arXiv:1612.00537v1 [math.DS] 2 Dec 2016

CHARACTERISTIC FREE MEASURE RIGIDITY FOR THE
ACTION OF SOLVABLE GROUPS ON HOMOGENEOUS SPACES
AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Abstract. We classify measures on a homogeneous space which are invariant
under a certain solvable subgroup and ergodic under its unipotent radical. Our
treatment is independent of characteristic. As a result we get the first measure
classification for the action of semisimple subgroups without any characteristic
restriction.

1. introduction
Ratner’s celebrated work, [R90b, R92, R95] see also [MT94], classifies all probability
measures invariant and ergodic under a one parameter unipotent subgroup in the
setting of homogeneous spaces over local fields of characteristic zero. In positive
characteristic setting, however, classification theorems in this generality are not yet
available.
Roughly speaking, the main technical difficulty arises from the fact that the image
of a polynomial map over a field of positive characteristic can lie in a proper subfield,
and hence the image may be small. This simple fact enters our study as follows. The
divergence of the orbits of two nearby points under a unipotent group is governed
by a certain polynomial like map, see §5. Now slow growth of polynomial maps and
Birkhoff ergodic theorem imply that µ is invariant under the image of a polynomial
like map, see §5.3. This way, we construct a higher dimensional subgroup which
leaves µ invariant. In the positive characteristic setting, however, this construction
only guarantees the dimension is increased by a (possibly non-constant) fraction at
each step. Thus, there is no a priori reason for this process to terminate.
In recent years there have been some partial results in this direction, e.g. [EG10,
M11, EM12]. In [EG10] measure rigidity for the action of semisimple groups was
proved under the assumption that the characteristic is large. Our account here, in
particular, removes the characteristic restriction from the main result in [EG10].
It is plausible that for large characteristics, one can carry out the proof in [MT94]
and get a measure rigidity for the action of unipotent groups.
2000 Mathematics Subject Classification. 37D40, 37A17, 22E40.
Key words and phrases.
Invariant measures, measure rigidity, local fields, positive
characteristic.
A.M. was supported in part by NSF Grants #1200388, #1500677 and #1128155, and Alfred
P. Sloan Research Fellowship.
A.S-G. was supported in part by NSF Grants #1001598, #1160472, #1303121 and #1602137,
and by Alfred P. Sloan Research Fellowship.
1

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AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Removing the characteristic restriction introduces serious technical difficulties. Resolving these difficulties requires not only a comprehensive use of the main ideas
and techniques from [MT94], but also an extensive use of the theory of algebraic
groups. In contrast, the proof in [EG10] relies on a simpler argument which goes
back to [E06].
In order to properly state our main results we need to make use of Weil’s restriction
of scalars for possibly non separable extensions. But before we set up the general
formulation, we start with an example
which is essentially as hard as the general

case. Consider G = SLd Fq ((t)) and let Γ ⊂ G be a discrete subgroup. Let

B = SU be the group of upper triangular matrices in a copy of SL2 Fq ((tq )) in G.
Our result classifies measures on G/Γ which areinvariant under B and ergodic for
U. Note, however, to get a copy of SL2 Fq ((tq )) in G it is inevitable to view G as
the set of Fq ((tq ))-points of
RFq ((t))/Fq ((tq )) (SLd )

where R denotes Weil’s restriction of scalars, see Definition 2.4. This is an indication of the subtlety involved in the setup.
Q
Let T be a finite set, and for any v ∈ T let kv be a local field; set kT = v∈T kv .
For any v ∈ T let Gv be a kv -algebraic group and let Gv = Gv (kv ). Define
Q
Q
G = v∈T Gv and G = v∈T Gv ,

and let Γ be a discrete subgroup of G.

Fix an element w ∈ T once and for all and let k ′ be a closed subfield of kw . Suppose
k ′′ /k ′ is a finite extension of k ′ , note that k ′′ is not necessarily assumed to be a
subfield of kw . Let H′ be a connected k ′′ -almost simple, k ′′ -group which is isotropic
over k ′′ , and put
H = Rk′′ /k′ (H′ ).
′
In particular, we have H(k ) = H′ (k ′′ ).
Fix a non central cocharacter λ of H, that is λ : Gm → H is a non central homomorphism defined over k ′ ; such homomorphism exists thanks to the fact that H′ is
k ′′ -isotropic, [CGP10, App. C]. Put S = λ(Gm ). Let s′ ∈ S(k ′ ) be an element which
′
generates an unbounded subgroup and set U = W+
H (s ), see §2 for the notation.
For the base change H ×k′ kw , we fix a kw -homomorphism ι : H ×k′ kw → Gw with
a solvable kernel, and put
H = ι(H(k ′ )), S = ι(S(k ′ )), and U = ι(U(k ′ )).
We recall the following definition. Let M be a locally compact second countable
group and suppose Λ is a discrete subgroup of M . Let µ be a probability measure
on M/Λ and let Σ = {g ∈ M : g∗ µ = µ}. We call µ homogeneous if µ is the
Σ-invariant probability measure on a closed orbit Σx for some x ∈ M/Λ.
Theorem 1.1 (Solvable case). Let µ be a probability measure on G/Γ which is
invariant under the action of SU and is U -ergodic. Then µ is a homogeneous
measure.
The following presents an interesting special case which is a direct corollary of
Theorem 1.1 based on the generalized Mautner phenomenon.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

3

Theorem 1.2 (Semisimple case). Let
• k ′ ⊂ kw be a closed subfield,
• let H0 be a connected, simply connected, k ′ -almost simple, k ′ -isotropic semisimple group,
• let ι : H0 ×k′ kw → Gw be a kw -homomorphism with a finite central kernel.
Set H0 = ι(H0 (k ′ )). If µ is an H0 -invariant ergodic measure on G/Γ, then µ is a
homogeneous measure.
Proof. Since H0 is k ′ -isotropic, by [Mar90b, Ch. I, Prop. 1.6.3] we have that there
exists a k ′ -homomorphism
j : SL2 → H0

with a finite central kernel. Let L = ι j(SL2 (k ′ )) and let B = SU be the image
under j ◦ ι of the group of upper triangular matrices in SL2 (k ′ ).

Since H0 is simply connected, k ′ -simple, and k ′ -isotropic, it follows from [Mar90b,
Ch.I, Thm. 2.3.1 and Ch. II, Lemma 3.3] that µ is S-ergodic. In particular, µ
is L-ergodic. Now by [Mar90b, Ch. II, Lemma 3.4] we get that µ is U -ergodic.
Therefore, the claim follows from Theorem 1.1.

For a homogeneous measure µ, the group Σ = {g : g∗ µ = µ} has an algebraic
description. The statement of this refinement involves definitions and notation
which will be developed later, see Theorem 6.9. When Γ is an arithmetic lattice,
we also give a connection between the arithmetic structure of Γ and the algebraic
description of Σ, see Theorem 7.1 and Theorem 7.2. This connection and Theorem 1.2 are crucial ingredients in [ELM16] where a measure rigidity for the action
of diagonalizable groups is proved.
Now we give a brief description of our approach and highlight the main difficulties.
We construct extra invariance for the measure using quasi-regular maps and utilize
entropy, similar to the strategy in [MT94]. As we described above, however, several
problems arise along the way. The source of the main technical difficulties is the
fact that topologically closed unipotent subgroups in positive characteristic are far
from being algebraic. This issue appears in our proof, as we need to control the
group generated by the image of a polynomial map.
In our setting, since µ is invariant under S, we need to understand the structure
of topologically closed unipotent subgroups which are normalized by S. In §4, we
show that such unipotent groups have an algebraic structure which is controlled
by the weights of the conjugation action of S. Our argument is based on a recent
treatment, [CGP10, App. B], of the fundamental work of Tits on unipotent groups
in positive characteristic.
As was mentioned above, §7 is devoted to providing a connection between the
arithmetic structure of Γ and the algebraic description of Σ. Our argument is
based on the structure theory of pseudo reductive groups developed in [CGP10], to
get a local model, and a descent argument from [P98], to get a global model.
Let us briefly recall the definition of a pseudo reductive group and how it appears
in our work. An l-group with no nontrivial, normal, unipotent l-subgroup is called
pseudo reductive. In the setting of homogeneous dynamics we have to work with

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AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

topologically closed subgroups of G. As we mentioned above these groups are not
necessarily algebraic, but under some conditions, they become algebraic after using
Weil’s restriction of scalars and viewing G as an algebraic group over a smaller field.
So groups of the form Rk/l (L) where k/l is a finite extension naturally appear in our
work. When L is a connected, reductive, l-group and k/l is inseparable, Rk/l (L) is
pseudo reductive but not reductive.
It is worth mentioning that, in our work, the structure theory of pseudo reductive
groups is used only to reveal a more precise local description of Σ, however, our
measure classification result, Theorem 1.1, does not rely on this structure theory.
We close the introduction with the following remark. It is desirable to classify all
SU -orbit closures. Since SU is amenable, such a topological rigidity would follow
from a combination of linearization techniques and a classification of SU -invariant,
ergodic probability measures. Here, however, we have made and used, in a crucial
way, the stronger assumption that µ is SU -invariant and U -ergodic. Indeed if
we assume µ is SU -invariant and only SU -ergodic, then µ is not necessarily U ergodic, even though, by Mautner phenomenon, it is S-ergodic. In particular, S
acts ergodically on the space of U -ergodic components of µ and the invariance group
of a U -ergodic component of µ is not necessarily normalized by S. So we can not
get a good algebraic description of this invariance group.
Acknowledgement: We would like to thank M. Einsiedler, A. Eskin, M. Larsen,
E. Lindenstrauss, G. Margulis, and G. Prasad for their interest in this project and
several insightful conversations.

2. Notation and preliminary lemmas
Q
Let T be a finite set and let kv be a local field for all v ∈ T ; define kT = v∈T kv
as in the introduction. We endow kT with the norm | · | = maxv∈S | · |v where | · |v
is a norm on kv for each v ∈ T .
2.1. kT -algebraic groups. A
Q variety
Q kT -algebraic group M (resp. kT -algebraic
M) is the formal product of v∈T Mv of kv -algebraic groups (resp. v∈T Mv of
kv -algebraic varieties). The usual notions from elementary algebraic geometry e.g.
regular maps, rational maps, rational point etc. are defined fiberwise. We will use
these notions without further remarks.
There are two topologies on M(kT ), the Hausdorff topology and the Zariski topology. We will make it clear when referring to the Zariski topology. Hence, if in a
topological statement we do not give reference to the particular topology used, then
the one which is being considered is the Hausdorff topology.
Let M be a kT -group. An element e 6= g ∈ M(kT ) is an element of class A if
g = (gv )v∈T is diagonalizable over kT , and for all v ∈ T the component gv has
eigenvalues which are integer powers of the uniformizer ̟v of kv .
Given a subset B ⊂ M(kT ) we let hBi denote the closed (in the Hausdorff topology)
group generated by B.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

5

2.2. Pseudo-parabolic subgroups. Let k be a local field of positive characteristic. Suppose M is a connected k-group, and let λ : Gm → M be a non central
homomorphism defined over k. Define −λ(a) = λ(a)−1 for all a ∈ k × . As in [Sp98,
§13.4] and [CGP10, Ch. 2 and App. C], we let PM (λ) denote the closed subgroup
of M formed by those elements x ∈ M such that the map λ(a)xλ(a)−1 extends to
a map from Ga into M.
Let W+
M (λ) be the normal subgroup of PM (λ), formed by x ∈ P(λ) such that
λ(a)xλ(a)−1 → e as a goes to 0. The centralizer of the image of λ is denoted
−
by ZM (λ). Similarly define W+
M (−λ) which we will denote by WM (λ).
The multiplicative group Gm acts on Lie(M) via λ, and the weights are integers.
The Lie algebras of ZM (λ) and W±
M (λ) may be identified with the weight subspaces
of this action corresponding to the zero, positive and negative weights. It is shown
in [CGP10, Ch. 2 and App. C], see also [Sp98, §13.4] and [BT78], that PM (λ),
±
ZM (λ) and W±
M (λ) are k-subgroups of M. Moreover, WM (λ) is a normal subgroup
of PM (λ) and the product map
ZM (λ) × W+
M (λ) → PM (λ) is a k-isomorphism of varieties.
A pseudo-parabolic k-subgroup of M is a group of the form PM (λ)Ru,k (M) for some
λ as above where Ru,k (M) denotes the maximal, connected, normal unipotent ksubgroup of M, [CGP10, Def. 2.2.1].
We also recall from [CGP10, Prop. 2.1.8(3)] that the product map
(2.1)

+
W−
M (λ) × ZM (λ) × WM (λ) → M is an open immersion of k-schemes.

