit. some equivalent
conditions for a subgroup to be observable were given.
Then Grosshans (see [Gr1], [Gr2, Chap. 1] and reference therein) has
added several other conditions. It turned out later that for closed subgroups
the property of being observable for a subgroup H is equivalent to the equality H = (k[G]H ) . Up to now there are known several equivalent conditions
for a subgroup to be observable, which are more or less simple to verify and
they are gathered in Theorem 2.2.1 below.
Epimorphic subgroups. On the opposite side, a closed subgroup H ⊆ G
may satisfy the equality (k[G]H ) = G. If it is so, H is called an epimorphic
subgroup of G. In fact, under an equivalent condition, this notion was first
introduced and studied by Bien and Borel in [BB1] [BB2] (see also [Gr2, Sec.
23, p. 132] for recent treatment), which in turn, is based on similar notion
for Lie algebras, given by Bergman (unpublished). There were given several

equivalent conditions for a closed subgroup to be epimorphic (see Theorem
2.2.10 below).
Grosshans subgroups. In the connection with the solution of the 14th
Hilbert problem, the following well-known problem is of great interest. Assume that X is an affine variety, G is a reductive group acting upon X
morphically, H is a closed subgroup of G and consider the G-action on the
regular function ring k[X] by left translation: (lg .f )(x) = f (g −1 .x). It is
natural to ask when k[X]H is a finitely generated k-algebra.
H
For a closed subgroup H ⊂ G, we have k[X]H = k[X](k[G] ) (see [Gr1],
[Gr2, Chap. 1]). On the other hand, it is well-known (loc. cit) that (k[G]H )
8


is the smallest observable subgroup of G containing H. So the problem is reduced to the case when H is an observable subgroup. To solve this problem,
Grosshans ([Gr1], [Gr2, Chap. 1, Sec. 4, 5]) introduced the codimension 2
condition for observable subgroups, and the subgroups satisfying this condition are called subsequently Grosshans subgroups of G (see below).
Quasi-parabolic and subparabolic subgroups. Closely related to observable subgroups is the following class of subgroups. Recall that ([Gr2,
p. 17]) a closed subgroup H of an affine algberaic group G is called quasiparabolic if H ⊂ G◦ and H is the isotropy subgroup of a highest weight vector
for some (finite-dimensional) absolutely irreducible representation of G◦ . A
connected closed subgroup H is called subparabolic if H ⊂ Q, Ru (H) ⊂
Ru (Q) for some quasi-parabolic subgroup Q of G, and in general a closed
subgroup H is subparabolic if H ◦ is so.
In this section we are interested in some questions of rationality related
to observable, epimorphic, quasi-parabolic, subprabolic and Grosshans subgroups. The first rationality results regarding observable (resp. epimorphic)
subgroups were already given in [BHM], and then in [Gr2], [W] (resp. [BB1],
[BB2] and [W]), where some arithmetical applications to ergodic actions were
also given. We give some other new results related to rationality properties of
observable, epimorphic and Grosshans subgroups (which were stated initially
for algebraically closed fields). Some arithmetic and geometric applications
will be considered in another paper under preparation.

Some rationality properties for observable groups. First we recall
well-known results over algebraically closed fields. For an algebraic group G
we denote by G◦ the identity connected component subgroup of G.
2.2.1. Theorem ([BHM], [Gr2, Theorems 2.1 and 1.12]) Let G be a linear algebraic group defined over an algebraically closed field k and let H be
a closed k-subgroup of G. Then the following conditions are equivalent.
a) H = (k[G]H ) .
b) There exists a finite dimensional rational representation ρ : G → GL(V )
and a vector v ∈ V , all defined over k such that
H = Gv = {g ∈ G : ρ(g).v = v}.
c) There are finitely many functions f ∈ k[G/H] which separate the points
in G/H.
9


d) G/H is a quasi-affine k-variety.
e) Any finite dimensional rational k-representation ρ : H → GL(V ), can be
extended to a finite dimensional rational k-representation ρ : G → GL(V ),
where V → V , i. e., every finite dimensional rational H-module is a Hsubmodule of a finite dimensional rational G-module.
f ) There is a finite dimensional rational k-representation ρ : G → GL(V )
and a vector v ∈ V such that H = Gv , the isotropy group of v, and
G/H ∼
= G.v = {ρ(g).v : g ∈ G}
(as algebraic varieties).
g) The quotient field of the ring of G◦ ∩ H-invariants in k[G◦ ] is equal to the
field of G◦ ∩ H-invariants in k(G◦ ).
h) If 1-dimensional rational H-module M is a H-submodule of a finite dimensional rational G-module then the H-dual module M ∗ of M is also a
H-submodule of a finite dimensional rational G-module.
Examples. 1) If H is a normal closed subgroup of G, then H is observable in G.
2) Let χ be a character of G and let Hχ := Ker(χ). Then H is observable
in G. In particular, if H has no character, then H is observable in G.

