HAL Id: hal-01230558
https://hal.archives-ouvertes.fr/hal-01230558v2
Submitted on 15 Jul 2020HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Almost algebraic actions of algebraic groups and
applications to algebraic representations
Uri Bader, Bruno Duchesne, Jean Lécureux
To cite this version:
Uri Bader, Bruno Duchesne, Jean Lécureux. Almost algebraic actions of algebraic groups and applications to algebraic representations. Groups Geom. Dyn, 2017, 11 (2), pp.705 - 738. �10.4171/GGD/413�. �hal-01230558v2�
Almost algebraic actions of algebraic groups
and applications to algebraic representations
Uri Bader, Bruno Duchesne, and Jean Lécureux
Abstract. Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis–Zimmer super-rigidity phe-nomenon [2].
Mathematics Subject Classification (2010).20G15, 12J25, 37C40.
Keywords.Complete separable valued fields, probability measures on algebraic varieties, algebraic representations of amenable ergodic actions, Margulis–Zimmer super-rigidity.
Contents
1 Introduction . . . .705
2 Preliminaries . . . .711
3 Almost algebraic groups and actions . . . .717
4 On bounded subgroups . . . .728
5 The space of norms and seminorms . . . .732
6 Existence of algebraic representations . . . .735
References . . . .737
1. Introduction
This work concerns mainly the dynamics of an algebraic group acting on the space of probability measures on an algebraic variety. Most (but not all) of our results are known for local fields (most times, under a characteristic zero assumption). Our main contribution is giving an approach which is applicable also to a more general class of fields: complete valued fields. On our source of motivation, which stems from ergodic theory, we will elaborate in §1.2, and in particular Theorem1.16.
First we describe our objects of consideration and our main results, put in some historical context.
Setup 1.1. For the entire paper.k; j j/will be a valued field, which is assumed to be complete and separable as a metric space, andkO will be the completion of its
algebraic closure, endowed with the extended absolute value.
Note thatkOis separable and complete as well (see the proof of Proposition2.2).
The most familiar examples of separable complete valued fields are of courseR
and C, but one may also consider the p-adic fields Qp, as well as their finite
extensions. Considering k D Cp D yQp one may work over a field which is
simultaneously complete, separable and algebraically closed. Other examples of a complete valued field are given by fields of Laurent seriesK..t //, whereK is
any field (this field is local if and only ifKis finite, and separable if and only if Kis countable), or more generally the field of Hahn seriesK..t//, whereis a
subgroup ofR(see for example [17]). This field is separable if and only ifK is
countable andis discrete (see [13]).
Convention 1.2. Algebraic varieties overkwill be identified with theirkO-points
and will be denoted by boldface letters. Theirk-points will be denoted by corre-sponding Roman letters. In particular we use the following.
Setup 1.3. We fix ak-algebraic groupGand we denoteG D G.k/.
We are interested in algebraic dynamical systems, which we now briefly de-scribe. For a formal, pedantic description see §2.1 and in particular Proposi-tion2.2. By an algebraic dynamical system we mean the action ofGonV, where V is the space ofk-points of ak-algebraic varietyV on whichGactsk
-morphi-cally. Such a dynamical system is Polish: G is a Polish group,V a Polish space
and the action mapG V ! V is continuous (see §2.1for proper definitions).
The point stabilizers of such an action are algebraic subgroups, and by a result of Bernstein and Zelevinski [3], the orbits of such an action are locally closed (see Proposition2.2).
Following previous works of Furstenberg and Moore, Zimmer found a sur-prising result: for the action of an algebraic groupG on an algebraic varietyV,
all defined overR, consider now the action ofG on the space Prob.V /of
prob-ability measures onV. Then the point stabilizers are again algebraic subgroups
and the orbits are locally closed. However, this result does not extend trivially to other fields. For example, withk D C, consider the Haar measure on the circle S1< C. For the action of
Con itself, the stabilizer of that measure is
S1, which
is not aC-algebraic subgroup. Similarly, fork D Qp, consider the Haar measure
on thep-adic integers Zp < Qp. For the action ofQp on itself, the stabilizer of
Definition 1.4. A closed subgroup L < G is called almost algebraic if there exists ak-algebraic subgroupH< Gsuch thatLcontainsH D H.k/as a normal
cocompact subgroup. A continuous action of G on a Polish space V is called almost algebraicif the point stabilizers are almost algebraic subgroups ofGand the collection ofG-invariant open sets separates the G-orbits, i.e. the quotient topology onGnV isT0.
Remark 1.5. Ifkis a local field thenGis locally compact and by [8, Theorem 2.6]
the conditionGnV isT0is equivalent to the (a priori stronger) condition that every G-orbit is locally closed inV.
Remark 1.6. Ifk D Rthen every compact subgroup ofGis the real points of a
real algebraic subgroup ofG (see e.g. [22, Chapter 4, Theorem 2.1]). It follows
that every almost algebraic subgroup is the real points of a real algebraic subgroup ofG. We get that a continuous action ofGon a Polish spaceV is almost algebraic
if and only if the stabilizers are real algebraic and the orbits are locally closed.
Two obvious classes of examples of almost algebraic actions are algebraic ac-tions (by the previously mentioned result of Bernstein and Zelevinski) and proper actions (as the stabilizers are compact and the space of orbits isT2, that is,
Haus-dorff). The notion of almost algebraic action is a natural common generalization. It is an easy corollary of Prokhorov’s theorem (see Theorem2.3below) that if the action ofG onV is proper then so is its action on Prob.V /, see Lemma2.7. The main theorem of this paper is the following analogue.
Theorem 1.7. If the action ofGon a Polish spaceV is almost algebraic then the action ofGonProb.V /is almost algebraic as well.
The following corollary was obtained by Zimmer, under the assumptions that
kis a local field of characteristic 0 andV is homogeneous, see [24, Chapter 3].
Corollary 1.8. AssumeG has ak-action on a k-variety V. Then the induced
action ofG D G.k/onProb.V .k//is almost algebraic.
In the course of the proof of Theorem 1.7 we obtain in fact a more precise information. Ak-G-variety is ak-variety with ak-action of G.
Proposition 1.9. Fix a closed subgroupL < G. Then there exists ak-subgroup
H0 < G which is normalized by L such that L has a precompact image in
the Polish group.NG.H0/=H0/.k/ and such that for everyk-G-varietyV, any L-invariant finite measure on V.k/ is supported on the subvariety of H0-fixed
This proposition is a generalization of one of the main results of Shalom [19], who proves it under the assumptions thatk is local and L D G. For the case
L D Gthe following striking corollary is obtained.
Corollary 1.10. If for every strictk-algebraic normal subgroupHGG,G.k/=H.k/
is non-compact, then everyG-invariant measure on any k-G-algebraic variety V.k/is supported on theG-fixed points.
In particular we can deduce easily the Borel density theorem.
Corollary 1.11. LetGbe ak-algebraic group and < G D G.k/ be a closed
subgroup such that G= has aG-invariant probability measure. If for every properk-algebraic normal subgroupHG G, G.k/=H.k/ is non-compact, then is Zariski dense inG.
To deduce the last corollary from the previous one, consider the map
G= ! .G=xZ/.k/;
wherexZdenotes the Zariski closure of, and push forward the invariant measure
fromG= to obtain aG-invariant measure on .G=xZ/.k/. The homogeneous
spaceG=xZmust contain aG-fixed point, hence must be trivial. That isxZ D G.
1.1. Applications: ergodic measures on algebraic varieties. A classical theme in ergodic theory is the attempt to classify all ergodic measures classes, given a continuous action of a topological group on a Polish space. In this regard, the axiom that the space of orbits isT0 has strong applications. Recall that, given
a groupLacting by homeomorphisms on a Polish spaceV, a measure onV is
L-quasi-invariant if its class is L-invariant. The following proposition is well known.