It is worth mentioning that these results are generalization to arbitrary groups of
analogous and well known statements for reductive groups.
Let M = M(k), and put
±
(λ) = W±
WM
M (λ)(k), and ZM (λ) = ZM (λ)(k).
−
+
From (2.1) we conclude that WM
(λ)ZM (λ)WM
(λ) is a Zariski open dense subset
of M, which contains a neighborhood of identity with respect to the Hausdorff
topology.

For any λ as above define
(2.2)

+
−
M + (λ) := hWM
(λ), WM
(λ)i.

Lemma 2.1. The group M + (λ) is a normal subgroup of M for any λ as above.
Moreover, M + (λ) is unimodular.
−
+
Proof. Since D := WM
(λ)ZM (λ)WM
(λ) is a Zariski open dense subset of M, we
−
−
have DD = M. In particular, hWM (λ), ZM (λ), WM
(λ)i = M. This together with
±
the fact that WM (λ) is normalized by ZM (λ) implies the first claim.

We now show that M + (λ) is unimodular. The modular function is a (continuous)
homomorphism from M + (λ) into the multiplicative group R+ . However, M + (λ)
is generated by unipotent subgroups and since char(k) is positive, unipotent subgroups are torsion. Hence, the modular function is trivial; the claim follows.


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AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Given an element s ∈ M from class A. There is λ : Gm → M so that s = λ(a) for
some a ∈ k with |a| > 1. Then we define
(2.3)

±
WM
(s) := W ± (λ).

2.3. When working with algebraic groups over a non perfect field k, say with
characteristic p > 0, it is convenient to use the language of group schemes. With
the exceptions of §3 and §7 which are independent of the rest of the paper, we have
tried to avoid this language. However, one should note that certain natural objects,
e.g. kernel of a k-morphism, are not necessarily defined over the base field as linear
algebraic groups in the sense of [B91] or [Sp98]. They are so called k-closed1; a
notion which we now define.
Definition 2.2 ([B91], AG, §12.1). Let Ω be an algebraically closed field which
contains k and let M = Spec(Ω[x1 ,· · · , xn ]/I) be a variety. The variety M is called
k-closed if I = rad Ω[x1 , · · · , xn ]J where J is an ideal in k[x1 , · · · , xn ].

A subset of k n will be called k-closed if it may be realized as the k-points of a
k-closed subset of Gna .
If M ⊂ k n is a set which is the zero set of an ideal J in k[x1 , · · · , xn ], then M is a
k-closed set; this is how the k-closed sets arise in our study. We also note that if
−∞
M is k p -closed, then it is also k-closed.

If we start with a subset of the k-points of a variety and take the Zariski closure
of this set, then we get a variety defined over k, see [Sp98, Lemma 11.2.4(ii)]. The
next lemma is a more general formulation.
Lemma 2.3 (Cf. [CGP10], Lemma C.4.1). Let M be a scheme locally of finite
type over a field k. There exists a unique geometrically reduced, closed subscheme
M′ ⊂ M such that M′ (k ′ ) = M(k ′ ) for all separable field extensions k ′ /k. The
formation of M′ is functorial in M, and commutes with the formation of products
over k and separable extensions of the ground field. In particular, if M is a k-group
scheme (not necessarily smooth), then M′ is a smooth k-subgroup scheme.
Let us also recall the definition of the Weil’s restriction of scalars.
Definition 2.4. Let k be a field and k ′ a subfield of k such that k/k ′ is a finite extension, and let M be an affine k-variety. The Weil’s restriction of scalars Rk/k′ (M)
is the affine k ′ -scheme satisfying the following universal property
(2.4)

Rk/k′ (M)(B) = M(k ⊗k′ B)

for any k ′ -algebra B.
2.4. Ergodic measures on algebraic varieties. Let M be a kT -group and let
M = M(kT ). Suppose B ⊂ M is a group which is generated by one parameter
kT -split unipotent algebraic subgroups and by one dimensional kT -split tori. Let Λ
be a discrete subgroup of M and put π : M → M/Λ to be the natural projection.
1Let us remark that over a perfect field the notation of k-closed and that of a variety defined
over k coincide.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

7

Lemma 2.5 (Cf. [MT94], Proposition 3.2). Let µ be a B-invariant and ergodic
Borel probability measure on M/Λ. Suppose D is a kT -closed subset of M and put
D = D(kT ). If µ(π(D)) > 0, then there exists a connected kT -algebraic subgroup
E of M such that B ⊂ E := E(kT ), and a point g ∈ D such that Eg ⊂ D and
µ(π(Eg)) = 1. Moreover, E ∩ gΛg −1 is Zariski dense in E.
Proof. First note that thanks to Lemma 2.3 we may assume that D is the kT points of a kT -variety. Now since the Zariski topology is Noetherian we may and
will assume that D is minimal – in the sense that µ(π(D′ (kT ))) = 0 for all proper
kT -varieties D′ ⊂ D. This, in particular, implies that D is irreducible.
Put B ′ := {g ∈ B : gπ(D) = π(D)}. The minimality assumption, and applying
Lemma 2.3 again if necessary, imply that µ(π(D) \ gπ(D)) = 0 for all g 6∈ B ′ .
Since µ is a probability measure, we get that B ′ has finite index in B. Let now
g ∈ B ′ . Then gD ⊂ DΛ and since Λ is countable, we get that there exists some
λ ∈ Λ so that µ(π(gD ∩Dλ)) > 0. Using our minimality assumption and Lemma 2.3
one more time, we get that gD ⊂ Dλ; therefore, gD = Dλ. Since Λ is countable,
we get that B ′ /B ′′ is countable where B ′′ := {g ∈ B ′ : gD = D}.
All together we have shown that B ′′ has countable index in B. Recall that B = hBi i
where each Bi = Bi (kT ) and Bi is either a one dimensional kT -split unipotent
algebraic subgroup, or a one dimensional kT -split tori; in particular, Bi is connected
for all i. This implies Bi ∩ B ′′ is Zariski dense in Bi . Hence Bi D = D for all i,
which implies BD = D.
Put Λ0 = {λ ∈ Λ : Dλ = D}, and Y = D \ ∪Λ\Λ0 Dλ. Minimality of D implies that
µ(π(Y )) = µ(π(D)). Moreover, BY Λ0 = Y and the natural map Y /Λ0 → M/Λ is
injective. We thus get a B-invariant ergodic probability measure µ0 on Y /Λ0 .
Let F be the Zariski closure of Λ0 in M. Then F is a kT -group, see [Sp98, Lemma
11.2.4(ii)], and DF = D. The push forward of µ0 gives a B-ergodic invariant measure
on D/F ⊂ (D/F)(kT ) where F = F(kT ). Now by [Shl99, Thm. 1.1 and Thm. 3.6],
this measure is the Dirac mass at one point. That is there is some z ∈ D so that
µ(π(zF )) = 1. Since zF ⊂ D, our minimality assumption implies zF = D; in
particular, we have F is connected. Therefore, g = z and E = gFg −1 satisfy the
claims in the lemma.

Let the notation be as in the beginning of §2.4. We will say a Borel probability
measure µ on M/Λ is Zariski dense if there is no proper kT -closed subset M of M
such that µ(π(M )) > 0, where M = M(kT ). Two kT -subvarieties L1 and L2 of M
are said to be transverse at x if they both are smooth at x and
Tx (L1 ) ⊕ Tx (L2 ) = Tx (M),
where Tx (•) denotes the tangent space of • at x. Thanks to Lemma 2.3, we also
have the following, see [MT94, Prop. 3.3].
Lemma 2.6. Suppose B = B(kT ) for a kT -subgroup B of M. Assume that µ is
a Zariski dense B-invariant Borel probability measure on M/Λ. Suppose L is a
connected kT -algebraic subvariety of M containing e which is transverse to B at e.
Let D ( L be a kT -closed subset of L containing e. Then, there exists a constant
0 < ε < 1 so that the following holds. If Ω ⊂ M/Λ is a measurable set with

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AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

µ(Ω) > 1−ε, then one can find a sequence {gn } of elements in M with the following
properties
(i) {gn } converges to e,
(ii) gn Ω ∩ Ω 6= ∅, and
(iii) {gn } ⊂ L(kT ) \ D(kT ).
2.5. Homogeneous measures. Let M be a locally compact second countable
group and let Λ be a discrete subgroup of M . Suppose µ is a Borel probability
measure on M/Λ. Let Σ = {g ∈ M : g∗ µ = µ}. The measure µ is called homogeneous if there exists x ∈ M/Λ such that Σx is closed and µ is the Σ-invariant
probability measure on Σx.
Lemma 2.7 (Cf. [MT94], Lemma 10.1). Let M be a locally compact second countable group and Λ a discrete subgroup of M. Suppose B is a normal and unimodular
subgroup of M. Then any B-invariant, ergodic measure on M/Λ, is homogeneous.
Moreover, Σ = BΛ.
2.6. Modulus of conjugation. Suppose M is a kT -algebraic group. Then M =
M(kT ) is a locally compact group. Let a ∈ M and suppose B ⊂ M is a closed,
with respect to the Hausdorff topology, subgroup which is normalized by a. We let
α(a, B) denote the modulus of the conjugation action of a on B, i.e. if Y ⊂ B is a
measurable set
θ(aY a−1 ) = α(a, B)θ(Y )
where θ denotes a Haar measure on B.
Note that α(·, M ) is the modular function of M. In particular, if a ∈ [M, M ], the
commutator subgroup of M, then α(a, M ) = 1.
3. Structure of pseudo reductive groups
In this section we will record a corollary of the main results in [CGP10]; this section
is required only for our study in §7.
We begin by fixing some notation. Let k be a local field. Throughout this section we
let M be a connected, simply connected, semisimple group defined over k. Moreover,
we assume that either char(k) > 3 or if char(k) = 2, 3, then all of the absolutely
almost simple factors of M are of type A.
Let B ⊂ M(k) be a Zariski dense subgroup of M. Let l ⊂ k be a closed subfield and
put
The irreducible component of the
F :=
identity in the Zariski closure of B in Rk/l (M)
We will investigate the structure of F in this section.
Lemma 3.1.
(1) F is a connected l-subgroup of Rk/l (M).
(2) F(l) ⊂ M(k) and F(l) is Zariski dense in M.
Proof. The group F is connected by its definition. Also since
B ⊂ M(k) = Rk/l (M)(l),

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

9

the Zariski closure of B in Rk/l (M) is defined over l, see [Sp98, Lemma 11.2.4(ii)].
This implies part (1).
The first claim in part (2) follows from the definition. To see the second claim, note
that M is connected and B ∩ F(l) has finite index in B.

Put F′ := [F, F], the commutator subgroup of F.
Lemma 3.2.
(1) F′ is a connected l-group.
′
(2) F (l) is Zariski dense M.
(3) F is an l-pseudo reductive group. That is Ru,l (F) = {e} where Ru,l (F) is
the largest connected, normal, unipotent l-subgroup.
(4) F′ is a perfect group. That is it equals its own commutator group.
Proof. Part (1) follows from [B91, §2.1].
Part (2) is a consequence of Lemma 3.1(2) and [B91, Ch. I, §2.1(e)] since M is
semisimple.
Part (3) is a special case of a more general statement [CGP10, Prop. 4.2.5]. In the
case at hand, however, a simpler proof is available as we now explicate. Contrary to
our claim, suppose that Ru,l (F) is nontrivial. Then by [Mar90b, Ch. 1, Prop. 2.5.3]
we have Ru,l (F)(l) is Zariski dense in Ru,l (F); in particular, Ru,l (F)(l) is nontrivial.
This, in view of Lemma 3.1(2), implies that M has a nontrivial unipotent radical,
contradicting the fact that M is semisimple.
Part (4) follows from part (3) and [CGP10, Prop. 1.2.6].