3) If H1 , H2 are observable subgroups of G, then so is H1 ∩ H2 .
4) If H is observable in K and K is observable in G then so is H in G.
5) Let B be a Borel subgroup of SL2 . Then the quotient SL2 /B is isomorphic
to the projective line P1 , so B is not observable in SL2 .
Now let k be any field. If a closed k-subgroup H of a linear algebraic
k-group G satisfies the condition b) (resp. e)) in Theorem 1 where v ∈ V (k)
and the corresponding representation ρ is defined over k, then we say that
H is an isotropy k-subgroup of G (resp. has extension property over k).
First we recall the following rationality results proved in [BHM, Theorems 5, 8].
2.2.2. Theorem. ([BHM, Theorem 5]) Let G be a linear algebraic k-group,
H a closed k-subgroup of G, K an algebraic field extension of k. Then H has
extension property over k if and only if it has one over K.
2.2.3. Theorem. ([BHM, Theorem 8]) If H is a closed k-subgroup of a linear algebraic k-group G with extension property over k, then H is an isotropy
10


k-subgroup of G. Conversely, if k is algebraically closed and H is a isotropy
k-subgroup then it has extension property over k.
From Theorem 2.2.2 and Theorem 2.2.3, we derive the following.
2.2.4. Proposition. ([TB]) Let k be an arbitrary field and let H be a
closed k-subgroup of a k-group G. The following two conditions are equivalent.
a) H is an isotropy subgroup of G over k.
b) H is an isotropy subgroup of G over k, i.e., there exists a finite dimensional k-rational representation ρ : G → GL(V ) and a vector v ∈ V (k) such
that H = Gv .
Remark. In [W], another proof of Proposition 2.2.4 was given, which is
based on some ideas of Grosshans [Gr1], under the condition (which is not
essential) that k = Q and H is connected.
We consider
k[G]H(k) = {f ∈ k[G] : rh .f = f, ∀h ∈ H(k)},
and

(k[G]H(k) ) = {g ∈ G : rg .f = f, ∀f ∈ k[G]H(k) }.
Then k[G]H(k) and k[G]H := {f ∈ k[G] : rh .f = f, ∀h ∈ H} are ksubalgebras of k[G]. In general we have the following diagram
k[G]H(k)

¯ H(k)
⊆ k[G]

|
k[G]H

|
⊆

¯ H
k[G]

so we have
(k[G]H(k) )

¯ H(k) )
⊇ (k[G]

|
(k[G]H )

|
⊇
11

¯ H) .

(k[G]


If, moreover, H(k) is Zariski dense in H then we have
k[G]H(k) = k[G]H = k[G]H ∩ k[G].
We say that H is relatively observable over k if H = (k[G]H(k) ) , and H is
k-observable, if (k[G]H ) = H. It is clear that if k is algebraically closed, then
these notions coincide with the observability. We have the following obvious
implication
H is k-observable ⇒ H is observable.
2.2.5. Proposition. ([TB]) Let k be a field, and let H be a closed k-subgroup
of a k-group G. Then
a) H = k[G]H = k ⊗k k[G]H ;
b) H is observable if and only if H is k-observable;
c) Assume that H(k) is Zariski dense in H. Then H is observable ⇔ H is
k-observable ⇔ H is relatively observable over k.
2.2.6. Proposition. ([TB]) Let H be a k-subgroup of a k-group G. The
following are equivalent:
a) There exist finitely many functions in k[G/H] which separate the points
in G/H.
b) There exist finitely many functions in k[G/H] which separate the points
in G/H.
2.2.7. Proposition. ([TB]) Let G be a k-group, H a closed k-subgroup
of G. Assume that, there exists finite dimensional k-rational representation
ρ : G → GL(V ), and v ∈ V (k) such that H = Gv . Then there is a finite
dimensional k-rational representation ρ : G → GL(W ) and w ∈ W (k) such
that H = Gw and G/H ∼
=k G.w.
The following fact is also very useful, which allows one to reduce to the
case of connected groups.