Proposition 1.12. LetV be a PolishG-space and assume that the quotient topol-ogy on GnV is T0. Let L < G be a subgroup and an L-quasi-invariant
ergodic probability (or -finite) measure. Then there exists v 2 V such that
.V Gv/ D 0.
Indeed, GnV is second countable, as V is, and for a countable basis Bi,
denoting the push forward oftoGnV byN, the set \
¹Bi j N.Bi/ D 1º \ \
¹Bicj N.Bi/ D 0º
In particular, we get that for a subgroupL < G and an algebraic dynamical system ofG, everyL-invariant measure is supported on a singleG-orbit. Another striking result is that an algebraic variety cannot support a weakly mixing proba-bility measure. Recall that anL-invariant probability measureis weakly mixing if and only if isL-ergodic.
Corollary 1.13. Assume G has a k-action on the k-variety V. Fix a closed
subgroupL < G and letbe anL-invariant weakly mixing probability measure onV D V .k/. Then there exists a pointx 2 VLsuch that D ıx.
This corollary follows at once from Proposition 1.9, as the action of L on VH0\ V .k/is via a compact group.
We end this subsection with the following useful application, obtained by composing Proposition1.12with Theorem1.7. This corollary is in fact our main motivation for developing the material in this paper. It deals with measure on spaces of measures, and is the main tool in deriving Theorem1.16below.
Corollary 1.14. AssumeGhas ak-action on thek-varietyV. DenoteV D V .k/.
LetL < G be a subgroup and be anL-ergodic quasi-invariant measure on Prob.V /. Then there exists 2Prob.V /such that.Prob.V / G/ D 0.
1.2. Applications to algebraic representations of ergodic actions. A main motivation for us to extend the foundation outside the traditional local field zone is the recent developments in the theory of algebraic representations of ergodic actions, and in particular its applications to rigidity theory. In [2] the following theorem, as well as various generalizations, are proven.
Theorem 1.15([2, Theorem 1.1], Margulis super-rigidity for arbitrary fields). Let
lbe a local field. LetT to be thel-points of a connected almost-simple algebraic group defined overl. Assume that thel-rank ofT is at least two. Let < T be a lattice.
Letkbe a valued field. Assume that as a metric spacekis complete. LetGbe thek-points of an adjoint simple algebraic group defined overk. LetıW ! Gbe a homomorphism. Assumeı./is Zariski dense inGand unbounded. Then there exists a continuous homomorphismd W T ! Gsuch thatı D d j.
The proofs in [2] are based on the following, slightly technical, theorem which will be proven here.
Theorem 1.16. LetRbe a locally compact group andY be an ergodic, amenable Lebesgue R-space. Let .k; j j/ be a valued field. Assume that as a met-ric spacek is complete and separable. LetG be a simple k-algebraic group.
Then either there exists ak-algebraic subgroupH ΠGand anf-equivariant measurable map W Y ! G=H.k/, or there exists a complete and separable metric spaceV on whichG acts by isometries with bounded stabilizers and an
f-equivariant measurable map 0W Y ! V
.
A more friendly, cocycle free, version is the following.
Corollary 1.17. LetRbe a locally compact, second countable group. LetY be an ergodic, amenableR-space. Suppose that G is an adjoint simplek-algebraic group, and there is a morphismR ! G DG.k/. Then
either there exists a complete and separable metric spaceV, on whichGacts by isometries with bounded stabilizers, and an R-equivariant measurable mapY ! V or
there exists a strictk-algebraic subgroupHand anR-equivariant
measur-able mapY !G=H.k/.
TakingY to be a point in the above corollary, we obtain the following.
Corollary 1.18. SupposeR <GLn.k/is a closed amenable subgroup. Then the
image ofRinRxZ modulo its solvable radical is bounded.
Indeed, upon modding out the solvable radical ofRxZ, the latter is a product of
simple adjoint factors, and by the previous corollary the image ofRin each factor
is bounded.
Note that over various fields, such asCp andFNp..t //, every bounded group is
amenable, being the closure of an ascending union of compact groups, while for other fields there exist bounded groups which are not amenable. For example SL2.QŒŒt /, which is bounded in SL2.Q..t ///, factors over the discrete group
SL2.Q/which contains a free group.
1.3. The structure of the paper. The paper has two halves: the first half consist-ing of §2,§3 is devoted to the proof of Theorem1.7and the second half is devoted to the proof of Theorem1.16.
In §2 we collect various needed preliminaries, in particular we discuss the Polish structure on algebraic varieties, and on spaces of measures. The most im-portant results in this section are Proposition2.2that discusses algebraic varieties and and Corollary2.14that uses disintegration as a replacement for a classical er-godic decomposition argument (which is not applicable in our context, due to the lack of compactness). The heart of the paper is §3, where the concept of almost algebraic action is discussed. Theorem1.7is proven at §3.4.
In §4, we give a thorough discussion of bounded subgroups of algebraic groups, and in §5, we discuss a suitable replacement of a compactification of coset spaces. In §6, we prove Theorem1.16.
2. Preliminaries
2.1. Algebraic varieties as Polish spaces. Recall that a topological space is called Polish if it is separable and completely metrizable. For a good survey on the subject we recommend [14]. We mention that the class of Polish spaces is closed under countable disjoint unions and countable products. AGı-subset of a
Polish space is Polish so, in particular, a locally closed subset of a Polish space is Polish. A Hausdorff space which admits a finite open covering by Polish open sets is itself Polish. Indeed, such a space is clearly metrizable (e.g. by Urysohn metrization theorem [14, Theorem 1.1]) so it is Polish by Sierpinski theorem [14, Theorem 8.19] which states that the image of a continuous open map from a Polish space to a separable metrizable space is Polish.
A topological group which underlying topological space is Polish is called a Polish group. Sierpinski theorem also implies that for a Polish groupK and a closed subgroupL, the quotient topology onK=Lis Polish. Effros Lemma [8, Lemma 2.5] says that the quotient topology on K=Lis the unique K-invariant Polish topology on this space. Another important result of Effros concerning Polish actions (that are continuous actions of Polish groups on Polish spaces) is the following.
Theorem 2.1(Effros theorem [8, Theorem 2.1]). For a continuous action of a Polish groupG on a Polish spaceV the following are equivalent.
(1) The quotient topology onGnV isT0.
(2) For everyv 2 V, the orbit mapG=StabG.v/ ! Gvis a homeomorphism.
Our basic class of Polish actions will be given by actions of algebraic groups on algebraic varieties. As mentioned in Setups1.1&1.3, we fixed a complete and separable valued field.k; j j/, that is a fieldk with an absolute valuej jwhich is complete and separable (in the sense of having a countable dense subset). See [9,6]1for a general discussion on these fields. It is a standard fact that a complete absolute value on a field F has a unique extension to its algebraic closure Fx
[6, §3.2.4, Theorem 2] and Hensel lemma implies that the completionFy of this
algebraic closure is still algebraically closed [6, §3.4.1, Proposition 3].
Recall that we identify eachk-variety V with its set ofkO-points. In particular,
this identification yields a topology on V. Identifying the affine spaceAn. Ok/with O
kn, any affinek-variety can be seen as a closed subset ofAn. Ok/. More generally,
a k-variety has a unique topology making its affine charts homeomorphisms.
Observe that with this topology, the set ofk-pointsV of V is closed.
Topological notions, unless otherwise said, will always refer to this topology. In particular, for thek-algebraic group G we fixed,GandG DG.k/are
topolog-ical groups. We note thatV actually carries a structure of ak-analytic manifold, 1In the second reference, the word valuation is used for what we call an absolute value.