The following is the main result of this section; the proof is based on [CGP10,
Thm. 1.5.1 and Thm. 5.1.1].
Theorem 3.3. Let the notation be as above. Then,
(a) there is a subfield l ⊂ l′ ⊂ k with k/l′ a separable extension, and l′ /l a
purely inseparable extension,
(b) there is some m ≥ 1 and for every 1 ≤ i ≤ m, there is a field l ⊂ li ⊂ l′ , in
particular, li /l is a purely inseparable extension,
(c) for every 1 ≤ i ≤ m, there is an li -simple, connected, simply connected,
li -group Li , and
Q
′
(d) there is an isomorphism ι : m
i=1 Li ×li l → L, where L is the irreducible
component of the identity in the Zariski closure of B in Rk/l′ (M),
so that the following hold.

Q
(1) F′ (l) = ι m
i=1 Li (li ) ,
(2) F(l)/F′ (l) is a compact, abelian group.
Proof. We give the proof in some steps.
n

Step 1. The field l′ . Let l′ = l1/p where n is the largest positive integer so that
n
l1/p ⊂ k. Then l/l′ is purely inseparable and k/l′ is a separable extension.

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AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Since B is Zariski dense in M and k/l′ is separable, we have that L is a connected,
simply connected, semisimple group defined over l′ . Moreover, if char(k) = 2, 3,
then all of the absolutely almost simple factors of L are of type A.
Note also that since F is connected, we have F ⊂ Rl′ /l (L).
Step 2. The structure of F. Recall from Lemma 3.2(3) that F is l-pseudo reductive.
The map F → F/Ru (F) factors as
q

F ֒→ Rl′ /l (L) −
→L
where the map q is the natural projection, see [CGP10, §A.5]. Therefore, if
char(k) = 2, 3, then all of the absolutely almost simple factors of F/Ru (F) are
of type A.
The structure theory of F is extensively studied in [CGP10]. The following is a
corollary of [CGP10, Thm. 1.5.1 and Thm. 5.1.1] in the case at hand. The group
F is a standard pseudo reductive group, see [CGP10, Def. 1.4.4]. That is:
• for all 1 ≤ i ≤ m, there is a (finite) purely inseparable extension li /l,
• for all 1 ≤ i ≤ m, there is an li -simple, connected, simply connected, li group Li , and a maximal li -torus Ti ,
• a commutative pseudo reductive l-group A which gives a factorization
m
Y

φ

Rli /l (Ti ) −
→A→

m
Y

Rli /l (T̄i ),

i=1

i=1

of the natural map induced from Ti → T̄i := Ti /Z(Li ),
so that F is isomorphic to
m
Y

(3.1)

Rli /l (Li ) ⋊ A/

m
Y

Rli /l (Ti ).

i=1

i=1

Moreover, F′ = [F, F] is the image of the natural map
(3.2)

j:

m
Y

Rli /l (Li ) →

m
Y

Rli /l (Li ) ⋊ A/

i=1

i=1

m
Y

Rli /l (Ti ),

i=1

and ker(j) = ker(φ) is central, see [CGP10, Prop. 4.1.4 and Cor. A.7.11 ].
Step 3. The proofs. We now show that the collections {li } and {Li } satisfy the
claims in the proposition.
For all 1 ≤ i ≤ m, let us denote by fi : Rli /l (Li ) ×l l′ → L the map
q

j

Rli /l (Li ) ×l l′ −
→ F ×l l′ ֒→ Rl′ /l (L) ×l l′ −
→L
where q is the natural map.
Since l′ /l is purely inseparable, we have ker(q) is unipotent, see [CGP10, Prop.
A.5.11]. Also recall that ker(j) is central. Therefore, ker(fi ) is a solvable group for
all i. Put
m
Y
Y
Rli /l (Li ) ×l l′ → L
fi :
f=
i

i=1

Then by Lemma 3.2(2) and (3.2), f is surjective with a solvable kernel.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

11

Recall that li /l is a purely
inseparable extension for all 1 ≤ i ≤ m. Therefore,

ker(qi ) = Ru Rli /l (Li ) where qi : Rli /l (Li ) → Li is the natural surjection,
see [CGP10, Prop. A.5.11]. Moreover, L is a semisimple l′ -group, hence,

ker(fi ) ⊃ Ru Rli /l (Li ) .

Since ker(fi ) is a solvable and hence proper subgroup, we get from [CGP10, Prop.
A.7.8](1) that
li ⊂ l′ for all 1 ≤ i ≤ m;
here we also used the uniqueness of embedding of a purely inseparable extension.
As we mentioned above,


Rli /l (Li ) ×l l′ /Ru Rli /l (Li ) ×l l′ = Li .

Therefore, the map f factors through a surjection
Y
Li ×li l′ → L
ι:
i

with central kernel. Recall, however, that L is simply connected; therefore, ι is
an isomorphism. This establishes claims (a)–(d). Note also that we get: j is an
isomorphism.
To see part (1), note that

Q
m
F′ (l) = j
i=1 Rli /l (Li ) (l)
Q

m
=j
R
(L
)(l)
i
l
/l
i
i=1

Qm
= j i=1 Li (li )

Qm
= ι i=1 Li (li ) .

by (3.2)
since j is an isomorphism

since Ru,l Rli /l (Li ) = {1}

We now turn to the proof of part (2). Indeed, F(l)/F′ (l) is abelian as F/F′ is abelian.
To se it is also compact, consider F(l) as a subgroup of L(l′ ). Then for Q
every g ∈ F(l),
conjugation by g defines an automorphism of L(l′ ) which preserves i Li (li ). This
automorphism is fiberwise an li -algebraic automorphism. Since Li (li ) is Zariski
dense in Li , we get that this automorphism is fiberwise an inner li -automorphism.
Part (4) follows from this, in view of [Mar90b, Ch. 1, Thm. 2.3.1].

4. An algebraic statement
One of the remarkable features of Ratner’s theorems is that they connect objects
which are closely connected to the Hausdorff topology of the underlying group, like
closure of a unipotent orbit or a measure invariant by a unipotent group, to objects
which are described using the Zariski topology, e.g. algebraic subgroups. In positive
characteristic setting these two topologies are rather far from each other2. From a
philosophical stand point, this is one reason why the existing proofs in characteristic
zero do not easily generalize to this case.
Let us restrict ourselves to unipotent groups in order to highlight one of the differences. In characteristic zero, the group generated by one unipotent matrix already
2See [P98] where structure of compact subgroups of semisimple groups of adjoint type is
described.

12

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

carries quite a lot of information, e.g. it is Zariski dense in a one dimensional group.
In positive characteristic, however, all unipotent elements are torsion. The situation improves quite a bit in the presence of a split torus action. In a sense, such an
action can be used “to redefine a notion of Zariski closure” for the group generated
by one element. This philosophy is used in this section where we prove the following
proposition which is of independent interest.
Proposition 4.1. Let k be a local field of characteristic p > 0 and W a unipotent
k-group equipped with a k-action by GL1 . Assume that all the weights are positive
integers. Let U be a subgroup of W(k) which is invariant under the action of k × .
Then, there is q a power of p, which only depends on the set of weights, such that
U = W′ (k q ), where W′ is the Zariski-closure of U ⊂ W(k) = Rk/kq (W)(k q ) in
Rk/kq (W). Moreover, W′ is connected.
Groups like U arise naturally in our study, see §6 for more details.
We shall prove Proposition 4.1 in several steps. First we prove it when W is a
commutative p-torsion k-group. In the next step, the general commutative case is
handled. In the final step, the general case is proved by induction on the nilpotency
length.
In order to prove the first step, we shall start with a few auxiliary lemmas. In
Lemmas 4.2 and 4.3, we assume that the weights are powers of p. In Lemmas 4.4
and 4.5, we get a convenient decomposition of U into certain subgroups, and finally
in Lemma 4.6, we prove the first step. Let begin with the following
Lemma 4.2. Let F be an infinite field of characteristic p and 0 < l1 < · · · < ln
positive integers. Assume GL1 acts linearly on a standard F -vector group W with
weights equal to pli . Let Wi be the weight space of pli and suppose U is a subgroup
of W(k) which is invariant under GL1 (k). If U does not intersect ⊕ni=2 Wi , then
l

U = W′ (F p ),
for any l ≥ l1 , where W′ is the Zariski-closure of U in RF/F pl (W).
Proof. Via the action of GL1 (F ), and in view of our assumption of the weights,
we can view W(F ) as an F -vector space and U as a F -subspace. Since U does
not intersect ⊕ni=2 Wi , we get an F -linear map θ from pr1 (U) to ⊕ni=2 Wi , where
pr1 : W → W1 is the projection map, and we have
U = {(x, θ(x))| x ∈ pr1 (U)}.
l1

It is clear that pr1 (U) is a F p -subspace of W1 (F ) with respect to the standard
scalar multiplication. It is also clear that θ can be extended to an F -morphism from
l1
W1 to ⊕ni=2 Wi . Hence there is a standard F p -vector subgroup W′1 of RF/F pl1 (W1 )
such that
l1
U = {(x, y)| x ∈ W′1 (F p ), y = RF/F pl1 (θ)(x)}.
Since F is an infinite field, the Zariski-closure W′ of U in RF/F pl1 (W) is equal to
{(x, y)| x ∈ W′1 , y = RF/F pl1 (θ)(x)},
l1

which shows that U = W′ (F p ). Now, one can easily deduce the same result for
any l ≥ l1 .


MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

13

Lemma 4.3. Let F be an infinite field of characteristic p and 0 < l1 < · · · < ln
positive integers. Assume GL1 acts linearly on a standard F -vector group W with
weights equal to pli . Let U be a subgroup of W(F ) which is invariant under GL1 (F ).
Then
l
U = W′ (F p ),
for any l ≥ ln , where W′ is the Zariski-closure of U in RF/F pl (W).
Proof. We denote the weight space of pli by Wi . Let Ui = U ∩ (⊕nj=i Wj ) and define
Ui′ to be a GL1 (F )-invariant complement of Ui in Ui−1 . So we have
Ui′ ⊂ ⊕nj=i−1 Wj
Ui′ ∩ (⊕nj=i Wj ) = {0}
Ui−1 = Ui ⊕ Ui′
U = U2′ ⊕ U3′ ⊕ · · · ⊕ Un′ ⊕ Un .
l

By Lemma 4.2, we have that Ui′ = W′i (F p ), for all i and any l ≥ li−1 , where W′i is
the Zariski-closure of Ui′ in Rk/kpl (⊕nj=i−1 Wj ). Moreover, Un is a subspace of Wn
n

with respect to the standard action of F p ; one can easily conclude.



Lemma 4.4. Let m1 , . . . , md be distinct positive integers which are coprime with
p. Let g(x) = (xm1 , . . . , xmd ) be a morphism from A1 to Ad . Then
G(x1 , . . . , xd ) := g(x1 ) + · · · + g(xd )
is a separable function from Ad to Ad at a F -point, for any infinite field F of
characteristic p.
Proof. It is enough to show that the Jacobian of G is invertible at some F -point.
Thus, thanks to our assumption: all of mi are coprime with p, it suffices to show
that the kernel of D = [xjmi −1 ] is trivial for some xj ∈ F . Now since F is an infinite
field, there is an element x ∈ F × of multiplicative order larger than maxi mi . Set
xj = xj−1 ; if D has a non-trivial kernel, then there is non-zero polynomial Q of
degree at most d − 1 with coefficients in F , such that
Q(xm1 −1 ) = Q(xm2 −1 ) = · · · = Q(xmd −1 ) = 0.
This is a contradiction as xmi −1 are distinct and the degree of Q is at most d−1. 
Lemma 4.5. Let k be a local field of characteristic p and m1 , · · · , md distinct
positive integers which are coprime with p. Let W be a standard k-vector group
equipped with a linear k-action by GL1 . Assume that the set of weights Φ = Φ1 ⊔
· · · ⊔ Φd , Φi = {pl mi ∈ Φ| l ∈ N}, and moreover Φi is non-empty. Let U be a
subgroup of W(k), which is invariant under the action of GL1 (k). Then
U = U1 ⊕ · · · ⊕ Ud ,
where Ui = U ∩ (⊕α∈Φi Wα ) and Wα is the weight space corresponding to α. Furthermore, if Φi = {pli1 mi , pli2 mi , . . . , plini mi } and x = (x1 , . . . , xni ) ∈ Ui , then
li1

(λp

lin

x1 , . . . , λp

i

xni ) ∈ Ui , for any λ ∈ k × .