2.2.8. Lemma. With above assumption, H is k-observable in G if and
only if H ∩ G◦ is k-observable in G◦ .
From results proved above, we have the following theorem, which is an
analog of Theorem 2.2.1 for arbitrary fields.
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2.2.9. Theorem. ([TB]) Let G be a linear algebraic group defined over
a field k and let H be a closed k-subgroup of G. Then the following are equivalent.
¯ H ) , i. e., H is observable.
a) H = (k[H]
a ) H = (k[G]H ) , i.e., H is k-observable.
b ) There exists a k-rational representation ρ : G −→ GL(V ) and a vector
v ∈ V (k) such that
H = Gv = {g ∈ G : g.v = v}.
c ) There are finitely many functions f ∈ k[G/H] which separate the points
in G/H.
d ) G/H is a quasiaffine variety defined over k.
e ) Any k-rational representation ρ : H −→ GL(V ), can be extended to a
k-rational representation ρ : G −→ GL(V ).
f ) There is a k-rational representation ρ : G −→ GL(V ) and a vector
v ∈ V (k) such that H = Gv and
G/H ∼
=k G.v = {ρ(g)v : g ∈ G}.
g ) The quotient field of the ring of G◦ ∩ H-invariants in k[G◦ ] is equal to
the field of G◦ ∩ H-invariants in k(G◦ ).
If, moreover, H(k) is Zariski dense in H, then the above conditions are equivalent to the relative observability of H over k.
Some further extensions. Recently, in a series of papers, J. Alper et
al. (see e.g. [AE]) have generalized some of the results considered above for
observable subgroup schemes basing on a very general framework of stacks.

The following passage is based on [Al1, Al2, AE].
Let S be any scheme, G → S a flat finitely presented quasi-affine group
scheme. Denote by OX the structure sheaf of the scheme X, BX the classifying stack of X. Then a flat, finitely presented quasi-affine subgroup
scheme H ⊆ G is called observable if every quasi-coherent OS [H]-module
is a quotient of a quasi-coherent OS [G]-module. In the case S = Spec(k),
k is a field, this definition is equivalent to ours, namely every finite dimensional H-representation is a subH-representation of a finite dimensional Grepresentation. We have the following
13


2.2.9. Theorem (bis) [AE, Thm. 1.3] Let S be any scheme, G → S
a flat finitely presented quasi-affine group scheme. Consider a flat, finitely
presented quasi-affine subgroup scheme H ⊆ G. Then the following are equivalent.
a) H is observable;
b) For every quasi-coherent OS [H]-module F, the induced map IndG
HF → F
is a surjection of OS [H]-modules;
c) BH → BG is quasi-affine;
d) The quotient G/H → S is quasi-affine.
If, moreover, S is a noetherian scheme, then all the above are equivalent to
the following
e) Every coherent OS [H]-module F is a quotient of a coherent OS [G]-modules;
f ) For every coherent OS [H]-module F, the induced map IndG
H F → F is a
surjection of OS [H]-modules.
Some rationality properties for epimorphic subgroups. We recall
¯
(after Grosshans [Gr2, p. 132]) that (k-)epimorphic
subgroups H ⊆ G are
¯ H ) = G. We have
those closed subgroups of G satisfying the condition (k[G]

the following characterizations of epimorphic subgroups over an algebraically
closed fields.
2.2.10. Theorem ([BB1, Th´eor`eme 1], [Gr2, Lemma 23.7]) Let H be a
closed subgroup of G, all defined over an algebraically closed field k. Then
the following are equivalent.
a) H is epimorphic, i. e., (k[G]H ) = G.
b) k[G/H] = k.
c) k[G/H] is finite dimensional over k.
d) If V is any rational G-module then the spaces of fixed points of G and H
in V coincide.
e) If V is a rational G-module such that V = X ⊕ Y , where X, Y are Hinvariant, then X, Y are also G-invariant.
f ) Morphisms of algebraic groups from G to another L are defined by their
values on H.
Remark. The initial definition of epimorphic subgroups was given in [BB1],
by only requiring that the condition f ) above hold.

