On Affinely Closed, Homogeneous Spaces

Abstract

Affinely closed, homogeneous spaces G/H, i.e., affine homogeneous spaces that admit only the trivial affine embedding, are characterized for an arbitrary affine
algebraic group G. A description of affine G-algebras with finitely generated invariant subalgebras is obtained.

Full text

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arXiv:math/0406493v1 [math.AG] 24 Jun 2004
ON AFFINELY CLOSED HOMOGENEOUS SPACES
IVAN V. ARZHANTSEV AND NATALIA A. TENNOVA
To V. N. Latyshev on his 70-th anniversary
Abstract. Affinely closed homogeneous spaces G/H, i.e., affine
homogeneous spaces that admit only the trivial affine embedding,
are characterized for any affine algebraic group G. As a corollary,
a description of affine G-algebras with finitely generated invariant
subalgebras is obtained.
1. Introduction
Let G be an affine algebraic group over an algebraically closed field k
of characteristic zero and H an algebraic subgroup of G. By Chevalley’s
Theorem, the homogeneous space G/H admits the canonical structure
of a quasiprojective variety. An embedding of the homogeneous space
G/H is an algebraic G-variety X with a base point x ∈ X such that
the orbit Gx is dense (and open) in X and the stabilizer Gxequals H.
We denote an embedding as G/H ֒→ X. We say that an embedding
is trivial if Gx = X. An embedding G/H ֒→ X is said to be affine
if the variety X is affine. It is easy to show (for example, see [PV89,
Th.1.6]) that a space G/H admits an affine embedding if and only if
G/H is a quasi-affine variety, or, equivalently, H may be realized as
the stabilizer of a vector in a finite-dimensional G-module. In this case
the subgroup H is said to be observable in G. An effective description
of observable subgroups in an affine algebraic group G was obtained by
A. Sukhanov [Su88].
The following definition was introduced in [AT01].
Definition 1. A homogeneous space G/H is called affinely closed if it
admits only the trivial affine embedding.
An affinely closed homogeneous space is automatically affine. The
consideration of this class of homogeneous spaces is motivated by the
Date: June 24, 2004.
2000 Mathematics Subject Classification. Primary 13A50, 14M17; Secondary
14R20, 14L30, 32M10.
Key words and phrases. Affine algebraic groups, observable subgroups, homoge-
neous spaces, affine embeddings, G-algebras.
Supported by RFBR grant MAC 03-01-06252,by CRDF grant RM1-2543-MO-03
and by the RF President grant MK-1279.2004.1.
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2IVAN V. ARZHANTSEV AND NATALIA A. TENNOVA
following question: when does ”the stabilizer of a point x on an affine
G-variety X equals H” imply ”the orbit Gx is closed” ?
For a reductive group G a homogeneous space G/H is an affine
variety if and only if the subgroup H is reductive (Matsushima’s Crite-
rion). Note that for an arbitrary affine algebraic group G a description
of affine homogeneous spaces G/H in group-theoretic terms for the pair
(G,H) is an open problem, for more information see [Gr97, Ch.2].
For reductive G a description of affinely closed homogeneous spaces
follows directly from the result due to D. Luna [Lu75]:
Theorem 1. Let G be a reductive group. A homogeneous space G/H
is affinely closed if and only if the subgroup H is reductive and has a
finite index in its normalizer NG(H). Moreover, if G acts on an affine
variety X and the stabilizer of a point x ∈ X contains a reductive
subgroup H such that the group NG(H)/H is finite, then the orbit Gx
is closed.
For example, for a maximal torus T of a reductive group G the
Weyl group W = NG(T)/T is finite, hence G/T is affinely closed.
If ρ : H → SL(V ) is an irreducible representation of a semisimple
group, then SL(V )/ρ(H) is affinely closed (by the Schur Lemma, the
group NSL(V )(ρ(H))/ρ(H) is finite). Thus the class of affinely closed
homogeneous spaces is wide.
In [Ar03], affinely closed homogeneous spaces of a reductive group
G play a key role in a classification of affine G-algebras such that any
invariant subalgebra is finitely generated. Characterizations of com-
plex affinely closed homogeneous spaces of reductive groups in terms
of compact transformation groups and invariant algebras on compact
homogeneous spaces are given in [La99] and [GL01].
The aim of this paper is to generalize the result of D. Luna to the
case of an arbitrary affine algebraic group G (Theorem 2) and to obtain
a classification of affine G-algebras with finitely generated invariant
subalgebras (Theorem 3). Note that a characteristic-free variant of
these results for a solvable group G is given in [Te04].
2. A description of affinely closed spaces
Let us fix a Levi decomposition G = LGuof the group G in a semidi-
rect product of a reductive subgroup L and the unipotent radical Gu.
By φ denote the homomorphism φ : G → G/Gu. We shall identify the
image of φ with L. Put K = φ(H).
Theorem 2. The following conditions are equivalent:
(1) G/H is affinely closed;
(2) L/K is affinely closed.
Proof. The subgroup H is observable in G if and only if the subgroup
K is observable in L [Su88], [Gr97, Th.7.3].