14

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Proof. Take an arbitrary element x = (xα )α∈Φ ∈ U. Since U is invariant under the
action of GL1 (k), we have (λα xα )α∈Φ is also in U, for any λ ∈ k × . On the other
hand, as U is a group,
(4.1)

α
α
((±λα
1 ± λ2 ± · · · ± λn )xα )α∈Φ ∈ U,

for any λ1 , . . . , λn ∈ k × . On the other hand, by Lemma 4.4 and the Inverse Function
Theorem, the image of G − G has an open neighborhood of the origin. Therefore,
thanks to scale invariance of the image, we have: for any λ′1 , . . . , λ′d ∈ k, one can
find λi , µi ∈ k such that
Pd
P
mi
i
(4.2)
λ′i = dj=1 λm
j −
j=1 µj ,

for any 1 ≤ i ≤ d. Now since p is the characteristic of k, using (4.1) and (4.2) one
can easily finish the argument.

Lemma 4.6. Let k be a local field of characteristic p. Let W be a p-torsion commutative unipotent k-group equipped with a linear k-action by GL1 ; further, assume
that all the weights are positive. Let U be a subgroup of W(k), which is invariant
under the action of GL1 (k). Then, there exists some l0 , depending only on the
weights, such that for any integer l ≥ l0
l

U = W′ (k p ),
where W′ is the Zariski-closure of U in Rk/kpl (W).
Proof. By [CGP10, Prop. B.4.2], we can assume that W is a k-vector group equipped
with a k-linear action of GL1 . Applying Lemma 4.5, we can decompose U into subgroups Ui and get a new GL1 action on ⊕α∈Φi Wα such that all the new weights
are powers of p and Ui is invariant under this new action of GL1 (K). The lemma
now follows from Lemma 4.3.

Lemma 4.7. Let k be a local field of characteristic p. Let W be a commutative
unipotent k-group equipped with a k-action by GL1 such that ZGL1 (W) = {1}.
Suppose U is a subgroup of W(k), which is invariant under the action of GL1 (k).
Then, there is some l0 , depending only on the weights of the action of GL1 on
Lie(W), such that for any integer l ≥ l0
l

U = W′ (k p ),
where W′ is the Zariski-closure of U in Rk/kpl (W).
Proof. Since W is unipotent, it is a torsion group. We now proceed by induction
on the exponent of W; if it is p, by Lemma 4.6, we are done. Thus, assume that
the exponent of W is pl . Let U[p] = {u ∈ U|up = 1}; since U is commutative,
U[p] is a subgroup of U which is clearly invariant under the action of GL1 (k).
Let V be the Zariski-closure of U[p] in W; then, V is a p-torsion commutative
unipotent group which in view of Lemma 2.3 is defined over k. Therefore, by
Lemma 4.6 for a large enough power of p (depending only on the weights), which
′
we denote by q ′ , we have U[p] = W(p) (k q ), where W(p) is the Zariski-closure of U[p]
in Rk/kq′ (V) ⊂ Rk/kq′ (W).
Let W′′ be the Zariski-closure of U in Rk/kq′ (W). Note that GL1 acts on W′′ with
no trivial weights, and W(p) is invariant under this action. Hence both of these

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

15

′

groups and their quotient are k q -split unipotent groups. We now consider the
′
following exact sequence of k q -split unipotent groups,
π

→ W′′ /W(p) → 1.
1 → W(p) → W′′ −
Note that π(U) is Zariski-dense in W′′ /W(p) and pl−1 -torsion, which implies W′′ /W(p)
is pl−1 -torsion. Hence, by induction hypothesis, there exists q ≥ q ′ which is a large
enough power of p, depending only on the weights, such that
′

π(U) = W (k q ),

(4.3)
′

where W is the Zariski-closure of π(U) in Rkq′ /kq (W′′ /W(p) ). On the other hand,
(4.4)

′

U[p] = W(p) (k q ) = Rkq′ /kq (W(p) )(k q )

and, by [Oe84, Cor. A.3.5], Rkq′ /kq (W(p) ) is k q -split unipotent group. Thus U[p]
is Zariski-dense in Rkq′ /kq (W(p) ). By [Oe84, Prop. A.3.8], we also know that the
following is exact
π′

1 → Rkq′ /kq (W(p) ) → Rkq′ /kq (W′′ ) −→ Rkq′ /kq (W′′ /W(p) ) → 1.
Now let W′ be the Zariski-closure of U in Rk/kq (W′′ ). By the above discussion, it
is easy to get the following short exact sequence and show that all of the involved
groups are k q -split unipotent groups,
π

′

1 → Rkq′ /kq (W(p) ) → W′ −
→ W → 1.
By (4.3) and (4.4) and the fact that these groups are k q -split unipotent groups, we
get the following exact sequence,
(4.5)

π

1 → U[p] → W′ (k q ) −
→ π(U) → 1.

So, by (4.5) and U ⊂ W′ (k q ), one can easily deduce that U = W′ (k q ), which finishes
the proof.

Proof of Proposition 4.1. We proceed by induction on the nilpotency length of W.
If it is commutative, by Lemma 4.7, we are done. Assume W is of nilpotency
length c. Then [U, U] ⊂ [W, W](k), where [•, •] is the derived subgroup of •. The
nilpotency length of [W, W] is c − 1; hence, by induction hypothesis, for any q ′
which is a large enough power of p (depending only on the weights), we have
′
[U, U] = f
W(k q ),

where f
W is the Zariski-closure of [U, U] in Rk/kq′ (W). Let W′′ be the Zariskiclosure of U in Rk/kq′ (W); since GL1 acts on W′′ with no trivial weights and since
f
W is invariant under this action, both of these groups and the quotient group are
′
k q -split groups. We consider the following short exact sequence

(4.6)

π

W → 1;
→ W′′ /f
W → W′′ −
1→f

since π(U) is commutative and Zariski-dense in W′′ /f
W, we get that W′′ /f
W is
′
commutative. Therefore, by Lemma 4.7, if q ≥ q is a large enough power of p
(depending only on the weights), we have
(4.7)

′

π(U) = W (k q ),

16

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

′
W). We also have
where W is the Zariski-closure of π(U) in Rkq′ /kq (W′′ /f

(4.8)

′

[U, U] = f
W(k q ) = Rkq′ /kq (f
W′ )(k q ).

By [Oe84, Prop. A.3.8] and (4.6), we have the exact sequence
′

(4.9)

π
1 → Rkq′ /kq (f
W) → Rkq′ /kq (W′′ ) −→ Rkq′ /kq (W′′ /f
W) → 1.
′

Let W′ be the Zariski-closure of U in Rkq′ /kq (W′′ ). Since f
W is a k q -split unipotent
group, by (4.7) and (4.9), we have the following exact sequence
′

′
π
1 → Rkq′ /kq (f
W) → W′ −→ W → 1

and so, by (4.8), (4.9) and the fact that all the involved groups are k q -split unipotent
groups, we have
(4.10)

1 → [U, U] → W′ (k q ) → π(U) → 1;

this together with U ⊂ W′ (k q ) finishes the proof except connectedness.
To see W′ is connected, note that in view of our assumption that all the weights
are positive, there exists some r ∈ k so that every element in U is contracted to the
identity by r.

5. Polynomial like behavior and the basic lemma
In this section we assume µ is a probability measure on X = G/Γ which is invariant
under the action of some kT -split, unipotent kT -subgroup of G.
We will recall an important construction based on the slow divergence of two nearby
unipotent orbits in X. Then, we will use this to acquire new elements in the stabilizer of µ. Investigating the polynomial like behavior of two diverging unipotent orbits in the intermediate range dates back to several important works, e.g.
Margulis’ celebrated proof of the Oppenheim conjecture [Mar86], using topological arguments, and Ratner’s seminal work on the proof of the measure rigidity
conjecture [R90a, R90b, R91].
5.1. Construction of quasi-regular maps. This section follows the construction
in [MT94, §5]. It is written in a more general setting than what is needed for the
proof of Theorem 1.1, namely µ is not assumed to be ergodic for the action of the
unipotent group U which is used in the construction. We first recall the definition of
a quasi-regular map. Here the definition is given in the case of a local field, which
is what we need later, the T -arithmetic version is a simple modification. It is worth
mentioning that we have a simplifying assumption here compare to the situation
in [MT94]: our group U is normalized and expanded by an element from class A.
This is used in order to define nice Folner sets in U. In view of this, we do not need
the construction of the group U0 in [MT94].
Definition 5.1 (Cf. [MT94], Definition 5.3). Let k be a local field.
(1) Let E be a k-algebraic group, U a k-subgroup of E(k), and M a k-algebraic
variety. A k-rational map f : M(k) → E(k) is called U-quasiregular if the

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

17

map from M(k) to V, given by x 7→ ρ(f (x))q, is k-regular for every krational representation ρ : E → GL(V), and every point q ∈ V(k) such that
ρ(U)q = q.
(2) Let E = E(k) and suppose U ⊂ E is a k-split unipotent subgroup. A map
φ : U → E is called strongly U-quasiregular if there exist
(a) a sequence gn ∈ E such that gn → e,
(b) a sequence {αn : U → U} of k-regular maps of bounded degree,
(c) a sequence {βn : U → U} of k-rational maps of bounded degree, and
(d) a Zariski open, dense subset X ⊂ U,
such that φ(u) = limn→∞ αn (u)gn βn (u), and the convergence is uniform
on the compact subsets of X .
We note that if φ is strongly U-quasiregular, then it indeed is U-quasiregular. To
see this, let ρ : E → GL(V ) be a k-rational representation, and let q ∈ V be a
U-fixed vector. For any u ∈ X we have
(5.1)

ρ(φ(u))q = lim ρ(αn (u)gn )q.
n→∞

Thanks to the fact that U is split we can identify U with an affine space. Then
ψn : U → V given by ψn (u) = ρ(αn (u)gn )q
is a sequence of polynomial maps of bounded degree. Moreover, this family is
uniformly bounded on compact sets of X . Therefore, it converges to a polynomial
map with coefficients in k. This shows φ is U-quasiregular.
For the rest of this section we assume the following
•
•
•
•

k is a local field,
G is the group of k-points of a k-group,
U is a connected k-split, unipotent k-subgroup of G,
there is an element s ∈ G from class A so that U ⊂ WG+ (s) and U is
normalized by s.

In view of these assumptions, [BS68, Prop. 9.13] implies that there a regular cross
section, V, for U in WG+ (s) which is invariant under conjugation by s. Put
L := WG− (s)ZG (s)V.
Then L is a rational cross section for U in G.
We fix, B+ and B− , relatively compact neighborhoods of e in WG+ (s) and WG− (s)
respectively, with the property that
B+ ⊂ sB+ s−1 and B− ⊂ s−1 B− s.
Using these, we define a filtration in WG+ (s) and WG− (s) as follows
n + −n
−n − n
B+
and B−
B s .
n =s B s
n =s

Define ℓ± : W ± (s) → Z ∪ {−∞} by
+
+
• ℓ+ (x) = j if x ∈ B+
j \ Bj−1 , and ℓ (e) = −∞,
−
−
−
• ℓ (x) = j if x ∈ Bj \ Bj−1 , and ℓ− (e) = −∞.