Gis ak-analytic group and the action ofG onV isk-analytic. We will not make an explicit use of the analytic structure here. The interested reader is referred to the excellent text [18], in which the theory of analytic manifolds and Lie groups over complete valued fields is developed (see in particular [18, Part II, Chapter I]). We will discuss the category ofk-G-varieties. Ak-G-varietyis ak-variety
endowed with an algebraic action ofGwhich is defined overk. A morphism of
such varieties is ak-morphism which commutes with theG-action.
Proposition 2.2. Ak-variety V and its set ofk-pointsV are Polish spaces. In particular, G andG are Polish groups.
IfVis ak-G-variety then theG-orbits inV are locally closed and the quotient
topology onGnV isT0. Forv 2 V, the orbitGv is ak-subvariety ofV. There
exists ak-subgroup H < G contained in the stabilizer of v such that the orbit
map G=H ! Gv is defined over k and the induced map G=H ! Gv is a
homeomorphism, whereH D H.k/, G=H is endowed with the quotient space topology andGvis endowed with the subspace topology.
Proof. Let us first explain how the extended absolute value makeskO Polish. In our situationk has a countable dense subfieldk0. The algebraic closurekN0ofk0
is still countable and thus its completionkO0is separable and algebraically closed.
By the universal property of the algebraic closure,kNembeds inkO0and by
unique-ness of the extension of the absolute value, this embedding is an isometry. Thus
O
kis algebraically closed, complete and separable.
Since kO is Polish, so is the affine space An. Ok/ ' Okn. It follows that V
(respectivelyV) is a Polish space, as this space is a Hausdorff space which admits a finite open covering by Polish open sets — the domains of itsk-affine charts (respectively theirk-points).
The fact that the G-orbits in V are locally closed is proven in the appendix
of [3]. Note that in [3] the statement is claimed only for non-Archimedean local fields, but the proof is actually correct for any field with complete non-trivial absolute value, which is the setting of [18, Part II, Chapter III] on which [3] relies. Another proof can be found in [11, §0.5]. It is then immediate that the quotient topology onGnV isT0.
Forv 2 V the orbitGv is ak-subvariety ofV by [5, Proposition 6.7]. We
setHD StabG.v/Z (note that if char.k/ D 0thenH D StabG.v/). By [5, AG,
Theorem 14.4],His defined overk, and it is straightforward thatH D H.k/ D
StabG.v/. By [5, Theorem 6.8] the orbit mapG=H ! Gv is defined over k,
thus it restricts to a continuous map fromG=H ontoGv. The fact that the latter map is a homeomorphism follows from Effros theorem (Theorem2.1) sinceGnV
We emphasize that, as a special case of Proposition2.2, we get that for every
k-algebraic subgroup H of G, the embeddingG=H ! G=H.k/is a homeomor-phism on its image. We will use this fact freely in the sequel.
2.2. Spaces of measures as Polish spaces. In this subsectionV denotes a Polish
space. We let Prob.V /be the set of Borel probability measures onV, endowed
with the weak*-topology (also called the topology of weak convergence). This topology comes from the embedding of Prob.V /in the dual of the Banach space
of bounded continuous functions onV. If d is a complete metric on V which
is compatible with the topology (the metric topology coincides with the original topology onV), the corresponding Prokhorov metric d on Prob.V /is defined as
follows: for; 2Prob.V /, d.—;/is the infimum of" > 0such that for all Borel
subsetA V,.A/ .A"/ C "and symmetrically.A/ .A"/ C ", where A"is the"-neighborhood (ford) aroundA. The following theorem summarizes
some standard results, see Chapter 6 and Appendix III of [4].
Theorem 2.3(Prokhorov). The metric space.Prob.V /;d/is complete and sep-arable and the topology induced by d onProb.V /is the weak*-topology. In par-ticular the spaceProb.V /endowed with the weak*-topology is Polish.
A subset C in Prob.V / is precompact if and only if it is tight: for every
> 0there exists compactK V such that for every 2 C, .K/ > 1 . In particularProb.V /is compact ifV is.
Remark 2.4. Replacing if necessaryd by a bounded metric, we note that there
is another metric on Prob.V / with the same properties (metrizing the
weak*-topology and being invariant under isometries): the Wasserstein metric [21, Corol-lary 6.13].
We endow Homeo.V /with the pointwise convergence topology. The follow-ing is a standard application of the Baire category theorem, see [14, Theorem 9.14].
Theorem 2.5. AssumeG is acting by homeomorphisms onV. Then the action mapG V ! V is continuous if and only if the homomorphismG !Homeo.V /
is continuous.
Lemma 2.6. If G acts continuously on V then it also acts continuously on Prob.V / and if the action G Õ .V; d / is by isometries, the action G Õ .Prob.V /; d/is also by isometries.
Proof. The fact thatGacts by isometries on Prob.V /whenGacts by isometries
onV is straightforward from the definition of the Prokhorov metric. In order to
prove thatG acts continuously on Prob.V /when it acts continuously onV it is
gn in G, gn ! e in G implies gn ! in Prob.V /. Fix 2 Prob.V / and
assumegn ! e inG. For every bounded continuous functionf onV, we have
by Lebesgue bounded convergence theorem
Z f .x/d.gn/.x/ D Z f .gnx/d.x/ ! Z f .x/d.x/
as for everyx 2 V, gnx ! x. Thus, by the definition of the weak*-topology
gn ! .
We observe that Lemma2.6and Proposition2.2show that if V is ak-G-variety
thenGacts continuously onV D V .k/and on Prob.V /. The following is a nice application of Prokhorov theorem (Theorem2.3).
Lemma 2.7. If the action ofGonV is proper then the action ofGonProb.V /is proper as well.
Proof. For a compact C Prob.V / we can find a compact K V with
.K/ > 1=2for every 2 C by Theorem 2.3. Then forg 2 G and 2 C
such thatg 2 C we get that both.K/ > 1=2and.gK/ D g.K/ > 1=2, thus
gK \ K ¤ ;. We conclude that¹g 2 G j gC \ C ¤ ;ºis precompact, as it is a subset of the precompact set¹g 2 G j gK \ K ¤ ;º.
2.3. Polish extensions and disintegration
Definition 2.8. A Polish fibration is a continuous mappW V ! U whereU is a T0-space andV a Polish space. An action ofGon such a Polish fibration is a pair
of continuous actions onV andU such thatpis equivariant.
LetpW V ! U be a Polish fibration. Let ProbU.V /be the set of probability
measures onV which are supported on one fiber. We denotepWProbU.V / ! U
the natural map.
Lemma 2.9. The mappis a Polish fibration. If the groupG acts on the Polish
fibrationV ! U, then it also acts onp.
Proof. Since U is T0, fibers of p are separated by a countable family.Cn/of
closed saturated subsets ofV. A probability measureis supported on one fiber
if and only if for alln,.Cn/.V n Cn/ D 0. The set¹ 2Prob.V /; .Cn/ D 1º
is closed and¹ 2 Prob.V /; .V n Cn/ D 1º is Gı since for all 0 < r < 1, ¹ 2Prob.V /; .V n Cn/ > r ºis open. So ProbU.V /is aGı-subset of Prob.V /
and thus Polish.
Let us show that p is continuous. Assume n ! in ProbU.V /. Let u D p./ andun D p.n/. LetO U be an open set containingu. For nlarge enough,n.p 1.O// > 1=2and thusun2 O.
IfG acts onV ! U, it is clear thatGacts on ProbU.V /. The continuity of
the action on ProbU.V /follows from Lemma2.6.