14


Examples. 1) Let H be a closed subgroup of an affine algebraic group
G. Then H is epimorphic in (k[G]H ) (k is algebraically closed).
2) If H is a closed subgroup epimoprhic in G then so is H ∩ G◦ in G◦ .
Let notation be as above and let k be an arbitrary field. Then for a ksubgroup H of a k-group G we say that H is relatively epimorphic over k if
(k[G]H(k) ) = G, and that H is k−epimorphic if (k[G]H ) = G. Recall that
we have the following inclusions
(k[G]H(k) )

¯ H(k) )
⊇ (k[G]

|
(k[G]H )

|
⊇

¯ H) ,
(k[G]

therefore, the following implications holds
H is epimorphic ⇒ H is k-epimorphic,
H is k-epimorphic ⇐ H is relatively epimorphic over k.
In fact we have
2.2.11. Proposition. ([TB]) With above notation, if H is either (a) relatively epimorphic over k or (b) k-epimorphic, then it is also epimorphic.
We have the following analog of Theorem 2.2.10 over an arbitrary field.
2.2.12. Theorem. Let k be any field and let H be a closed k-subgroup
of a k-group G. Then the following are equivalent.
a ) H is k-epimorphic, i.e., (k[G]H ) = G.
b ) k[G/H] = k.
c ) k[G/H] is finite dimensional over k.
d ) For any rational G-module V defined over k, the spaces of fixed points of
G and H in V coincide.
e ) For any rational G-module V defined over k, if V = X ⊕ Y , where X, Y
are H-invariant, then X, Y are also G-invariant.
f ) Morphisms defined over k of algebraic k-groups from G to another one
are defined by their values on H.
15


Remark. It was mentioned in [W, p. 195], that Bien and Borel (unpublished) have also proved that if G is connected, then d) ⇔ d ). (Here d) from

Theorem 2.2.10 and d’) from 2.2.12.)
Some rationality properties for Grosshans subgroups. One of the
main results related with the finite generation problem (hence also with the
Hilbert’s 14-th problem) mentioned in the Introduction is the following result of Grosshans (Theorem 2.2.14). First we recall the following very useful
result which reduces to the case of connected groups.
2.2.13. Theorem. ([Gr2, Theorem 4.1]) Let k be an algebraically closed
field. For any closed subgroup H of G, if one of the following k-algebras
◦
◦
◦
k[G]H , k[G]H , k[G◦ ]H∩G , k[G◦ ]H is a finitely generated k-algebra, then the
same holds for the other.
2.2.14. Theorem. ([Gr2, Theorem 4.3]) For an observable subgroup H
of a linear algebraic group G, all defined over an algebraically closed field k,
the following are equivalent.
a) There is a finite dimensional rational representation ϕ : G → GL(V ),
an element v ∈ V , such that H = Gv and each irreducible component of
G.v − G.v has codimension ≥ 2 in G.v.
b) The k-algebra k[G]H is a finitely generated k-algebra.
If b) holds, let X be an affine variety with k[X] = k[G]H , and with G-action
via left translations of G on G/H. There is a point x ∈ X such that G.x is
open in X and G.x G/H via gH → g.x and each irreducible component of
X \ G.x has codimension ≥ 2 in X.
The observable subgroups which satisfy one of the equivalent conditions in
Theorem 2.2.14 are called Grosshans subgroups (see [Gr2, Chap. 1, Sec. 4]).
There are some nice geometrical characterizations and examples of Grosshans
subgroups presented in there and the reference therein.
For a field k, a k-group G and an observable k-subgroup H ⊂ G, we say
that H satisfies the codimension 2 condition over k if H satisfies condition
a) above where V, ϕ are all defined over k and v ∈ V (k).

We call H a Grosshans subgroup relatively over k (resp. k-Grosshans subgroup) of G if k[G]H(k) (resp. k[G]H ) is a finitely generated k-algebra.