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ON AFFINELY CLOSED HOMOGENEOUS SPACES3
Suppose that L/K admits a non-trivial affine embedding.
there are an L-module V and a vector v ∈ V such that the stabi-
lizer Lvequals K and the orbit boundary Y = Z \ Lv, where Z = Lv,
is nonempty. Let I(Y ) be the ideal in k[Z] defining the subvariety
Y . Recall that for an action of an algebraic group G on an affine
variety X any element f ∈ k[X] belongs to a finite-dimensional invari-
ant subspace, or, equivalently, k[X] is a sum of its finite-dimensional
G-submodules. Thus there exists an L-submodule V1⊂ I(Y ) that gen-
erates I(Y ) as an ideal. The inclusion V1⊂ k[Z] defines L-equivariant
morphism ψ : Z → V∗
phism ξ : Z → V2 = V∗
Y to the origin and is injective on the open orbit in Z. Hence we
obtain an embedding of L/K in an L-module such that the closure
of the image of this embedding contains the origin. Put v2 = ξ(v).
By the Hilbert-Mumford Criterion, there is a one-parameter subgroup
λ : k∗→ L such that limt→0λ(t)v2= 0. Consider the weight decompo-
sition v2= v(i1)
2
+ ··· + v(is)
2
of the vector v2, where λ(t)v(ik)
Here all ikare positive.
By the identification G/Gu= L, one may consider V2 as a G-
module. Let W be a finite-dimensional G-module with a vector w
whose stabilizer equals H.Replacing the pair (W,w) by the pair
(W ⊕ (W ⊗ W),w + w ⊗ w), one may suppose that the orbit Gw
intersects the line kw only at the point w. For a sufficiently large N in
the G-module W ⊗V⊗N
2
one has limt→0λ(t)(w ⊗v⊗N
be considered as a subgroup of G). On the other hand, the stabilizer
of w ⊗ v⊗N
2
coincides with H. This implies that the space G/H is not
affinely closed.
Conversely, suppose that G/H admits a non-trivial affine embedding.
This embedding corresponds to a G-invariant subalgebra A ⊂ k[G/H]
containing a non-trivial G-invariant ideal I. Note that the algebra
k[L] may be identified with the subalgebra in k[G] of (left- or right-)
Gu-invariant functions, k[G/H] is realized in k[G] as the subalgebra of
right H-invariants, and k[L/K] is the subalgebra of left Gu-invariants
in k[G/H]. Consider the action of Guon the ideal I. By the Lie-
Kolchin Theorem, there is a non-zero Gu-invariant element in I. Thus
the subalgebra A∩k[L/K] contains the non-trivial L-invariant ideal I∩
k[L/K]. If the space L/K is affinely closed then we get a contradiction
with the following lemma.
Then
1and ψ−1(0) = Y . Then L-equivariant mor-
1⊕ (V ⊗ V∗
1), z → (ψ(z),z ⊗ ψ(z)) maps
2
= tikv(ik)
2
.
2
) = 0 (λ(k∗) may
Lemma 1. Let L/K be an affinely closed space of a reductive group L.
Then any L-invariant subalgebra in k[L/K] is finitely generated and
does not contain non-trivial L-invariant ideals.
Proof. Let B ⊂ k[L/K] be a non-finitely generated invariant subal-
gebra. For any chain W1 ⊂ W2 ⊂ W3 ⊂ ... of finite-dimensional
L-invariant submodules in k[L/K] with ∪∞
i=1Wi= k[L/K], the chain