Let .U; / be a probability space and X be a Polish space, we denote by
L0.U; X /the space of classes of measurable maps fromU toX, under the
equiv-alence relation of equality-almost everywhere. Note that the dependence onis
implicit in our notation. We endow that space with the topology of convergence in probability. Fixing a compatible metricd onX, this topology is metrized as
follows: for ; 02L0.U; X /, the distance between and 0is
ı. ; 0 / D Z X min.d. .v/; 0 .v//; 1/d.v/:
This topology can be also defined using sequences: n! if for any" > 0, there
isA U such that.A/ > 1 "and for allnsufficiently large and all v 2 A,
d. .v/; n.v// < ". We note that this topology on L0.U; X /does not depend on
the choice of an equivalent metric onV. This turns L0.U; X /into a Polish space.
Lemma 2.10. Assume .˛n/ is a sequence converging to ˛ in probability in
L0.U; X /. Then there exists a subsequence˛
nk which convergence-a.e. to˛,
that is for-almost everyu 2 U,˛nk.u/converges to˛.u/inX.
The proof of the lemma is standard, but in most textbooks it appears only for the casesX D RorX D C, see for example [10, Theorem 2.30]. Even though the standard proof works mutatis mutandis, we give below a short argument, reducing the general case to the caseX D R.
Proof. The sequenced.˛n; ˛/(which denotes the mapu 7! d.˛n.u/; ˛.u//)
con-verges in probability to 0 in L0.U; R/. Thus there exists a subsequenced.˛ nk; ˛/
converging to 0 a.e, and we get that˛nk converges to˛a.e.
IfpW V ! Uis a Polish fibration, andis a measure onU, we denote L0 p.U; V /
the space of measurable (identified if agree almost everywhere) sections ofp, i.e. maps which associates tou 2 U a point inp 1.U /, endowed with the induced topology from L0.U; V /. If G acts on the Polish fibration p, it also acts on
L0
p.U; V /via the formula.gf /.u/ D gf .g 1u/whereu 2 U andf 2Lp0.U; V /.
The following theorem is a variation of the classical theorem of disintegration of measures. It is essentially proven in [20].
Theorem 2.11. Let pW V ! U be a Polish fibration and be a probability measure on U. Let P D ¹ 2 Prob.V / j p D º. For every ˛ 2
L0
p.U;ProbU.V //the formula
R
U˛.u/ddefines an element ofP. The map thus
obtainedLp0
Definition 2.12. For 2 P, the element of L0
p.U;ProbU.V // obtained by
applying to the inverse map of ˛ 7! R
U˛.u/d is denoted u 7! u. It is
called the disintegration ofwith respect topW V ! U.
Proof. We first claim that the map ˛ 7! R
U˛.u/dis continuous, and then we
argue to show that it is invertible, and its inverse is continuous as well.
For the continuity, given a converging sequence˛n! ˛in Lp0.U;ProbU.V //
with n D RU˛n.u/d, D RU˛.u/d, it is enough to show that every
subsequence ofnhas a subsequence that converges to. Since every sequence
that converges in measure has a subsequence that converges almost everywhere, abusing our notation and denoting again ˛n and n for the resulting
sub-sub-sequences, we may assume that˛nconverges to˛ -almost everywhere. Picking
an arbitrary continuous bounded functionf onV, we obtain that for-a.eu 2 U, R
V d˛n.u/f ! R
V d˛.u/f. Thus by Lebesgue bounded convergence theorem we
get Z V dnf D Z U d Z V d˛n.u/f ! Z U d Z V d˛.u/f D Z V df:
This shows that indeedn! .
We now argue that the map ˛ 7! R
U˛.u/d is invertible and its inverse is
continuous. Without loss of generality, we can assume that p is onto. Hence
U is second countable. Since it is also T0, it follows thatU is countably
sep-arated. By [24, Proposition A.1], there exists a Borel embedding W U ! Œ0; 1. We considerŒ0; 1with the measure . Precomposition by gives a
homeomor-phism L0
. ıp/ Œ0; 1;ProbŒ0;1.V / !L
0
p.U;ProbU.V //. Thus, in what follows
we may and do assume thatU Œ0; 1.2Under this assumption [20, Theorem 2.1]
guarantees that the map L0
p.U;ProbU.V // ! P is invertible. We denote the
preimage of 2 P byu 7! u. We are left to show that this association is
con-tinuous. To this end we embedV in a compact metric spaceV0
and extendpby settingp.v0
/ D 1forv0 2 V0
V. Then [20, Theorem 2.2] proves that for almost
everyu 2 U, u is obtained as the weak*-limit of the normalized restrictions,
denoted byu;, ofonp 1.u ; u C /as ! 0.
Assume thatn ! is a converging sequence inP. We know that for-a.e. u,d.u;; u/ ! 0when ! 0and similarly for alln 2 N,d.n
u;; nu/ ! 0
when ! 0. Fix" > 0. Forn 2 N, we set
AnD ¹u 2 U jthere exists0 > 0such that d.k
u;; u/ "for allk nand 2 .0; 0/º:
2Since the embeddingU ! Œ0; 1is only Borel, when we assumeU Œ0; 1, the fibration
V ! Ucannot be assumed to be Polish anymore. Since our argument does not depend on the topology ofU, this does not matter here.
Then.S An/ D 1andAn AnC1. Thus there isnsuch that.An/ 1 "and
foru 2 An,d.u; ku/ "for allk n. This shows that the image sequence of .n/in L0
p.U;ProbU.V //indeed converges to the image of.
We note that ifGacts on the fibrationV ! U (that is,Gacts onU andV and
pis equivariant) then the disintegration homeomorphism is also equivariant with respect to the natural action ofGon L0
p.U; V /given by.gf /.u/ D g.f .g 1u//.
Lemma 2.13. Let pW V ! U be a Polish fibration with an action of G such that theG-action on U is trivial. Let be a probability measure onU, and let
f 2Lp0.U; V /. Then there existsU1 U of full measure such that
Stab.f / D \ u2U1
Stab.f .u//:
Proof. LetLbe the stabilizer off inG. IfL0is a countable dense subgroup of L, then there is a full measure subsetU1 U such thatL0 Tu2U1Stab.f .u//
(for anyg 2 L0, there is such a subspace
Ug. ChooseU1to be the intersection over L0). Since all these stabilizers are closed, and
L0is dense in
L, we actually have L T
u2U1Stab.f .u//. Since the reverse inclusion is clear, we conclude that
L D \ u2U1
Stab.f .u//:
Corollary 2.14. Assume G acts continuously on the Polish space V and the quotient topology on GnV is T0. Let L < G be a closed subgroup and be
anL-invariant probability measure onV. Then there exist a pointv 2 V and an
L-invariant probability measure onG v ' G=Stab.v/.
Proof. Let be the pushforward measure of on U. By Theorem 2.11, we
may consider the disintegration of as an element .u/ 2 Lp0.U;ProbU.V //
and this element is clearlyL-invariant. By Lemma2.13, the stabilizer of.u/
is an intersection of stabilizers of the measures u, for u in a subset of U.
In particularLstabilizes someu, which is a measure supported on an orbitG v.
The latter is equivariantly homeomorphic toG=Stab.v/thanks to Effros theorem
(Theorem2.1).
3. Almost algebraic groups and actions
The goal of this section is the proof of Theorem 1.7. Starting with an almost algebraic action ofGon a PolishV, we aim to prove that the actionG ÕProb.V /
is algebraic as well. So we have to prove that stabilizers of probability measures onV are almost algebraic and the quotientGnProb.V /isT0. Going toward wider
3.1. Almost algebraic groups. Recall that by our setup1.1,.k; jj/is a fixed com-plete and separable valued field and G is a fixed k-algebraic group.
By Proposition2.2,G D G.k/ has the structure of a Polish group. Recall that a closed subgroupL < G is called almost algebraic if there exists ak-algebraic subgroupH < Gsuch thatLcontainsH D H.k/as a normal cocompact
sub-group (Definition1.4).
Lemma 3.1. An arbitrary intersection of almost algebraic subgroups is again almost algebraic.