16


Examples. 1) Let H be a closed subgroup of G. Then the following are
equivalent: a) H is a Grosshans subgroup of G; b) H ∩ G◦ is a Grosshans
subgroup of G◦ ; c) H ◦ is a Grosshans subgroup of G◦ .
2) If K < H are observable subgroups of G and K is a Grosshans subgroup
of G, then so is K in H.
3) If H is a reductive subgroup of G, then H is a Grosshans subgroup in G.
Now let k be any field We have a result similar to Theorem 2.2.14 for
k-Grosshans subgroups.
2.2.15. Theorem. ([TB]) Let k be an infinite perfect field, G a connected
k-group. Assume that H is a observable k-subgroup of G. Consider the following conditions.
a ) H satisfies codimension 2 condition over k.
◦
◦
◦
b ) One of the k-algebras k[G]H , k[G]H , k[G◦ ]H∩G , k[G◦ ]H is a finitely generated k-algebra.
c ) H is a Grosshans subgroup relatively over k of G (i.e k[G]H(k) is finitely
generated k-algebra).
Then, together with conditions in Theorem 15, we have the following implications
a) ⇔ a ) ⇔ b) ⇔ b ) ⇒ c ).
If, moreover, H(k) is Zariski dense in H, then all these conditions are
equivalent.
Remark. It is of interest to find examples where the condition c ) holds
but the other conditions do not. It will, perhaps, ultimately lead to another counter-examples to (generalized) 14-th Hilbert’s Problem in the case
of char.k > 0. (Various extensions of classical results in (geometric) invariant theory to the case of characteristic p > 0 were discussed at length in
[MFK, Appendices].) It will be more interesting to have examples with G, H

connected groups.
A relation with the subalgebra of invariants of a Grosshans subgroup of
a reductive group acting rationally upon a finitely generated commutative
algebra is given in the following
2.2.16. Theorem. [Gr2, Theorem 9.3].) Let k be an algebraically closed
field. For any closed subgroup H of a reductive group G, all defined over k,
17


the following are equivalent.
a) k[G]H is a finitely generated k-algebra.
b) For any finitely generated, commutative k-algebra A on which G acts rationally, the algebra of invariants AH is a finitely generated k-algebra.
We consider the following relative version of this theorem.
2.2.17. Theorem. ([TB]) Let k be an infinite perfect field, H a closed
k-subgroup of a connected reductive k-group G. Consider the following conditions.
a ) k[G]H is a finitely generated k-algebra.
b ) For any finitely generated, commutative k-algebra Ak on which G acts
k-rationally, the algebra of invariants AH
k is a finitely generated k-algebra.
c ) For any finitely generated, commutative k-algebra Ak on which G acts
H(k)
k-rationally, the algebra of invariants Ak
is a finitely generated k-algebra.
Then with notations as in Theorem 2.2.16 we have
a) ⇔ a ) ⇔ b) ⇔ b ) ⇒ c ).
If, moreover, H(k) is Zariski dense in H, then all conditions above are equivalent.
Some rationality properties of subparabolic subgroups. We consider
the following relative notions of quasi-parabolic and subparabolic subgroups.
a) For a k-group G, a subgroup Q of G0 is said to be k-quasiparabolic in G if
Q = G0v for a highest weight vector v ∈ V (k) of some absolutely irreducible

k − G0 -module V . Here V (k) denotes the set of k-points of V with respect
to a fixed k-structure of V ([Bo2, Section 11.1]).
b) For a k-group G, a subgroup H of G is called k-subparabolic if it is defined
over k and there is a k-quasiparabolic subgroup Q of G0 such that H 0 ⊆ Q
and Ru (H) ⊆ Ru (Q).
Note that in the literature, a closed subgroup Q of G0 is called quasiparabolic
¯
if it is k-quasiparabolic
and a closed subgroup H of G is called subparabolic
¯
if it is k-subparabolic
and then we are back to the usual notions introduced
above.
a’) For a k-group G, a subgroup Q of G0 is said to be quasiparabolic over k
(or quasiparabolic k-subgroup) if it is defined over k and quasiparabolic.
b’) For a k-group G, a subgroup H of G is called subparabolic over k (or
18


subparabolic k-subgroup) if it is defined over k and subparabolic. H is called
strongly subparabolic over k if there is a quasiparabolic k-subgroup Q of G0
such that H 0 ⊆ Q and Ru (H) ⊆ Ru (Q). (Thus, being strongly subparabolic
over k is a priori stronger than just being subparabolic over k.)
One of important theorems in geometric invariant theory is due to Bogomolov which relates the stabilizer subgroup of an unstable vector to some
quasiparabolic subgroup. Its relative version below provides the abundance
of k-quasiparabolic subgroups. It is also one of main results of this paper.
¯
2.2.18. Theorem. (Cf. [Bog1, Theorem 1], [Gr2, Theorem 7.6] when k = k,
[BaT3] when k is perfect) Let k be a perfect field, G a connected reductive
k-group and let V be a finite dimensional k-G-module. Let v ∈ V (k) \ {0}. If