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4IVAN V. ARZHANTSEV AND NATALIA A. TENNOVA
of subalgebras B1⊂ B2⊂ B3⊂ ... generated by Widoes not stabi-
lize. Hence one may suppose that all inclusions here are strict. Let
Zibe the affine L-variety corresponding the algebra Bi. The inclusion
Bi⊂ k[L/K] induces the dominant morphism L/K → Ziand Theo-
rem 1 implies that Zi= L/Ki, K ⊂ Ki. But B1⊂ B2⊂ B3⊂ ..., and
any Kiis strictly contained in Ki−1, a contradiction. This shows that
B is finitely generated and, as proved above, L acts transitively on the
affine variety Z corresponding to B. But any non-trivial L-invariant
ideal in B corresponds to a proper L-invariant subvariety in Z.
?
Theorem 2 is proved.
?
Corollary 1. Let G/H be an affinely closed homogeneous space. Then
for any affine G-variety X and a point x ∈ X such that Hx = x, the
orbit Gx is closed.
Proof. The stabilizer Gxis observable in G, hence φ(Gx) is observable
in L. The subgroup φ(Gx) contains K = φ(H), and Theorems 1 and 2
imply that the space L/φ(Gx) is affinely closed. By Theorem 2, the
space G/Gxis affinely closed.
?
In particular, we get
Corollary 2. If X is an affine G-variety and a point x ∈ X is T-fixed,
where T is a maximal torus of the group G, then the orbit Gx is closed.
3. G-algebras with finitely generated invariant
subalgebras
Below an affine algebra over a field k means a finitely generated
associative commutative k-algebra with unit. Let F be a subgroup of
the automorphism group of an affine algebra A and rad(A) the set of
nilpotent elements of the algebra A. Clearly, rad(A) is an F-invariant
ideal in A.
Lemma 2. The following conditions are equivalent:
(1) any F-invariant subalgebra in A is finitely generated;
(2) any F-invariant subalgebra in A/rad(A) is finitely generated and
dimrad(A) < ∞.
Proof. Any finite-dimensional subspace in rad(A) generates a finite-
dimensional subalgebra in A. Hence if dimrad(A) = ∞, then the
subalgebra generated by this subspace is not finitely generated. On the
other hand, the preimage in A of any non-finitely generated subalgebra
in A/rad(A) is not finitely generated.
Conversely, suppose that (2) holds. Then any subalgebra in A is gen-
erated by elements whose images generate the image of this subalgebra
in A/rad(A), and by a basis of the radical of the subalgebra.
?

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ON AFFINELY CLOSED HOMOGENEOUS SPACES5
By definition, a G-algebra is an affine algebra A with an action (by
automorphisms) of an algebraic group G such that any element a ∈ A
is contained in a finite-dimensional G-invariant subspace, where G acts
rationally. Our aim is to describe all G-algebras such that any G-
invariant subalgebra is finitely generated. By Lemma 2, we may assume
that rad(A) = 0.
Let X = Spec(A) be the affine variety corresponding to affine algebra
A without nilpotents. To fix a structure of G-algebra on A = k[X]
is nothing else but to fix an (algebraic) G-action on X. Define the
dimension dimA of the algebra A as the dimension of the variety X.
Lemma 3. If dimA ≤ 1, then any subalgebra in A is finitely generated.
Proof. The case when X is irreducible is considered in [Ar03, Prop. 2].
If X = X1∪···∪Xmis the decomposition on irreducible components,
then A is embedded into the direct sum of the algebras k[Xi], and
any subalgebra in a summand is finitely generated. Now it is easy to
finish the proof by induction on m considering the projection of A on
k[X1].
?
Lemma 4. Suppose that X = Z1∪ Z2, where Z1 and Z2 are closed
invariant subvarieties. Then the following conditions are equivalent:
(1) any invariant subalgebra in A is finitely generated;
(2) any invariant subalgebra in k[Z1] and in k[Z2] is finitely gener-
ated.
Proof. If there is a non-finitely generated subalgebra in k[Z1], then one
may consider its preimage with respect to the restriction homomor-
phism k[X] → k[Z1]. To prove the converse, embed k[X] in k[Z1] ⊕
k[Z2] and use the arguments from the proof of the previous lemma.
?
By Lemma 4, one may assume that G acts transitively on the set of
irreducible components of the variety X.
Below we generalize a construction from [Ar03] to the case of non-
connnected groups and reducible varieties. Let Y be a closed subvariety
of X. Consider a subalgebra
A(X,Y ) = {f ∈ k[X] | f(y1) = f(y2) ∀y1,y2∈ Y }.
Lemma 5. If Y contains an irreducible component of positive dimen-
sion that does not coincide with any irreducible component of X, then
A(X,Y ) is not finitely generated.
Proof. Note that A(X,Y ) = k ⊕ I(Y ). If A(X,Y ) is finitely gener-
ated, then one may assume that generators f1,...,fkare in I(Y ). Any
monom in f1,...,fkof degree s is in I(Y )s. Hence it is sufficient to
prove that for some l the space I(Y )/I(Y )lis infinite-dimensional.
Let Y = Y1∪···∪Ynand X = X1∪···∪Xmbe the decompositions on
irreducible components, and Y1⊂ X1, Y1?= X1, dimY1> 0. Suppose
that f ∈ I(Y ) and f is not identically zero on X1. Let OX1,Y1be