More precisely, let.Li/i 2I be a collection of almost algebraic subgroups and Hi algebraic subgroups such thatHi D Hi.k/is normal and cocompact inLi.
Then one can find a finite subsetI0such that, definingHDTi 2I0Hi, we have
thatHDT
i 2IHiandH.k/is normal and cocompact inTi 2I Li.
Proof. LetL DT Li andH DT Hi which coincides with.Ti 2IHi/.k/. Then
it is straightforward to check thatH C L. Thanks to the Noetherian property of
G, there exists a finite subsetI0 I such thatT
iHi coincides with T
i 2I0Hi.
Let L be the Zariski closure of Land Li the one of Li. The diagonal
im-age ofL.k/ inQ
i 2I0Li.k/=Hi is locally closed by Proposition2.2 and it is a
group. Thus it is actually closed. Moreover it is homeomorphic to L.k/=H.
To conclude, it suffices to observe that L=H is closed in L.k/=H and lies in .L.k/=H /T Q
i 2I0Li=Hi
which is compact.
Remark 3.2. Actually the proof of this lemma shows that any almost algebraic subgroupLhas a minimal subgroup among all cocompact normal subgroupsN
which can be written N D N.k/ for some algebraic subgroup N G. This group is actually the intersection of all such subgroups and it is invariant under the normalizerNG.L/ofLinG.
Lemma 3.3. LetH; Lbe closed subgroups ofGsuch thatH is almost algebraic,
H C LandL=H is compact. ThenLis almost algebraic.
Proof. There is a algebraic subgroup N of G such thatN D N.k/is normal and
cocompact inH. Moreover thanks to Remark3.2,N may be chosen to be invariant
underNG.H /and thusN is cocompact and normal inL.
3.2. Almost algebraicity of stabilizers of probability measures. LetV be a
Polish space endowed with a continuousG-action. Recall that the actionG Õ V
is called almost algebraic if the stabilizers are almost algebraic subgroups ofG
Remark 3.4. For a continuous action ofG on a Polish spaceV, the action is almost algebraic if and only if the stabilizers are almost algebraic and for every
v 2 V and any sequencegn 2 G, gnv ! v implies gn ! e in G=StabG.v/.
This equivalent definition is much easier to check, and we will allow ourselves to use it freely in the sequel. The two definitions are indeed equivalent by Effros’ Theorem2.1.
Example 3.5. LetIbe ak-algebraic group and W G ! Iak-morphism. LetL
be an almost algebraic group inI D I.k/. Then the action ofGonI =Lis almost
algebraic. This fact is proved after Lemma3.7.
Lemma 3.6. LetK be a compact group acting continuously on a T0-spaceX.
Then the orbit spaceKnXisT0 as well.
Proof. Continuity of the action means that the action mapK X ! K Xwhich
associates.k; kx/to.k; x/is a homeomorphism. Compactness ofKimplies that
the projection.k; x/ 7! xfromK X toXis closed. Composing the two yields
closedness of the map.k; x/ 7! kx. This implies that ifF X is closed, then KF is again closed.
Letx; y 2 X in differentK-orbits. Let us considerY D Kx [ Ky with the
induced topology. This is a compactT0-space. Now, consider the set of closed
non-empty subspaces ofY with the order given by inclusion. By compactness
any decreasing chain has a non-empty intersection and thus Zorn’s Lemma implies there are minimal elements, that are points sinceY isT0. ThusY has at least a
closed point.
Without loss of generality we may and shall assume that¹xº is closed inY.
This means that there exists a closed subsetF ofX such thatF \ Y D ¹xº. In particularF \ Ky D ;, and thereforeKy \ KF D ;. Finally,KF is a closed K-invariant set separatingKxfromKy.
Lemma 3.7. LetJ be a topological group acting continuously on a topological spaceX. IfN is a closed normal subgroup ofJ, the induced action ofJ =N on
N nX is continuous and the orbits spacesJ nX and.J =N /n.N nX /are homeo-morphic.
Proof. The map .g; x/ 7! Ngx from J X to N nX is continuous and goes
through the quotient spaceJ =N N nXwhich is the orbit space ofN N acting
diagonally onJ X. Thus,.gN; N x/ 7! Ngxis continuous, that is the action of J =N onN nX is continuous.
By the universal property of the topological quotient, the continuous map
x 7! .J =N /N x fromX to.J =N /n.N nX / induces a continuous mapJ nX ! .J =N /n.N nX /. Conversely, the continuous map N nX ! J nX induces also a
continuous map .J =N /n.N nX / ! J nX which is the inverse of the previous
Proof of Example3.5. Since 1.L/and its conjugates are almost algebraic inG,
it is clear that the stabilizers are almost algebraic. So we are left to prove that the topology onGnI =L is T0. Let H be a cocompact normal subgroup in Lwith H D H.k/for somek-algebraic subgroupHofI. By Lemma3.7the orbit space GnI =Lis homeomorphic to the space of orbits of the action of G .L=H /on
I =H. Note that the action of G on I =H I=H.k/ has locally closed orbits (and thereforeGnI =H isT0) by Proposition 2.2, as the action of GonI=His k-algebraic. Now the T0 property ofGnI =L follows from Lemma 3.6 for the
compact groupL=H acting continuously on theT0-spaceGnI =H.
Lemma 3.8. Let J be a countable set, .Li/i 2J a family of almost algebraic
subgroups ofG. Then the diagonal action ofGonQ
i 2JG=Liis almost algebraic.
Proof. Stabilizers of points in Q
i 2JG=Li are intersections of almost algebraic
subgroups ofG. Hence by Lemma3.1they are almost algebraic. So we just have to prove thatGn Q
i 2JG=Li
isT0.
Fori 2 J, letHi be an algebraic subgroup of G such thatHi D Hi.k/is a
cocompact normal subgroup ofLi. ConsiderV D Qi 2JG=Hi. We first prove
that the topology onGnV isT0, by proving that orbit maps are homeomorphisms
(Theorem 2.1). Let.hiHi/i 2J be an element of V and .gn/ be a sequence of
elements ofGsuch thatgn .hiHi/converges to.hiHi/inV.
Let H D T
i 2J hiHihi1 D Stab..hiHi/i 2J/. We have to prove that gn
converges toe inG=H (see Remark3.4). By Noetherianity, there exists a finite J0 J such that H D Ti 2J0hiHih
1
i . Set V0 D Q
i 2J0G=Hi. We see that,
inV0, we have thatgn:.hiHi/i 2J0 converges to.hiHi/i 2J0. By Proposition2.2,
it follows thatgnconverges to the identity inG=H.
Now letKbe the compact groupQ
i 2J Li=Hi. The groupKacts also
contin-uously onV via the formula.liHi/ .giHi/ D .gili 1Hi/and this action
com-mutes with the action ofG. Thus we can apply Lemma3.6toK acting onGnV
and get that the space of orbits for theG-action onV =K ' Q
i 2J Gi=Li is T0,
as desired.
Our main goal in this subsection is proving the following theorem, which is an essential part of our main theorem, Theorem1.7.
Theorem 3.9. LetV be a Polish space with an almost algebraic action ofG. Then stabilizers of probability measures onV are almost algebraic subgroups ofG.
We first restate and prove Proposition1.9, discussed in the introduction. Proposition 3.10. Fix a closed subgroupL < G. Then there exists ak-subgroup
H0 < G which is normalized by L such that L has a precompact image in
the Polish group.NG.H0/=H0/.k/ and such that for everyk-G-varietyV, any L-invariant finite measure on V.k/ is supported on the subvariety of H0-fixed
Proof. Replacing G by the Zariski closure ofL, we assume thatLis Zariski-dense inGand consider the collection
¹H < G j His ak-algebraic subgroup; Prob.G=H.k//L¤ ;º:
By the Noetherian property of G there exists a minimal element H0 in this
collection. We let0be a correspondingL-invariant measure onG=H0.k/.