v is unstable for the action of G on V (i.e., 0 ∈ G.v), then Gv is contained
in a proper k-quasiparabolic subgroup Q of G.
Remark. We note that the original proof in [Bog1] (cf. also [Bog2], [Ro2]) is
given for algebraically closed fields and does not seem to extend to arbitrary
perfect fields. Since we make an essential use of Kempf - Rousseau results
(see below), which does not seem to be extended to the case of non-perfect
fields as noted in [Ro1] (cf. also [He]), our approach does not cover this case.
As an application of Theorem 2.2.18 and also of other results, we establish
the following second main result of this section about rationality properties of
quasiparabolic, subparabolic and observable subgroups of a linear algebraic
group G defined over a perfect field k.
2.2.19. Theorem. ([BaT3]) Let k be a perfect field, G a linear algebraic
k-group, H a closed k-subgroup of G. We consider the following statements.
1) H is k-quasiparabolic;
2) H is quasiparabolic over k;
3) H is observable over k;
4) H is k-subparabolic;
5) H is strongly subparabolic over k.
6) H is subparabolic over k.
Then we have 1) ⇒ 2) ⇒ 3) ⇔ 4) ⇔ 5) ⇔ 6). If, moreover, G is semisimple,
then 1) ⇔ 2).

19


Remarks. 1) In general, there are examples show that in Theorem 2.2.19,
3) ⇒ 2) ⇒ 1).
¯ 3) ⇔ 4) above is Sukhanov’s Theorem (cf. [Suk],
2) In the case k = k,
[Gr2]). The proof of Sukhanov’s Theorem in the absolute case (see [Suk],

or [Gr2, Theorem 7.3], with some refinements) makes an essential use of the
important theorem due to Bogomolov mentioned above. The same happens
while we prove the relative version: we make an essential use of Theorem
2.2.17 and other related results.

3
3.1

Stability in geometric invariant theory over
non-algebraically closed fields
Stability

To establish the relative version of Bogomolov’s theorem and also other versions of Sukhanov’s theorem, we need to use the Kempf’s-Rousseau’s Instability theory. This appears to be very important in the arithmetic invariant
theory. In fact, from the very beginning of the modern geometric invariant
theory, to study the closedness of G-orbits, one encounters immediately the
notion of stability, which plays a crucial role.
Let an affine algebraic group G act on a vector space V via a linear representation ρ : G → GL(V ), all are defined over a field k. We say (after Kempf
[Ke]) that a non-zero vector v ∈ V is stable (resp. unstable, semistable) if
G.v is closed (resp. G.v is not closed, Cl(G.v) does not contain 0). A stable
vector is called properly stable if moreover, the stabilizer Gv of v is finite. The
following important theorem has been proved by Kempf [Ke] and Rousseau
[Ro1], [Ro2] (independently and differently). Let Y (G) := M or(Gm , G) the
set of all one-parameter subgroups of G. Let f : Gm → V be a morphism
of algebraic varieties. If f can be extended to a morhism f˜ : Ga → V , with
f˜(0) = v, then we write f (t) → v while t → 0, or lim f (t) = v.
t→0

3.1.1. Theorem (Cf. [Ke, Thm. 1.4]) Let G be a reductive group acting on an affine variety V, all defined over a perfect field k. Let v ∈ V (k)
be a k-point, S a closed G-stable subvariety of V such that S ∩ Cl(G.v) = ∅.
Then there exists a one-parameter subgroup λ ∈ Y (G) defined over k, such

that the limit lim λ(t).v exists and belongs to S.
t→0

20


Equivalently, it can be stated by making use of the numerical criterion for
instability as follows. To each λ ∈ Y (G) there corresponds a representation
ϕ : Gm → GL(V ), ϕ = λ ◦ ρ, which is diagonalizable. We write V = ⊕Vn ,
where Vn := {x ∈ V | ϕ(t).x = tn .x, ∀ t ∈ Gm } and set v = n vn , vn ∈ Vn
and define µ(v, λ) := M ax{−n | vn = 0}.
3.1.1.(bis) Theorem. (Cf. [Ro2]) With above notation, v is an unstable vector for G if and only if there exists λ ∈ Y (G) defined over k, such that
µ(v, λ) < 0.
These theorems vastly generalizes a classical result due to Hilbert - Mumford (cf. e.g. [Kr, Chap. III, Sec. 2.1]), which says that if k is algebraically
closed, G a linearly reductive k-group acting on a k-vector space V via a
linear k-representation, and v ∈ V is an unstable vector. Then there exists
a one-parameter subgroup λ : Gm → G such that lim λ(t).v = 0.
t→0