18

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

For any integer n, set Un = B+
n ∩ U.
Let {gn } ⊂ L U \ NG (U) be a sequence with gn → e. Since L is a rational crosssection for U in G, we get rational morphisms
φen : U → L and ωn : U → U

such that ugn = φen (u)ωn (u) holds for all u in a Zariski open, dense subset of U.

Recall that by a theorem of Chevalley, there exists a kT -rational representation
ρ : G → GL(Ψ) and a unit vector q ∈ Ψ such that
U = {g ∈ G : ρ(g)q = q}.

(5.2)

According to this description we also have
ρ(NG (U))q = {z ∈ ρ(G)q : ρ(U)z = z}.

(5.3)

Fix a bounded neighborhood B(q) of q in Ψ such that
ρ(G)q ∩ B(q) = ρ(G)q ∩ B(q),

(5.4)

where the closure is taken with respect to the Hausdorff topology of Ψ.
Recall that gn ∈
/ NG (U). Thus, in view of (5.3), there is a sequence of integers
{b(n)} such that
• b(n) → ∞,
• ρ(Ub(n)+1 gn )q 6⊂ B(q), and
• ρ(Um gn )q ⊂ B(q) for all m ≤ b(n).
Define k-regular isomorphisms τn : U → U as follows. For every u ∈ U put
τn = λb(n) where λn (u) = sn us−n .

(5.5)

Given n ∈ N, we now define the k-rational map φn : U → L by φn := φen ◦ τn .
Let ρL be the restriction of the orbit map g 7→ ρ(g)q to L and define
φ′n := ρL ◦ φn : U → Ψ.

(5.6)

It follows from the definition of b(n) that φ′n (B0 ) ⊂ B(q), but φ′n (B1 ) 6⊂ B(q).
Note that φ′n (u) = ρ(αn (u)gn )q. Hence φ′n : U → Ψ is a k-regular morphism.
Since
• U is a connected k-group,
• U is normalized by S, and
• ZG (S) ∩ U = {e}
we get from [BS68, Cor. 9.12] that U and its Lie algebra are S-equivariantly isomorphic as k-varieties. Hence, {φ′n } is a sequence of equicontinuous polynomials
of bounded degree. Therefore, after possibly passing to a subsequence, we assume
that there exists a k-regular morphism φ′ : U → Ψ such that
φ′ (u) = lim φ′n (u) for every u ∈ U.

(5.7)

n→∞

′

The map φ is non-constant since φ′ (B1 ) in not contained in B(q); moreover, since
gn → e we have φ′n (e) → q, hence, φ′ (e) = q.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

19

Let M = ρ(L)q. Since L is a rational cross section for U in G which contains e, we
get that M is a Zariski open dense subset of ρ(G)q and q ∈ M.
Let now φ : U → L be the kT -rational morphism defined by
′
φ(u) := ρ−1
L ◦ φ (u).

(5.8)

It follows from the construction that φ(e) = e and that φ is non-constant.
Claim. The map φ constructed above is strongly U-quasiregular.
To see the claim, first note that by the definition of φn and in view of (5.7) and (5.8)
we have
φ(u) = lim φn (u) for all u ∈ φ′−1 (M).

(5.9)

n→∞

Now since the convergence in (5.7) is uniform on compact subsets and since ρ−1
L is
continuous on compact subsets of M, we get that the convergence in (5.9) is also
uniform on compact subsets of φ′−1 (M). Recall that
τn (u)gn = φn (u)wn (τn (u)).
′−1

Hence, for any u ∈ φ

(M) we can write

(5.10)

φ(u) = lim τn (u)gn (wn (τn (u)))−1 ;
n→∞

the claim follows.
5.2. Properties of quasi-regular maps and the Basic Lemma. We will need
some properties of the map φ constructed above. The proofs of these facts are mutandis mutatis of the proofs in characteristic zero in [MT94]; we will only highlight
the required modifications here.
Proposition 5.2 (Cf. [MT94], §6.1 and §6.3). The map φ is a rational map from U
into NG (U). Furthermore, there is no compact subset K of G such that Im(φ) ⊂ KU.
Proof. Recall from (5.3) that
NG (U) = {g ∈ G : ρ(U)ρ(g)q = ρ(g)q}.
Thus, we need to show that for any u0 ∈ U and any u ∈ φ′−1 (M), we have
ρ(u0 )ρ(φ(u))q = ρ(φ(u))q;
this suffices as φ′−1 (M) is a Zariski open, dense subset of U.
Let u ∈ φ′−1 (M), then by (5.10) we have
φ(u) = lim τn (u)gn (wn (τn (u)))−1 .
n→∞

On the other hand, we have ρ(u0 τn (u)gn )q = ρ(τn (τn−1 (u0 )u)gn )q.
Note now that τn−1 (u0 ) → e as n → ∞. This, in view of the above discussion,
implies that φ(u) ∈ NH (U) for all u ∈ φ′−1 (M). The first claim follows.
′
′
To see the second assertion, note that φ = ρ−1
L ◦ φ . The claim thus follows since φ
is a non-constant polynomial map and ρL is an isomorphism from L onto a Zariski
open, dense subset of the quasi affine variety ρ(G)q.


20

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

In the sequel we will utilize a quasi-regular map, φ, which is constructed using a
sequence of elements gn → e with the following property.
Definition 5.3 (Cf. [MT94], Definition 6.6). A sequence {gn } is said to satisfy the
condition (∗) with respect to s if there exists a compact subset K of G such that
for all n ∈ N we have s−b(n) gn sb(n) ∈ K.
This technical condition is used in the proof of the Basic Lemma. It is also essential
in the proof of Proposition 6.2.
We also recall the following
Definition 5.4. A sequence of measurable, non-null subsets An ⊂ U is called an
averaging net for the action of U on (X, µ) if the following analog of the Birkhoff
pointwise ergodic theorem holds. For any continuous, compactly supported function
f on X and for almost all x ∈ X one has
Z
Z
1
(5.11)
lim
f (h)dµy(x) (h),
f (ux)dθ(u) =
n→∞ µ(An ) A
X
n
where µy(x) denotes the U-ergodic component corresponding to x.
The proof of the following is standard.
Lemma 5.5 (Cf. [MT94], §7.2). Let A ⊂ U be open, relatively compact, and nonnull. Let An = λn (A). Then {An } is an averaging net for the action of U on
(X, µ).
Ergodic theorems hold on full measure subsets of the space (with the exception of
uniquely ergodic systems). The following is a uniform and quantitive version of full
measure sets, and it is better adapted to limiting arguments.
Definition 5.6. A compact subset Ω ⊂ X is said to be a set of uniform convergence
relative to {An } if the following holds. For every ε > 0 and every continuous,
compactly supported function f on X one can find a positive number N (ε, f ) such
that for every x ∈ Ω and n > N (ε, f ) one has
Z
Z
1
f (h)dµy(x) (h) < ε.
f (ux)dθ(u) −
θ(An ) An
X
The following is a consequence of Egoroff’s Theorem and the second countability
of the spaces under consideration, see [MT94, §7.3].
Lemma 5.7. For any ε > 0 one can find a measurable set Ω with µ(Ω) > 1 − ε
which is a set of uniform convergence relative to {An = λn (A)} for every open,
relatively compact, and non-null subset A of U.
5.3. The following is the main application of the construction of the quasi-regular
maps. It provides us with the anticipated extra invariance property.
Basic Lemma (Cf. [MT94], Basic Lemma, §7.5). Let Ω be a set of uniform convergence relative to all averaging nets {An = λn (A)} for all A ⊂ U which are
open, relatively compact, and non-null. Let {xn } be a sequence of points in Ω
with xn → x ∈ Ω. Let {gn } ⊂ G \ NG (U) be a sequence which satisfies condition

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

21

(∗) with respect to s. Assume further that gn xn ∈ Ω for every n. Suppose φ is
the U-quasiregular map corresponding to {gn } constructed above. Then the ergodic
component µy(x) is invariant under Im(φ).
Proof. The proof in [MT94, Basic Lemma] works the same here. Indeed, the analysis simplifies in our situation as U = U0 = U; we present a sketch here.
Let the notation be as in the construction of φn ; in particular, τn = λb(n) . Then,
the condition (∗) allows one to write
ωn ◦ τn = τn ◦ ηn
where ηn : U → U is a rational map and locally a diffeomorphism. Furthermore,
given u0 ∈ φ′−1 (M), the sequence {ηn } uniformly converges to a diffeomorphism,
η, on a neighborhood of u0 .
Let A = {u : kφn (u)f − φ(u0 )f k∞ < ε} for f ∈ Cc (X) and ε > 0 is small enough.
One then uses η to compare the ergodic average
Z

1
f τn (u)gn x dθ(u)
θ(A) A
Z

1
f φn (u)ωn (τu (u))x dθ(u).
with θ(A)
A

Passing to the limit, we get that µ is φ(u0 )-invariant, as was claimed.



We finish this section with the following remark which will be used in the proof of
Theorem 1.1, see Step 4 in the proof.
Remark 5.8. The construction above assumed {gn } 6∈ NG (U), however, we make
the following observation. Let U, µ and Ω be as in the Basic Lemma. Further, let
us assume that µ is U-ergodic. Suppose3
g ∈ NG (U) ∩ ZG (s)WG+ (s)
is so that gx ∈ Ω for some x ∈ Ω. Then µ is invariant by g.
To see this, put A = B0 ∩ U and let An = λn (A). For all n ≥ 0 and any continuous
compactly supported function f we have
Z
Z
1
1
f (ugx)dθ(u) =
f (λn (u)gx)dθ(u)
θ(An ) An
θ(A) A
Z
1
=
f (gg −1 λn (u)gx)dθ(u)
θ(A) A
Z
1
f (gλn (gn−1 ugn )x)dθ(u)
=
θ(A) A
Z
1
=
f (gλn (u)x)dθ(u),
θ(B(n)) B(n)
−1
where gn = λ−1
n (g) and B(n) = gn Agn . In the last equality we used the fact that
the Jacobian of the conjugation by gn is constant.
3In general, N (U ) ⊂ Z (s)W + (s) can be arranged by embedding G in some SL and choosing
n
G
G
G
the element s more carefully. However, the more restrictive statement given here already suffices
for our application.

22

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Now let g0 ∈ NG (U ) be so that gn → g0 as n → ∞. Put B = g0−1 Ag0 , then
θ(B(n) △ B) → 0 as n → ∞. Hence, for any ε > 0 and all large enough n, we have
Z
Z
1
1
f (gλn (u)x)dθ(u) −
f (gλn (u)x)dθ(u) ≤ ε
θ(B(n)) B(n)
θ(B) B
On the other hand since x, gx ∈ Ω, for all large enough n we have
R
R
1
θ(An ) An f (ugx)dθ(u) − X f (h)dµ(h) < ε
R
R
1
θ(Bn ) Bn f (gux)dθ(u) − X f (gh)dµ(h) < ε.

Putting all these together we get that |µ(f ) − gµ(f )| ≤ 3ε. This implies the claim
if we let ε → 0.
6. Proof of Theorem 1.1

Let us recall the setup from the introduction. We fixed
•
•
•
•

a
a
a
a

kT -algebraic group G,
closed subfield k ′ ⊂ kw ,
k ′ -group H, and
kw -homomorphism ι : H ×k′ kw → Gw .

Therefore, if we replace Gw by G′w := Rkw /k′ (Gw ), we get a k ′ -group, G′w , such
that G′w (k ′ ) = Gw (kw ). Furthermore, it follows from the universal property of
Weil’s restriction of scalars that H is a k ′ -subgroup of G′w . Hence, we may and
will assume that k ′ = kw and Gw = G′w . To simplify the notation, we will denote
kw = k for the rest of this section.
We also have fixed a non central k-homomorphism, λ : Gm → H. Recall the one
dimensional k-split tours S ⊂ G defined using λ. Let s = ι(s′ ), then s ∈ S is an
element from class A.
6.1. The subgroup U. As in the statement of Theorem 1.1, µ is a probability
measure on X = G/Γ which is SU -invariant and U -ergodic. Define
(6.1)

U ⊂ WG+ (s) to be the maximal subgroup which leaves µ invariant.