We first claim thatH0 is normal inG. Assuming not, we letN ΠGbe the
normalizer ofH0and consider the set
U D ¹.xH0; yH0/ j y 1x … Nº G=H0 G=H0:
This set is a non-empty Zariski-open set which is invariant under the diagonal
G-action, as its complement is the preimage of the diagonal under the natural
mapG=H0 G=H0 ! G=N G=N. Since the support of0 0 inG=H0 G=H0 is invariant under L L which is Zariski-dense in G G we get that .0 0/.U .k// ¤ 0. It follows from Corollary2.14that there existu 2 U .k/
and anL-invariant finite measure onG=StabG.u/ .G=StabG.u//.k/. By the
definition ofUwe get a contradiction to the minimality ofH0, as point stabilizers
inU are properly contained in conjugates ofH0. This proves thatH0is normal
inG.
Next we letV be ak-G-variety andbe anL-invariant measure onV.k/. We
argue to show thatis supported onVH0 \ V .k/. Indeed, assume not. Let V0
be the Zariski-closure ofV.k/ \ VH0, andV00D V V0. Then we see thatV0is
defined overk[5, AG, 14.4]. Furthermore, H0acts on V0trivially, so that we have
V0
.k/ D V .k/\VH0. Hence by assumption we get that.V00.k// > 0. Replacing
V byV00and restricting and normalizing the measure, we may and shall assume
thatVH0\ V .k/ D ;.
We consider the varietyG=H0 V as ak-G-variety. The measure0 is
anL-invariant measure on.G=H0 V /.k/. It follows from Corollary2.14that
there existsu 2 .G=H0 V /.k/ and anL-invariant measure on G=StabG.u/.
By Proposition 2.2 there exist a k-algebraic subgroup H < G with H D H.k/ D StabG.u/and an orbit map G=H ! Guinducing a homeomorphism G=H ! G=StabG.u/. Thus we obtain anL-invariant probability measure on G=H.k/. Now, H is contained in some conjugate gH0g 1, for someg 2 G.
Hence we get thatg 1Hg < H0is such thatG=g 1Hghas anL-invariant
proba-bility measure. By minimality, this implies thatg 1Hg D H0, hence by normality
of H0,H D H0. Thereforeubelongs toV.k/ \ V0H, which was assumed to be
empty. Hence we get a contradiction. This proves thatis supported onVH0.
We set S D .G=H0/.k/ and let T be the closure of the image of Lin S.
We are left to show thatT is compact. S is a Polish group andT is a closed
subgroup. The quotient topology onT nS is Hausdorff, and in particularT0. The
measure0 is anL-invariant finite measure onS, hence it is alsoT-invariant.
onS which is supported on a uniqueT-coset,T s. The measure.Rs/1, given
by pushing1 by the right translation by s 1 is then a T-invariant probability
measure onT. It is well-known result due to A. Weil (see [15] where the result is attributed to Ulam) that a Polish group that admits an invariant measure class is locally compact, and a locally compact group that admits an invariant probability measure is compact. ThusT is indeed compact.
Corollary 3.11. Fix ak-G-algebraic varietyV, and set V D V .k/. Let 2
Prob.V /. ThenStab./is almost algebraic.
Proof. LetL DStab./. We may and shall assumeLto be Zariski-dense inG, and we can findH0as in Proposition1.9. We know thatis supported on the set
ofVH0 thus H
0 D H0.k/ < L. SinceG=H0 is acting onVH0\ V .k/and the
stabilizer ofis closed inG=H0, we conclude thatLhas a closed image. We know
that the image ofLis precompact, thus it is actually compact, and we conclude
thatLis almost algebraic.
Lemma 3.12. Let L < G be an almost algebraic group, with H D H.k/
a normal cocompact algebraic subgroup of L. Then there is a G-equivariant continuous map WProb.G=L/ ! Prob.G=H /. Furthermore, we have, for every
2Prob.G=L/,Stab./ DStab. .//.
Proof. Letbe a Haar probability measure onL=H. For a continuous bounded
functionf onG=H letfNbe the continuous bounded function onG=Ldefined by N
f .gL/ DR
L=Hf .gh/d.h/and finally ./.f / D . Nf /.
Then it is clear that is equivariant, and we deduce that Stab./ Stab. .//. In the other direction, we note that ifW G=H ! G=Lis the projection, we have
. .// D . Hence the other inclusion is also clear.
To check the continuity, letn ! 2 Prob.G=L/, and takef a continuous
bounded function onG=H. Then .n/.f / D n. Nf / ! . Nf / D ./.f /.
Hence .n/converges to ./.
Proof of Theorem3.9. Choose 2 Prob.V /and denote L D StabG./, H D
FixG.supp.//. Set U D GnV, and let D p, where pW V ! U is the
projection. Note thatpis a Polish fibration. By Theorem2.11,Lis equal to the
stabilizer of an element f 2 Lp0
.U;ProbU.V //. By Lemma2.13 there exists
a -full measure set U1 U such that L D Tu2U1Stab.f .u//. For a fixed
u 2 U1,f .u/ is a measure on aG-orbit in V which we identify withG=L0 for
some almost algebraic subgroupL0
< G. LetH0
< Gbe ak-algebraic subgroup
such thatH0 D H0
.k/is a cocompact normal subgroup ofL0. By Lemma3.12,
Stab.f .u//is also the stabilizer of a probability measure onG=H0 G=H0 .k/.
By Corollary3.11, it follows that Stab.f .u//is almost algebraic. We conclude that
3.3. Separating orbits in the space of probability measures. In this subsec-tion, we prove the following theorem.
Theorem 3.13. LetL < Gbe an almost algebraic subgroup. Then the action of
GonProb.G=L/is almost algebraic.
The proof of Theorem3.13consists in several steps, proving particular cases of the theorem, each of them using the previous one. First we start with the case when
Lis trivial (Lemma3.14). Then we treat the case whenLis a normal algebraic
subgroup ofG(Lemma3.15). The main step is then to deduce the theorem when Lis any algebraic subgroup ofG (Proposition3.20), before concluding with the general case.
Lemma 3.14. TheG-action onProb.G/is almost algebraic.
Proof. The regular action ofGon itself is proper, so by Lemma2.7it follows that
the action ofGon Prob.G/is proper. Any proper action is almost algebraic.
Lemma 3.15. LetH< Gbe a normalk-algebraic subgroup. Then theG-action
onProb..G=H/.k//is almost algebraic.
Proof. DenotingID G=HandI D I.k/, we know that theI-action on Prob.I /
is almost algebraic (Lemma3.14). SinceG=H is a subgroup ofI,Gstabilizes each I-orbit. It is thus enough to show that G acts almost algebraically on each I-orbit. We know that such an orbit is of the formI =LwhereLis almost algebraic
(Theorem3.9), so this follows from Example3.5.
An essential technical tool for proving Theorem3.13and Theorem1.7is given by the following proposition.
Proposition 3.16. LetV be a Polish space, with a continuous action ofG. Assume that
the quotient topology onGnV isT0, and
for anyv 2 V, the action ofGonProb.G:v/is almost algebraic. Then the quotient topology onGnProb.V /isT0.
The proposition will directly follow from the following lemma.
Lemma 3.17. LetpW V ! U be a Polish fibration with an action ofG, and let
be a probability measure onU. Assume that the action ofGonU is trivial and that the action ofGonProb.p 1.u//is almost algebraic for almost everyu 2 U. LetP D ¹ 2Prob.V / j p D º. Then the topology onGnP isT0.
This proof is similar to the proof presented in [24, Proof of Proposition 3.3.1]; see also [1, Lemma 6.7].