3.2

Some further extensions

After the works by Kempf [Ke] and Rousseau [Ro1], [Ro2], on the one hand
one should note the works by Hesselink [He], Raman - Ramanathan [RR]
and Coia - Holla [CH], and [ADK] where some further more general results
over fields, which is possibly imperfect. On the other hand, in other directions, we note the works by M. Bate, G. Martin and G. R¨ohrle ([BMRT1],
[BMRT2]), where the investigation of stability in relation with the so-called
Center conjecture due to Tits have been made.

3.3

Some other notions and questions related with stability

In [Bo1], A. Borel raised several questions concerning the relation between
the closedness of the orbits (geometric property) and the anisotropicity of
the given group G (arithmetic property). We recall these questions below
and indicate some developments on the way of finding answers to them.
I) (Borel [Bo1, Sec.8.8]) Let k be a perfect field, G a connected reductive
21


k-group with X ∗ (G)(k) = {1}.
a) Is there a faithful representation G → GL(V ), all defined over k, such
that for all v ∈ V (k) \ {0}, cl(G.v) does not contain the 0 ?
b) Is there a faithful representation G → GL(V ), all defined over k, such
that for all v ∈ V (k) \ {0}, G.v is closed ?
c) Is it true that for any representation G → GL(V ), all defined over k, and
for all v ∈ V (k) \ {0}, cl(G.v) does not contain the 0 ?
d) Is it true that for any representation G → GL(V ), all defined over k, and
for all v ∈ V (k) \ {0}, G.v is closed ?
II) Let k be an imperfect field, G an anisotropic connected reductive k-group.
One considers the same questions as above, denoted by a’), b’), c’) and d’).
As Borel noted, c’) and d’) may have the answer ”no” in the imperfect case.
In connection with Borel’s questions, in one of the early papers on geometric
invariant theory over non-algebraically closed fields, D. Birkes [Bi] introduced
the following Properties A, B, C of representations of algebraic groups.
Property A. (Cf. [Bi]) Let ρ : G → GL(V ) be any linear representation
of an affine algebraic group G, all are defined over a field k. Let x ∈ V (k)
be a point of instability for G, Y a non-empty G-invariant closed subset in

Cl(G.x) \ G.x. Then there exists y ∈ Y , λ ∈ Y (G) defined over k, such that
limt→0 λ(t).x = y.
Thus, in the simplest case of action of algebraic groups on a vector space,
it simply says that Theorem 3.1.1 holds. Therefore, by Kempf - Rousseau
Theorem 3.1.1 (proved a bit later), any reductive group G has Property A
over any perfect field.
Property B. ([Bi]) Let ρ : G → GL(V ) be any linear representation of
an affine algebraic group G, all are defined over a field k. Let x ∈ V (k), such
that the stabilizer Gx contains a maximal k-split torus of G. Then the orbit
G.x is closed.
Property C. ([Bi]) Let ρ : G → GL(V ) be any linear representation of
an affine algebraic group G, all are defined over a field k. Let x ∈ V (k).
Then the orbit G.x is closed.
22


We want to sketch the proof of the fact that the Property B also holds
for any reductive group over perfect fields, as a consequence of Property A
(or Theorem 3.1.1). Thus, all the Borel’s questions I) have a positive answer.
The necessary reduction has already been done by Birkes. Recall that a reductive k-group G is called anisotropic over k, if G has no non-trivial split
k-subtori.
3.1.2. Lemma. ([Bi, Sec. 6]) 1) Let k be a field, G an affine algebraic
k-group. If G◦ the connected component of the identity of G has Property B,
then so does G.
2) If G is a connected k-group with unipotent radical Ru (G) defined over k.
If G/Ru (G) has Property B, then so does G.
3) If every connected k-anisotropic reductive algebraic group defined over a
field k has Property B, then so does every reductive group over k.
3.1.3. Lemma. ([Bi, Sec. 6,7]) Let k be a perfect field. If every connected reductive k-anisotropic group G over a perfect field k has Property B,
then so does any affine algebraic group over k.