Note that U ⊂ U. Since µ is S-invariant and WG+ (s) is normalized by S, the group
U is a closed, in Hausdorff topology, subgroup of WG+ (s) which is normalized by S.
Therefore, Proposition 4.1 implies the following. There exists some q = pn , deof k q -points of a
pending on the action of S on W+
Gw (s), such that U is the group

+
q
q
connected, k -split, unipotent, k -subgroup of Rk/kq WG (s) .
We fix a k q -homomorphism

λ′ : Gm → Rk/kq (H)
and an element s0 ∈ λ′ (Gb (k q )) so that WG+ (s) = WG+ (s0 ), see [CGP10, App. C].
Replacing Gw with Rk/kq (Gw ), which we continue to denote by Gw , we have U is
an algebraic subgroup of Gw .

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

23

Also replace S by λ′ (Gm ), S by Rk/kq (ι)(λ′ (Gm )(k q )) and s by s0 . Finally, we
replace k by k q in kT and continue to denote this by kT .
In particular, we have the following
• The group U is the set of kT -points of a connected, kT -split, unipotent,
kT -subgroup of WG+ (s).
• The group S is a k-split one dimensional k-torus, s ∈ S, and U ⊂ WG+ (s)
is normalized by S.
6.2. The subgroup F (s). Following [MT94], we define
(6.2)

z

F (s) = {g ∈ G : Ug ⊂ WG− (s)ZG (s)U };

as indicated, the closure is the Zariski closure. Since WG− (s)ZG (s) is a subgroup of
G the above can be written as
(6.3)

z

z

{g ∈ G : WG− (s)ZG (s)U g ⊂ WG− (s)ZG (s)U }.

Thus the inclusion in (6.3) may be replaced by equality. This implies
F (s) is a k-closed subgroup of G.
Note that S ⊂ F (s) and F (s) ∩ WG+ (s) = U.
Put U − = F (s) ∩ WG− (s). This a k-closed subgroup of WG− (s) which is normalized
by S.
Lemma 6.1. The group U − is the k-points of a connected, unipotent, k-subgroup
of WG− (s). Moreover, there is a regular cross-section, V − , for U − in WG− (s) and
V − is invariant under conjugation by s.
Proof. The second claim follows from the first claim and [BS68, Prop. 9.13].
To see the first claim, note that by Lemma 2.3, there is a smooth group scheme B
defined over k so that B(k) = U − . Hence, B ⊂ W−
G (s) is a unipotent group which is
normalized by s. Since s contracts every element of B to the identity, we get that
B is connected as was claimed.

Similarly, let V be a regular cross-section for U in WG+ (s) which is invariant under
conjugation by s.
6.3. Structure of µ along contacting leaves of s. In this section we will use
the maximality of U and the Basic Lemma to show that µ has a rather special
structure along WG− (s). The main result is Proposition 6.3. We first need some
more notation. Put
(6.4)

D = WG− (s)ZG (s)WG+ (s) = U − V − ZG (s)VU.

Then, D is a Zariski open dense subset of G containing e, see §2.1. Moreover, for
any g ∈ D we have a unique decomposition
(6.5)

g = w− (g)z(g)w+ (g) = u− (g)v − (g)z(g)v(g)u(g)

where u− (g) ∈ U − , v − (g) ∈ V − , z(g) ∈ ZG (s), u(g) ∈ U, v(g) ∈ V, w− (g) =
u− (g)v − (g), and w+ (g) = v(g)u(g).

24

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Note that for every w± ∈ WG± (s) we have
ℓ± (sm w± s−m ) = ℓ± (w± (g)) ± m.

(6.6)
We need the following

Proposition 6.2. Suppose {gn } is a sequence converging to e, and let s and U be
as above. Suppose one of the following holds
(1) the sequence ℓ− (v − (gn )) − ℓ− (u− (gn )) is bounded from below, or
(2) {gn } ⊂ ZG (s)WG+ (s) \ NG (U).
Then, {gn } satisfies the condition (∗). Furthermore, if we let φ be the quasi-regular
map constructed using U and this sequence {gn }, then Im(φ) ⊂ WG+ (s).
Proof. The fact that the conclusion holds under condition (1) is proved in [MT94,
Prop. 6.7]. We show (2) also implies the conclusion.
Under assumption (2), we have s−b(n) gn sb(n) → e, hence, {gn } satisfies the condition (∗). We now use an argument similar to [MT94, Prop. 6.7] to show Im(φ) ⊂
WG+ (s) when condition (2) above holds. By (5.9) we have
φ(u) = lim φn (u) = lim τn (u)gn ωn (τn (u))−1 for all u ∈ φ′−1 (M).
n→∞

n→∞

It follows from the choice of b(n) that
{s−b(n) • sb(n) } are bounded in U for • = τn (u) and ωn (τn (u))−1 .
In view of this and the condition (∗), we get the following. After possibly passing
to a subsequence, we have
lim s−b(n) φn (u)sb(n) = lim s−b(n) τn (u)gn ωn (τn (u))−1 sb(n) ∈ U.

n→∞

n→∞

WG+ (s)

for all u ∈ φ′−1 (M). Together with the fact that
This implies that φ(u) ∈
′−1
φ (M) is Zariski dense in U, this finishes the proof.

The following is an important consequence of the above proposition and the construction of quasi-regular maps in §5. It describes the local structure of the set of
uniform convergence. Our formulation here is taken from [MT94]; let us remark
that obtaining such description is also essential in [R90b].
Proposition 6.3. For every ε > 0, there exists a compact subset Ωε ⊂ X with the
following properties.
(1) µ(Ωε ) > 1 − ε, and
(2) if {gn } ⊂ G \ NG (U + (s)) is a sequence so that gn → e and
gn Ωε ∩ Ωε 6= ∅ for every n,
then the sequence {ℓ− (v − (gn )) − ℓ− (u− (gn ))} tends to −∞.
Proof. First note that U ⊂ U, therefore, µ is U-ergodic and invariant. Let ε > 0
be given. Let Ωε be a set of uniform convergence for the action of U, in the sense
of Definition 5.6, with µ(Ωε ) > 1 − ε. We will show that Ωε satisfies (2) in the
proposition as well.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

25

Assume contrary to our claim that there is a sequence {gn } so that (2) fails. Passing
to a subsequence we may assume the sequence
 − −
ℓ (v (gn )) − ℓ− (u− (gn ))

is bounded from below. Therefore, Proposition 6.2 guarantees that {gn } satisfies
condition (∗).
Now construct the quasi-regular map φ corresponding to {gn } as in §5. Then,

• in view of Proposition 5.2, the image of φ is not contained in KU for any
bounded subset K ⊂ G.
• It follows from the Basic Lemma that µ is invariant under Im(φ).
• By Proposition 6.2, we have Im(φ) ⊂ WG+ (s).
Therefore, µ is invariant under hU, Im(φ)i which is contained in WG+ (s) and strictly
contains U. This contradicts the maximality of U and finishes the proof.

We will use this proposition in the following form.
Corollary 6.4. There exists a subset Ω ⊂ X with µ(Ω) = 1 such that
WG− (s)x ∩ Ω ⊂ U − x
for every x ∈ Ω.
Proof. The proof follows the same lines as the proof of [MT94, Cor. 8.4] using
Proposition 6.3. We recall the proof here for the convenience of the reader.
First let us note that by Mautner’s phenomenon, every s-ergodic component of µ
is U-invariant, thus, µ is s-ergodic. For any ε > 0, let Ωε be as in Proposition 6.3.
Let Ω′ε ⊂ Ωε be a compact subset with µ(Ω′ε ) > 1 − 2ε so that the Birkhoff ergodic
theorem for the action of s and χΩε holds for every x ∈ Ω′ε .
Suppose x and y = w− x are in Ω′ε and assume that w− ∈
/ U − . Let ni → ∞ be a
ni
ni
subsequence so that both s x ∈ Ωε and s y ∈ Ωε , such sequence exists by Birkhoff
ergdoic theorem.
Let xi = sni x and
yi = sni y = sni ws−ni sni x = wi xi
where wi = sni ws−ni . Our assumption on w and (6.6) imply that
{ℓ− (v − (wi )) − ℓ− (u− (wi ))}
is bounded from below which contradicts Proposition 6.3.
The corollary now follows if we apply the above argument to a sequence εn → 0

and let Ω = ∪n Ω′εn .
6.4. A lemma on finite dimensional representations. We need certain properties of the subgroup F (s) which was defined in (6.2). These will be used when
we apply Theorem 6.8 in the proof of Theorem 1.1. The main property needed is
Lemma 6.7 below which is a consequence of Lemma 6.5. It is worth mentioning
that the latter is closely related to the notion of an epimorphic group which was
introduced by A. Borel.

26

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Retain the notation from §1. Recall also from the reductions in the beginning of
+
−
this section that k = k ′ . Put H + = hWH
(s), WH
(s)i. Since H′ is k ′′ -almost simple
′′
and k -isotropic we have

f′ )(k)
H + = ι̃ Rk′′ /k (H
f′ → H′ denotes the simply connected cover of H′ , and ι̃ is ι precomposed
where H
with the covering map, see [Mar90b, Prop. 1.5.4 and Thm. 2.3.1].

Lemma 6.5. Let (ρ, Φ) be a finite dimensional representation of Gw defined over
k and let k k denote a norm on Φ. Let q ∈ Φ.
(1) If hU, si ⊂ {g ∈ Gw : ρ(g)q ∈ k · q}, then kρ(sn )qk ≥ kqk for all n ≥ 1.
(2) If ρ(s)q = q and ρ(U )q = q, then ρ(H + )q = q.
Proof. The argument is similar to the one given in [Sh95, Lemma 5.2].
First note that since U is a unipotent subgroup of G our assumption in (1) implies
that ρ(U )q = q, and that ρ(s)q = χ(s)q for some k-character χ.
Assume the contrary to (1). Then since s acts by a character, we get that
lim ρ(sn )q = 0.

(6.7)

n→∞

Let Φ− denote the subspace of Φ corresponding to the negative weights of the action
of ρ(s) on Φ. Therefore, (6.7) is to say q ∈ Φ− .
We claim (6.7) implies the following
ρ(H + )q ⊂ Φ− .

(6.8)

Let us assume (6.8) and conclude the proof first. Indeed (6.8) implies that
Ψ := k-linear span of {ρ(H + )q} ⊂ Φ− .
Let ̺ : H + → GL(Ψ) be the corresponding representation. Now by Lemma 6.6
there is some n0 so that sn0 ∈ H + . Hence
|det(̺(sn0 ))| 6= 1.
This contradicts the fact that H + is a generated by k-unipotent subgroups and
finishes the proof of (1).
We now turn to the proof of (6.8). Recall that q is fixed by U, therefore, (6.8)
follows if we show that
−
ρ(WH
(s)) ⊂ Φ− .
−
Let w ∈ WH
(s) be arbitrary. Then sn ws−n → e as n → ∞. Using this and (6.7)
we get that
lim ρ(sn )ρ(w)q = lim ρ(sn ws−n )ρ(sn )q = 0.
n→∞

n→∞

−
Hence, ρ(w)q ∈ Φ− for all w ∈ WH
(s) as we wanted to show.

We now prove (2). The proof is similar to the above. Decompose Φ according to
the weights of the element s ∈ S. Hence
Φ = Φ− + Φ0 + Φ+ .
We claim that
(6.9)

−
ρ(WH
(s)ZH (s))q ⊂ Φ− + Φ0

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

27

−
To see this, let wz ∈ WH
(s)ZH (s) be arbitrary. Then sn ws−n → e as n → ∞ and
sz = zs. Since ρ(s)q = q we get that

lim ρ(sn )ρ(wz)q = lim ρ(sn ws−n )ρ(z)ρ(sn )q = ρ(z)q.

n→∞

n→∞

−
Hence, ρ(wz)q ∈ Φ + Φ for all wz ∈ WH
(s)ZH (s) as we wanted to show.
−

0

Our assumption that ρ(U )q = q together with (6.9) now implies that
−
+
ρ(WH
(s)ZH (s)WH
(s))q ⊂ Φ− + Φ0 .