Proof. The set P is Polish, as a closed subset of Prob.V /. By Theorem 2.1
we need to show that the orbit maps are homeomorphisms. By Theorem 2.11,
P is equivariantly homeomorphic to Lp0
.U;ProbU.V //.
Fixingf 2Lp0
.U;ProbU.V //and lettinggn 2 Gbe such thatgnf ! f, we
will show thatgn converges to the identity in G=Stab.f / by proving that every
subsequence of.gn/ has a sub-subsequence which converges to the identity in G=Stab.f /. Doing so, we are free to replace.gn/by any subsequence. Relying
on Lemma2.10, we replace.gn/by a subsequence such thatgnf .u/ ! f .u/for
everyuin some-full subsetU0 U. LetU1 U0be a full measure subset such
that the action ofGonp 1.u/is almost algebraic for everyu 2 U1.
Letu 2 U1. By definition, we know thatf .u/ 2 Prob.p 1.u//and that the
action ofG on Prob.p 1.u//is almost algebraic. By Proposition2.2, the orbit
mapG=Stab.f .u// ! Gf .u/is a homeomorphism thusgnf .u/ ! f .u/implies
thatgnconverges to the identity inG=Stab.f .u//. By Lemma2.13, there is also
a full measure subsetU2, that we may and do assume to be contained inU1, such
that
Stab.f / D \ u2U2
Stab.f .u//
and sinceG is second countable, one can findU3 countable inU2such that
Stab.f / D \ u2U3
Stab.f .u//:
By assumption, for every u 2 U3, the group Stab.f .u// is almost algebraic.
Hence by Lemma3.8, the action ofGonQ
u2U3G=Stab.f .u//is almost algebraic.
In particular, we see thatgnconverges toeinG=Stab.f /.
Proof of Proposition3.16. Let U D GnV and pW V ! U be the projection. Consider theG-invariant continuous mappWProb.V / ! Prob.U /. Clearly the
fibers ofpare closed andG-invariant, so it is enough to prove that for a given 2Prob.U /, the quotient spaceGnp1.¹º/has aT0-topology. This is precisely
Lemma3.17.
LetW V ! V0 be a continuous
G-map between Polish spaces, 2Prob.V /
and D . Then has a unique decomposition D c C d where c
andd are the continuous and discrete parts of. Moreoverd can be written P
2ƒ P
f 2Fıf, where
andF D ¹u 2 V0 j .¹uº/ D º. Defining to be the restriction of to 1.F
/andc D P2ƒ, we have a unique decomposition D cC P
2ƒ, where.c/is non-atomic and each./is a finitely supported,
uniform measure of the formP
f 2Fıf.
Lemma 3.18. LetW V ! V0
be a continuousG-map between Polish spaces and
2 Prob.V /. Using the above decomposition, we have Stab./ D Stab.c/ \ .T
Stab.//. Ifgn ! thengnc! c and for each 2 ƒ,gn ! .
Proof. The statement about Stab./is straightforward from the uniqueness of the
decomposition of. Let.gn/be a sequence such thatgn ! . Once again, we
use a sub-subsequence argument: we prove that any subsequence of.gn/contains
a sub-subsequence such thatgn! for every. Hence we start by replacing .gn/by an arbitrary subsequence.
Observe thatgn ! impliesgn ! becauseWProb.V / ! Prob.V0/
is continuous. LetK0be a compact metrizable space in which
V0is continuously
embedded as aGı-subset (see [14, Theorem 4.14]). Then Prob.V0/ embeds as
a Gı-subset in Prob.K0/ as well [14, Proof of Theorem 17.23]. We begin with
the following observation. Assume n is a sequence of probability measures
converging to 2 Prob.V0 /andn decomposes asıun C 0 n withun 2 V0 and 0 n 2Prob.V 0/. Up to extractionu
nconverges to somek 2 K0and thus.¹kº/ > 0
which implies thatk 2 V0.
Let 1 be the maximum of ƒ. The above observation implies that up to
extraction we may assume that for any f 2 F1, gnf converges to some
l.f / 2 V0. Since
gn ! , we have thatl.f / 2 F1 thus gn1 converges to
1, where1 D .1/. An induction onƒ(countable and well ordered with
the reverse order ofR) shows that (after extraction)gn ! for any 2 ƒ.
Once again, we embedV in some compact metrizable spaceK. Fix 2 ƒ
and let0be an adherent point of
.gn/in Prob.K/. AsisG-equivariant, we
have thatgn D gn D gn which converges to. Hence0 D .
Furthermore, we also see that0is supported on
1.F/, hence02Prob.V /.
The same argument proves that 0, which is an adherent point of
gn. /,
is supported onV n 1.F/.
As can be written uniquely as a sum of a measure supported on 1.F/
and a measure supported onV n 1.F/, we see, writing D . 0/ C 0 D . / C , that necessarily 0 D . This concludes the proof since
cD P2ƒ.
Lemma 3.19. Let H < G be a k-algebraic subgroup. Set N D NG.H/, H D H.k/ and N D N.k/. Let V D G=H, V0 D G=N
, V D V .k/and
V0 D V0
.k/. Consider the map W V ! V0
. Let F V0
be a finite set,
D 1=jF jP
f 2Fıf and 2 Prob.V / be a measure with D . Let .gi/
Proof. Denotem D jF j. We know that.V0/m=Sym.m/ is an algebraic variety,
hence by Proposition 2.2, every G-orbit in .V0/m=Sym.m/ is locally closed.
It follows in particular thatgi ! einG=Stab.F /.
Again, it is enough to show that every subsequence of .gi/ contains a
sub-sequence which tends toe modulo Stab./. We start by extracting an arbitrary subsequence of.gi/.
Let us numberf1; f2; : : : ; fmthe elements ofF and denoteF0D .f1; : : : ; fm/ 2 .V0/m. Since g
i converges to e in G=Stab.F /, it follows that, passing to a
subsequence, there exists 2 Sym.m/ such that giF0 tends to .F0/ D .f.1/; : : : ; f.m//. This meansGF0 G .F0/and thusGF0 G .F0/ Gn.F0/ D GF0for somen 2 N. In particularGF0 D G .F0/and since orbits
are locally closed we have thatGF0D G .F0 /.
This shows that there existsg 2 Stab.F /such thatgF0 D .F0
/. Hence we
havegiF0 ! gF0, and by almost algebraicity of the action on .V0/m it follows
thatgi tends togmodulo Stab.F0/ DTf 2FStab.f /.
Let us fix some notations. For f 2 F we denote by f the restriction of to 1.¹f º/ and fix f 2N G such that fNN D f and denote by Hf G
the conjugate of H byfN. Observe thatfNNfN 1 D Stab
G.f /, Hf C StabG.f /
and 1.¹f º/ ' StabG.f /=Hf where WG=H ! G=N is the projection and
StabG.f / is the stabilizer off under the action of G on G=N. We also denote
0 f D g 1 .f / and g0i D g 1gi. Since g 0 i ! e 2 G= T f 2F Stab.f / there
existsni 2Tf 2F Stab.f /such thatg 0
ini1converges toe(inG). We observe that nif D ni.gi0/ 1gi0f. Asgi0f tends tof0 andni.gi0/ 1tends toe, we have that nif converges tof0 .
Those measures are supported on 1.¹f º/ ' StabG.f /=Hf .k/. By
Lemma 3.15, Stab.f / acts almost algebraically on Prob..StabG.f /=Hf/.k//.
So we have thatni tends to somenin Stab.f /=Stab.f/.
We conclude thatg0 i D g
0
ini1ni tends toninG=Stab.f/. Arguing similarly
for everyf, it follows that gi tends to gn in G=Tf 2FStab.f/. Hence .gi/
converges also in G=Stab./, since T
f 2FStab.f/ Stab./. Let h be the
limit point of.gi/modulo Stab./. Then we have that giconverges tohby
continuity of the action. Henceh 2Stab./, meaning thath D emodulo Stab./.