Proof. ([Bi]) Let G be any k-group. By 3.1.2, 1), we may assume that
G is connected. The unipotent radical Ru (G) of G is k-closed and since k is
perfect, Ru (G) is also defined over k, so we are reduced to proving the same
thing for G/Ru (G) by 3.1.2,2), and by 3) we are done.
3.1.4. Proposition. ([Bi], [Ke]) Over a perfect field k, any k-group G has
Property B.
Proof. By 3.1.3, we may assume that G is a connected, reductive and kanisotropic k-group. Let ρ : G → GL(V ) be any linear representation,
all are defined over k. If there were an unstable vector x ∈ V (k), then
Y := Cl(G.x) \ G.x = ∅ and by Theorem 3.1.1, there would exist a nontrivial one-parameter subgroup λ : Gm → G defined over k. The image of
λ would then be a non-trivial k-split subtorus of G, which contradicts our
assumption that G is anisotropic.

23


Regarding Property C, it was proved in [Bi, 10.1] that the converse statement
for I), b) holds. More precisely we have
3.1.5. Lemma. (Cf. [Bi, Lem. 10.1]) Let G be an affine algebraic kgroup. If for some faithful representation G → GL(V ) over k, the orbit G.x
is closed for all x ∈ V (k), then G is anisotropic over k.
From above, assuming that k is perfect, one finds the following necessary
and sufficient condition for G to have Property C.
3.1.6. Proposition. 1) (Cf. [Bi, Prop. 10.2]) Let G be an affine algebraic group defined over a field k. If G has Property B, then G has Property
C if and only if G is anisotropic.
2) If k is perfect, then G has Property C if and only if G is k-anisotropic.
Finally, we mention the following general result due to Birkes [Bi, Prop.
9.10], which says that the nilpotent groups has Property A over any field.
3.1.7. Proposition. ([Bi, Prop. 9.10.]) Let k be an arbitrary field, G a
smooth nilpotent group acting linearly on a finitely dimensional vector space
V via a representation ρ : G → GL(V ), all defined over k. If v ∈ V (k), Y
is a non-empty G-stable closed subset of Cl(G.v) \ Gv, then there exist an

element y ∈ Y ∩ V (k), a one-parameter subgroup λ : Gm → G defined over
k, such that λ(t).v → y while t → 0.
One should also note that for the Property A to hold, one cannot go beyond the class of nilpotent groups, or the class of reductive groups, as the
example of [Bi, p. 474] shows.

24


4

Topology of relative orbits for actions of algebraic groups over completely valued fields

Let G be a smooth affine (i.e. linear) algebraic group acting regularly on an
affine variety X, all are defined over a field k. Due to the need of numbertheoretic applications, the local and global fields and relared rings k are in
the center of research of arithmetic invariant theory. For example, let an
algebraic k-group G act on a k-variety V , x ∈ V (k). We are interested in the
set G(k).x, which is called relative orbit of x (to distinguish with geometric
orbit G.x). One of the main steps in the proof of the analog of Margulis’
super-rigidity theorem in the global function field case (see [Ve], [Li], [Ma])
was to prove the (locally) closedness of some relative orbits G(k).x, x ∈ V (k),
for some action of an almost simple simply connected group G on a k-variety
V . Thus one may pose the
General Problem: Study the orbits G.x, G(k).x, x ∈ V (k).
Questions:
a)(geometric case): When is G.x a Zariski closed subset of V ?
b) (relative case): If k has a Hausdorff topology (e.g. k is a local field),
x ∈ V (k), when is G(k).x is closed in Hausdorff topology in V (k) ?
c) Relation between a) and b) ?
Remarks. Question a) (geometric case) above is a subject of Geometric
Invariant Theory, question b) (relative case) is a subject of geometric invariant theory over non-algebraically closed fields. Finally, the question c), if k

is of arithmetic nature (say k is the ring of integers in a local or global field,
or ad`ele ring of the later, it is a subject of Arithmetic Invariant Theory.

4.1

On Borel - Harish-Chandra and Bremigan’s results

In this section we assume that k is a field, complete with respect to a nontrivial valuation v of real rank 1, (e.g. p-adic field or the field of real numbers
R, i.e., a local field). Then for any affine k-variety V , we can endow V (k)
with the (Hausdorff) v-adic topology induced from that of k. Let an affine
algebraic group G act k-regularly on V , x ∈ V (k) be a k-point. We are
interested in a connection between the Zariski-closedness of the orbit G.x of
25