+
−
0
This is to say ρ ◦ ι W−
H (s)(k)ZH (s)(k)WH (s)(k) q ⊂ Φ + Φ . In view of (2.1) we
thus get

ρ(H + )q ⊂ ρ(H)q = ρ ι(H(k)) q ⊂ Φ− + Φ0 .

As above, define Ψ to be the k-span of {ρ(H)q}. Note that Ψ ⊂ Φ− + Φ0 . Let (̺, Ψ)
denote the corresponding representation of H on Ψ. Let n0 be so that sn0 ∈ H + .
Since H + is generated by kw -unipotent subgroups we get that
|det(̺(sn0 ))| = 1,
which implies that Ψ ⊂ Φ0 .

Let now p ∈ Ψ be any vector. For any compact subset B ⊂ U there is a compact
subset B ′ ⊂ Ψ so that ̺(B)p ⊂ B ′ . Since Ψ ⊂ Φ0 , we get
(6.10)

̺(snn0 Bs−nn0 )p = ̺(snn0 )̺(B)p ⊂ B ′ .

Letting n → ∞ we get from (6.10) that ρ(U )p ⊂ B ′ is contained in a compact
subset of Ψ. Note, however, that U is a k-split, unipotent, k-subgroup, therefore
̺(U )p = p.
Hence, U is in the kernel of ̺. Since the kernel is a normal subgroup of H + we get
from Lemma 6.6 that H + ⊂ ker(̺) which implies (2).

Lemma 6.6. The only normal subgroup of H + which contains U is H + . Moreover,
H/H + is a compact and torsion group.
Proof. Note that H(k) = H′ (k ′′ ) and s ∈ ι(H(k)). Therefore,

H + = ι H′ (k ′′ )+ .

Since H′ is k ′′ -almost simple and k ′′ -isotropic, it follows from [Mar90b, Ch. I,
Prop. 1.5.4, Thms. 1.5.6, and Thm. 2.3.1] that the claims in the lemma hold for
H′ (k ′′ )+ . Hence, they hold for H + .


Let F denote the Zariski closure of the k-closed group F (s). By Lemma 2.3 we have
F is a k-algebraic subgroup of G and F (s) = F(k). It is worth mentioning that F is
not necessarily connected we let F = F◦ (k). Using the adjoint action of s we have
Lie(F) = Lie(U − ) ⊕ Lie(F ∩ ZG (s)) ⊕ Lie(U) ⊂ Lie(G).
Recall from the introduction that the product map
(6.11)

U − × (F ∩ ZG (s)) × U → F

is a diffeomorphism onto a Zariski open dense subset which contains the identity.

28

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Let us fix a norm k k on Lie(G). Put Φ = ∧dim F Lie(G), ρ = ∧dim F Ad, and let
q ∈ k · ∧dim F Lie(F) be a nonzero vector. Then
F (s) ⊂ {g ∈ G : ρ(g)q ∈ k · q}.

(6.12)

Moreover, since U is a unipotent subgroup ρ(U)q = q. Recall now that U ⊂ U,
therefore,
(6.13)

ρ(U )q = q.

Lemma 6.7. We have
(6.14)

α(s, U) ≥ α(s−1 , U − )

where the function α denotes the modulus of the conjugation action, see §2.6.
Proof. We first note that in view of relations between the Haar measure and algebraic form of top degree, see [Bour, 10.1.6] and [Oe84, Thm. 2.4], the claim in the
lemma is equivalent to the fact that
(6.15)

|det(Ad(s))|Lie(U ) |w ≥ |det(Ad(s−1 ))|Lie(U − ) |w .

In view of the definition of ρ and q, (6.15) follows if we show kρ(s)qk ≥ kqk. The
latter holds thanks to Lemma 6.5(1) in view of (6.12) and (6.13).

6.5. Entropy argument and the conclusion of the proof. The following theorem is proved in [MT94]; it serves as one of the main ingredients in the proof of
the measure classification theorem in [MT94].
Given an element s from class A which acts ergodically on a probability measure
space (X, σ), let hσ (s) denote the measure theoretic entropy of s.
Theorem 6.8 (Cf. [MT94], Theorem 9.7). Assume s is an element from class A
which acts ergodically on a measure space (X, σ). Let V be an algebraic subgroup of
WG− (s) which is normalized by s and put α = α(s−1 , V ).
(1) If σ is V -invariant, then hσ (s) ≥ log α.
(2) Assume that there exists a subset Ω ⊂ X with σ(Ω) = 1 such that for every
x ∈ Ω we have WG− (s)x ∩ Ω ⊂ V x. Then hσ (s) ≤ log(α) and the equality
holds if and only if σ is V -invariant.
Proof of Theorem 1.1. Let µ be as in the statement of Theorem 1.1. First note
that µ is s-ergodic. Indeed by Mautner phenomenon any s-ergodic component of µ
is U -invariant. Therefore, s-ergodicity of µ follows its U -ergodicity.
Let U be as in (6.1), i.e. U is the maximal subgroup of WG+ (s) which leaves µ
invariant. We complete the proof in some steps.
Step 1. µ is invariant under U − .
By Corollary 6.3, there exists a full measure subset Ω ⊂ X such that
WG− (s)x ∩ Ω ⊂ U − x for every x ∈ Ω.
Recall that µ is s-ergodic. Applying Theorem 6.8, we get that
(6.16)

log α(s, U) ≤ hµ (s) = hµ (s−1 ) ≤ log α(s−1 , U − ).

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

29

Note, however, that by Lemma 6.7 we have
α(s, U) ≥ α(s−1 , U − ).
Therefore, the equality must hold in (6.16).
Now another application of Theorem 6.8(2) implies that µ is invariant under U − .
Step 2. Reduction to Zariski dense measures.
We now apply Lemma 2.5 with B = hU − , S, Ui and M = G. Hence, we get a
connected kT -subgroup G′ ⊂ G and a point gΓ = x ∈ X such that B ⊂ G′
and µ(G′ x) = 1 where G′ := G′ (kT ). Moreover, for every proper kT -closed subset
D ( G′ we have µ(π(D)) = 0, and G′ ∩ gΓg −1 is Zariski dense in G′ .
Abusing the notation we let V (resp. V − ) denote an S-invaraint cross section for U
(resp. U − ) in WG+′ (s) (resp. WG−′ (s)).
Step 3. µ is invariant under WG−′ (s).
We will show U − = WG−′ (s) which implies the claim in view of Step 1.
Assume the contrary. Then V − 6= {e}. The definition of U − , see (6.2), implies that
V − ZG′ (s)WG+′ (s) 6⊂ NG′ (U).
In particular,


D′ := ZG′ (s)WG+′ (s) ∪ NG′ (U) ∩ V − ZG′ (s)WG+′ (s)

is a proper kT -closed subset of V − ZG′ (s)WG+′ (s).

This together with Step 1 and Step 2 implies that conditions in Lemma 2.6 are
satisfied with M = G′ , B = U − , L = V − ZG′ (s)WG+′ (s), and D = D′ . Therefore, we
get the following from the conclusion of that lemma. If 0 < ε < 1 is small enough
and Ωε is a measurable set with µ(Ωε ) > 1 − ε, then there exists a sequence {gn }
converging to e such that

(6.17)
{gn } ⊂ V − ZG′ (s)WG+′ (s) \ ZG′ (s)WG+′ (s) ∪ NG′ (U) ,
and gn Ωε ∩ Ωε 6= ∅ for all n.

In particular, we have ℓ− (v − (gn )) > −∞ and ℓ− (u− (gn )) = −∞. This contradicts
Proposition 6.3 and shows that U − = WG−′ (s) as was claimed.
Step 4. µ is invariant under WG+′ (s).
We will show U = WG+′ (s) which implies the claim.
Assume the contrary. Then ZG′ (s)U is a proper subvariety of ZG′ (s)WG+′ (s). Apply
Lemma 2.6 with M = G′ , B = WG−′ (s), L = ZG′ (s)WG+′ (s), D = ZG′ (s)U, and Ωε
as in Step 3. Therefore, we find
{gn } ⊂ ZG′ (s)WG+′ (s) \ ZG′ (s)U
so that gn Ωε ∩ Ωε 6= ∅ and gn → e.
We consider two cases.
/ NG′ (U) for all i.
Case 1. Suppose there is a subsequence {gni } such that gni ∈
Abusing the notation, we denote this subsequence by {gn }. Construct the map φ
using this sequence {gn }. Then, µ is invariant by hU, Im(φ)i. On the other hand by

30

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Proposition 6.2(2) we have Im(φ) ⊂ WG+′ (s) \ U. This contradicts the maximality
of U and finishes the proof.
Case 2. Suppose there exists some n0 so that gn ∈ NG′ (U) for all n ≥ n0 . Taking
n ≥ n0 , we assume that gn ∈ NG′ (U) for all n. Now by Remark 5.8, µ is invariant
under gn for all n. Write
gn = z(gn )v(gn )u(gn ) ∈ ZG′ (s)VU.
Since µ is U-invariant, the above implies that µ is invariant under z(gn )v(gn ).
Moreover, in view of the choice of gn we have v(gn ) 6= e. Recall also that µ is
s-invariant. Therefore, µ is invariant under
sℓ z(gn )v(gn )s−ℓ = z(gn )sℓ v(gn )s−ℓ .
for all ℓ ∈ Z. For each n choose ℓn ∈ Z so that
vn = sℓn v(gn )s−ℓn ∈ (V ∩ B0 ) \ B−1 ;
such ℓn exists since v(gn ) 6= e and the cross section V is S-invariant. Now passing
to a subsequence, we get that z(gn )vn → v ∈ V and v 6= e. Therefore, µ is invariant
under v. This again contradicts maximality of U.
Step 5. Conclusion of the proof.
So far we have proved that µ(G′ x) = 1 and µ is invariant and ergodic under
G′′ := hWG−′ (s), WG+′ (s)i.
Hence, µ is a probability measure on G′ /G′ ∩ gΓg −1 which is invariant and ergodic
under G′′ . By Lemma 2.1 we have that G′′ is a normal and unimodular subgroup
of G′ , see (2.2) and (2.3). This and Lemma 2.7 now imply Theorem 1.1.

We now give a refinement of Theorem 1.1.
Theorem 6.9. Let the notation and the assumptions be as in Theorem 1.1. Then,
there exist
Q
(1) lT = v∈T lv ⊂ kT where lv = kv if v 6= w and lw = (k ′ )q for some q = pn ,
moreover, q depends only on the weights of the action of S by conjugation,
(2) a connected lT -subgroup F of RkT /lT (G) so that F(lT ) ∩ Γ is Zariski dense
in F,
(3) a point x = g0 Γ ∈ X,
such that µ is the Σ-invariant probability Haar measure on the closed orbit Σx with
Σ = g0 F + (λ)(F(lT ) ∩ Γ)g0−1
where
• the closure is with respect to the Hausdorff topology, and
• F + (λ) is defined in (2.2) for a non central lT -homomorphism λ : Gm → F.
Moreover, g0 F(lT )Γ is the smallest set of the form M(lT )Γ where M is an lT subvariety so that µ(M(lT )Γ/Γ) > 0.
Proof. Indeed the above assertions are proved in the course of the proof of Theorem 1.1. We give a more detailed discussion here for the sake of completeness.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

31

In view of the discussion in the beginning of §6, we have the following.
(a) There is some lT ⊂ kT as in (1) so that the group U which is defined in (6.1)
is an lT -split, unipotent, lT -subgroup of RkT /lT (G).
We thus replace G with RkT /lT (G) and have SU ⊂ G, see §6.1. By Step 2 in the
proof of Theorem 1.1 we have the following.
(b) There is a connected lT -subgroup, E, of minimal dimension such that
SU ⊂ E := E(lT )
and a point x = g0 Γ ∈ X with the following properties. µ is a probability
measure on E/E ∩ g0 Γg0−1 , moreover, E ∩ g0 Γg0−1 is Zariski dense in E.
By Step 5 in the proof of Theorem 1.1 we have the following.
(c) µ is the Σ-ergodic invariant measure on the closed orbit the closed orbit
Σx, where
Σ = E ′ (E ∩ g0 Γg0−1 ).
with E ′ := hWE+ (s), WE− (s)i.
Note that in view of (b) above g0−1 Eg0 ∩ Γ is Zariski dense in g0−1 Eg0 . Therefore,
F := g0−1 Eg0 ⊂ RkT /lT (G),
lT as in (a), and g0 as in (b) satisfy the claims in the theorem.
The final claim follows from Lemma 2.5 and Step 2 in the proof of Theorem 1.1. 
Corollary 6.10. The conclusion of Theorem 6.9 holds in the setting of Theorem 1.2.
We conclude this section with the following lemma.
Lemma 6.11. Let the notation be as in Theorem 6.9 and its proof. Assume further
that there is an lT -representation (ρ, Φ) and a vector q ∈ Φ so that
(6.18)

E = {g ∈ RkT /lT (G) : ρ(g)q = q}.