Proposition 3.20. LetH < G be ak-algebraic subgroup and setH D H.k/.
Then the action ofGonProb.G=H /is almost algebraic.
Proof. Assume the proposition fails for an algebraic subgroup H. We also
as-sume, as we may, thatHis minimal in the collection of k-subgroup ofG with
the property that theG-action on Prob.G=H /is not almost algebraic. By
Theo-rem3.9,Gacts on Prob.G=H /with almost algebraic stabilizers. Hence we have
to show that for every measure 2Prob.G=H /and sequencegnwithgn !
thengn tends toe inG=Stab./(Remark 3.4). We fix such a measureand a
sequencegn. We will achieve a contradiction by showing thatgn does tend toe
inG=Stab./.
We setN D NG.H/,N D N.k/, V D G=H, V0 D G=N,V D V .k/and V0 D V0
.k/. We consider the natural inclusion G=H V and view as a
measure onV. We consider the projection mapW V ! V0 and set
D .
We use the notation introduced in the discussion before Lemma3.18. The lemma gives: Stab./ DStab.c/ \ .T2ƒStab.//whereƒis a countable subset of Œ0; 1,gnc ! c and for each 2 ƒ,gn ! . By Lemma3.19, for each 2 ƒ,gi ! e 2 G=Stab./. Assume given also thatgn ! e 2 G=Stab.c/.
Since by Theorem3.9the groups Stab./and Stab.c/are almost algebraic, we
will get by Lemma3.8that the action ofGonG=Stab.c/ QG=Stab./is
almost algebraic. Hence,
gn ! e 2 G= Stab.c/ \ \ Stab./ D G=Stab./;
achieving our desired contradiction. We are thus left to show that indeed we have
gn ! e 2 G=Stab.c/.
For the rest of the proof we will assume as we may D c, that is 2Prob.V0/
is atom-free. We consider the measure 2Prob.V V /and the subset
U D ¹.xH; yH/ j y 1x … Nº G=H G=H D V V
defined and discussed in the proof of Proposition1.9. We setU D U .k/. Note
that the diagonal inV0 V0 is
-null asis atom-free, thusU is -full.
We view as we may as a probability measure onU.
We now consider the G-action onU and claim that theG-orbits are locally
closed and for everyu 2 U,G acts almost algebraically on Prob.Gu/. The fact
that theG-orbits are locally closed follows from Proposition2.2, as U is a k
-subvariety ofV. Fix now a pointu D .xH; yH/ 2 U for somex; y 2 G, and
consider theG-action on Prob.Gu/. By the definition ofU,H\ Hy 1xΠH, thus
by the minimality ofHtheG-action on Prob.G=H \ Hy 1x/ 'Prob.G=Hx\ Hy/is almost algebraic. Since by Proposition2.2 G=Hx\ Hy is equivariantly
homeomorphic toGu we conclude that indeed, G acts almost algebraically on
By Proposition 3.16, G acts on Prob.U /almost algebraically. Hence Effros’ Theorem2.1implies that gn ! e inG=Stab. /as gn. / ! .
Observing that Stab. / DStab./, the proof is complete.
Proof of Theorem3.13. By Theorem 3.9 we know that the point stabilizers in Prob.G=L/ are almost algebraic. We are left to show that for every 2
Prob.G=L/, for every sequencegn 2 G satisfyinggn ! we havegn ! e
modulo Stab./(see Remark3.4). Fix 2Prob.G=L/and a sequencegn2 G
sat-isfyinggn ! . LetH< Gbe ak-algebraic subgroup withH D H.k/normal
and cocompact inL, and recall that by Lemma3.12we can find aG-equivariant continuous map WProb.G=L/ ! Prob.G=H /such that Stab./ D Stab. .//. We get thatgn ./ ! ./. By Proposition3.20, theG-action on Prob.G=H /
is almost algebraic, thusgn ! emodulo Stab. .//. This finishes the proof, as
Stab./ DStab. .//.
3.4. Proof of Theorem1.7. For the convenience of the reader we restate Theo-rem1.7.
Theorem 3.21. If the action ofG onV is almost algebraic then the action ofG
onProb.V /is almost algebraic as well.
Proof. By Theorem 3.9, we know that the G-stabilizers in Prob.V /are almost
algebraic. We need to show that the quotient topology on GnProb.V / is T0.
By Proposition 3.16, it is enough to check that the quotient topology onGnV
isT0, which is guaranteed by the assumption that the action ofGonV is almost
algebraic, and, as we will see, that for anyv 2 V, the action ofG on Prob.Gv/
is almost algebraic. We note that by Effros theorem (Theorem 2.1), the orbit
Gv is equivariantly homeomorphic to the coset space G=StabG.v/, and thus
Prob.Gv/ 'Prob.G=StabG.v//. Since StabG.v/is an almost algebraic subgroup
ofG, the fact that theG-action on Prob.Gv/is almost algebraic now follows from
Theorem3.13.
4. On bounded subgroups
In this section, we essentially retain the setup1.1&1.3: we fix a complete.k; j j/
valued field and ak-algebraic group G. Nevertheless there is no need for us to
assume that.k; j j/is separable, so we will refrain from doing so.
Definition 4.1. A subset ofk is called bounded if its image underj jis bounded inR. For ak-variety V, a subset ofV.k/is called bounded if its image by any
Remark 4.2. Note that the collection of bounded sets on a k-variety forms a bornology.
Remark 4.3. For ak-variety V it is clear that a subset ofV.k/is bounded if and
only if its intersection with everyk-affine open set is bounded, so in what follows
we will lose nothing by considering exclusivelyk-affine varieties. We will do so.
Remark 4.4. Note that if .k0 ; j j0
/ is a field extension of k endowed with an
absolute value extension ofjjandVis ak-variety, we may regardV.k/as a subset
ofV.k0
/and, as one easily checks, a subset ofV.k/isk-bounded if and only if
it isk0-bounded. Thus it causes no loss of generality assuming
k is algebraically
closed sincekOis so. Nevertheless, we will not assume that.
It is clear that everyk-regular morphism ofk-varieties is a bounded map in the sense that the image of a bounded set is bounded. For ak-closed immersion
ofk-varieties f WU ! Valso the converse is true: a subset of U.k/is bounded
if and only if its image is bounded, asf
W kŒV ! kŒU is surjective. This is a
special case of the following lemma.
Lemma 4.5. For a finitek-morphismf WU !V a subset ofU.k/is bounded if
and only if its image is bounded.
Proof. Assume there exists an unbounded setLinU.k/withf .L/being bounded
inV.k/. Then we could findp 2 kŒU and a sequenceun2 Lwithjp.un/j ! 1.
The functionp is integral overf
kŒV so there existq1; : : : qm 2 fkŒV with pmCPm i D1qipm i D 0. Thus, 1 D ˇ ˇ ˇ ˇ ˇ m X i D1 qi.un/ pi.u n/ ˇ ˇ ˇ ˇ ˇ m X i D1 jqi.un/j jpi.u n/j ! 0;
as the sequencesqi.un/are uniformly bounded. This is a contradiction.
Recall that a seminorm on ak-vector spaceEis a functionk kW E ! Œ0; 1/
satisfying
(1) k˛vk D j˛jkvk, for˛ 2 k,v 2 Eand (2) ku C vk kuk C kvk, foru; v 2 E.
A seminorm onEis a norm if furthermore we have
(3) kvk D 0 () v D 0, forv 2 E.
Two norms on a vector space,k k; k k0
, are called equivalent if there exists someC 1such that
C 1k k k k0
C k k:
It is a general fact that any linear map between two Hausdorff topological
.k; j j/-vector spaces of finite dimensions is continuous [7, I, §2,3 Corollary 2]