Then, µ is invariant under H + .
Proof. Indeed we need to show that H + ⊂ Σ. Note first that since SU ⊂ E, we
have that
ρ(SU)q = q.
This together with Lemma 6.5(2) and the fact that U ⊂ U implies ρ(H + )q = q. In
view of (6.18) we thus get that H + ⊂ E. Hence,
−
WH
(s) ⊂ WE− (s) ⊂ E ′ .
−
+
Therefore, H + = hWH
(s), WH
(s)i ⊂ Σ as was claimed.



32

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

7. The arithmetic case
In this section we provide a more detailed description of the groups F and Σ in
Theorem 6.9 in the arithmetic setting.
We begin by fixing some notation. Let K be a global function field of characteristic
p > 0. Let G be a connected, simply connected, semisimple group defined over K.
Suppose T is a finite set of places of K, and put
Y
G=
Gv ,
v∈T

where Gv := G(Kv ) and Kv is the completion of K at v. Denote by OT the ring
of T -integers in K and let Γ be a finite index subgroup of G(OT ).
Q
Put kT = v∈T Kv . We use the notation in Theorem 6.9, in particular, lT ⊂ kT
and lv = Kv for all v 6= w.
Let F ⊂ RkT /lT (G) be as in Theorem 6.9(1) and (2), hence
F(lT ) ⊂ RkT /lT (G)(lT ) = G(kT ).

Let M denote the connected component of the identity in the Zariski closure of
F(lT ) ∩ Γ in G. Then M is a K-subgroup of G, see [Sp98, Lemma 11.2.4(ii)]. In
particular, M is a Kv -subgroup of G for all v ∈ T .
Recall from Theorem 6.9(2) and Lemma 3.1(2) that F(lT ) ⊂ M and that M equals
the connected component of the identity in the Zariski closure of F(lT ) in G.
The following is the standing assumption in this section.
(M) M is a connected, simply connected, semisimple group defined over K.
Moreover, either char(K) > 3 or if char(K) = 2, 3, then all of the absolutely
almost simple factors of M are of type A.
Assuming (M) above, our goal is to describe the structure of the group

(7.1)
D′ := F + (λ) F(lT ) ∩ Γ ,
where the closure is with respect to the Hausdorff topology.

Let {M∗i : 1 ≤ i ≤ r} denote the K-almost simple factors of M. Since M is simply
connected, see (M), we have
r
Y
M∗i .
(7.2)
M=
i=1

Therefore, for all 1 ≤ i ≤ r, there exists a separable extension Ki /K and a connected, simply connected, absolutely almost simple, Ki -group, M̌i , so that
M∗i = RKi /K (M̌i ).
Qbi σi,j
M̌i where {σi,j } are different
In particular, M̌i is naturally identified with j=1
Galois embeddings of Ki into the separable closure of K, see [Mar90b, Ch. 1, §1.7].
Theorem 7.1 (Special case). Assume that M is K-simple, i.e. M = RK1 /K (M̌1 ).
Moreover, assume that T = {w}. Then, there exist
(1) an infinite subfield K ′ ⊂ K1 ,

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

33

(2) a connected, simply connected, absolutely almost simple, K ′ -group, E′ , and
(3) a K1 -isomorphism f : E′ ×K ′ K1 → M̌1 ,
so that
D′ Γ = f (E′ (A))Γ
where A is the closure of K ′ in K1 ⊗K Kw .
Theorem 7.1 is a special case of the following more general statement. To state this
result we need some more notation.
We will work with (commutative) semisimple rings, Υ = ⊕j Υj , where Υj is a field
for each j. By a unital, semisimple, subring of Υ we always mean a subring with
the same multiplicative identity element.
`
By a linear algebraic group, B, over Υ = ⊕j Υj we mean B = j Bj where each Bj
is a linear algebraic group over Υj . The adjoint representation of B on Lie(B) =
⊕j Lie(Bj ) is the direct sum of the adjoint
representations of Bj on Lie(Bj ), and the
Q
group of Υ-points of B is B(Υ) = j Bj (Υj ). Similarly, other notions are defined
fiberwise.
Theorem 7.2 (The general case). Let the notation be as in (7.2) and Theorem 6.9.
Then, there exist
′

(1) a unital, semisimple, subring ⊕rα=1 Kα′ ⊂ ⊕ri=1 Ki ,
(2) a nonempty subset C ⊂ {1, . . . , r′ } and a subset J ⊂ {1, . . . , r} so that
⊕C Kα′ is a unital semisimple subring of ⊕J Kj ,
(3) a fiberwise connected,
simply connected, absolutely almost simple, ⊕α∈C Kα′ `
′
group, E = α∈C Eα , and
`
(4) a ⊕J Kj -isomorphism f : E′ ×⊕C Kα′ ⊕J Kj → J M̌j ,

so that

D′ Γ = f (E′ (A))Γ


where A is the closure of ⊕C Kα′ in ⊕J Kj ⊗K (⊕T Kv ) .

The proof of Theorem 7.2 occupies the rest of this section. Let us briefly outline
the strategy. First, we describe the structure of the group F. This is done using
the classification results in [CGP10]; we recalled what is required (and tailored it
for our application) in §3, see Theorem 3.3 and Lemma 7.3. In the next step, we
use a result of Pink, [P98], to provide a global model, see Lemma 7.6. In the third
and final step, we use strong approximation theorem and the fact that D′ ∩ Γ is a
lattice in D′ in order to tie the results from Lemma 7.3 and Lemma 7.6 together
and finish the proof. Let us now turn to the details of the argument.
Set F′ := [F, F] to be the commutator subgroup of F.
Since lv = Kv for all v 6= w, the definition of M implies that

(7.3)

Fv = F′v = M for all v 6= w.

Lemma 7.3. Put k := Kw and l := lw . The groups Fw , F′w , and Mw satisfy the
conclusions of Lemma 3.1, Lemma 3.2, and Theorem 3.3. In particular, we have

34

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

(a) there is a subfield l ⊂ l′ ⊂ k with k/l′ a separable extension, and l′ /l a
purely inseparable extension,
(b) there is some m ≥ 1 and for all 1 ≤ i ≤ m, there is a field l ⊂ li ⊂ l′ , in
particular, li /l is a purely inseparable extension,
(c) for all 1 ≤ i ≤ m, there is an li -simple, connected, simply connected, li group Li , and
Qm
(d) there is an isomorphism ι : i=1 Li ×li l′ → L, where L is the irreducible
component of the identity in the (fiberwise) Zariski closure of F(lT ) ∩ Γ in
Rk/l (M),
so that the following hold.

Q
(1) F′w (l) = ι m
i=1 Li (li ) ,
(2) Fw (l)/F′w (l) is a compact, abelian group.
Proof. Recall that Γ is diagonally embedded in G(kT ). By the definition, F is the
fiberwise Zariski closure of a subgroup B ⊂ Γ. Recall also from Theorem 6.9(2)
that F is connected. Moreover, M is defined to be the connected component of the
identity in the Zariski closure of F(lT ) ∩ Γ in G which satisfies the conditions in §3
by our assumption (M). The claims thus follow.

Corollary 7.4. Let the notation be as in Lemma 7.3; we add the subscript w to
′
emphasize the place w. Let ∆′w denote
` the projection of F (lT ) ∩ Γ to Gw . Then
−1
′
ι (∆w ) is fiberwise Zariski dense in i Li,w .


Q
Proof. Recall from Lemma 7.3(1) that F′ (l) = ι i Li (li,w ) . Let P denote the
`
(fiberwise) Zariski closure of ι−1 (∆′w ) in i Li,w . Then ∆w ⊂ ι(P).

Since L is semisimple and equals is the irreducible component of the identity in the
(fiberwise) Zariski closure of F(lT ) ∩ Γ in RKw /lw (M), we get that ∆′w is Zariski
dense in L. The claim thus follows in view of the fact that ι is an isomorphism. 
Since M is simply connected, we can write M =
simple for all 1 ≤ i ≤ nv . Therefore,

Qnv

i=1

Mi,v where Mi,v is Kv -almost

Mi,v = RKi,v /Kv (M†i,v )
where Ki,v /Kv is a finite separable extension and M†i,v is an absolutely almost
simple Ki,v -group, see [Mar90b, Ch. 1, §1.7], for all 1 ≤ i ≤ nv .
Similarly let us write
Qdw ∗
′
• Lw = i=1
Li,w , product of lw
-simple factors. Then L∗i,w = Rl′i,w /l′w (Ľi,w )
′
′
′
where li,w /lw is separable and Ľi,w is an absolutely almost simple li,w
-group.
†
†
• For any 1 ≤ i ≤ m = mw , we write Li,w = Rl† /li,w (Li,w ) where li,w /li,w
i,w

†
-group.
is separable and L†i,w is an absolutely almost simple li,w
†
= li′ = Ki,v ,
For all v 6= w, we have lv = lv′ = Kv and we put Li,v = L∗i,v = Mi,v , li,v
L†i,v = Ľi,v = M†i,v , and mv = dv = nv .

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

Lemma 7.5.

35

′
v
v
Ki,v and
lj,v
is a unital, semisimple, subring of ⊕ni=1
(1) ⊕dj=1
dv
a

Ľj,v ×⊕dv

j=1

′
j=1 lj,v

v
Ki,v
⊕ni=1

`nv

is isomorphic to i=1 M†i,v .
(2) There is a partition {1, . . . , dv } = J1,v ∪ · · · ∪ Jmv ,v so that ι = (ιi ), and
for every 1 ≤ i ≤ mv we have
Y
ιi : Rl† /li,v (L†i,v ) ×li,v lv′ →
Rl′j,v /l′v (Ľj,v )
i,v

Ji

is an isomorphism, see Lemma 7.3. In particular, for all 1 ≤ i ≤ mv and
all j ∈ Ji,v we have
†
′
,
⊂ lj,v
(a) li,v
† ′
′
, and
(b) the composite field li,v
lv equals lj,v
†
′
(c) Li,v ×li,v lj,v is isomorphic to Ľj,v .
nv
v †
(3) The isomorphism ι gives rise to an embedding of ⊕m
j=1 lj,v into ⊕i=1 Ki,v
as unital, semisimple, rings. More explicitly, this embedding is given as
follows. Let B be the total ring of quotients of the ring generated by


tr(ρ1 (g)), . . . , tr(ρnv (g)) : g ∈ Fv (lv ) ,
where ρi is the unique non-trivial subquotient of the adjoint representation
ad
v
of M†i,v
for each 1 ≤ i ≤ nv . Then, B = ⊕m
j=1 Bj and
(

†
Bj
if char(lj,v
), L†j,v 6= (2, SL2 )
†

lj,v :=
.
†
{c : c2 ∈ Bj } if char(lj,v
), L†j,v = (2, SL2 )

Proof. The lemma in the case v 6= w is clear, we thus assume v = w.
Using the transitivity of the restriction of scalars functor, we have
RKw /l′w (M) =

nw
Y

RKi,w /l′w (Mi,w ).

i=1

Now by the definition of Lw , see Lemma 7.3(d), we have the restriction to Lw ×l′w Kw
of the natural projection
q : RKw /l′w (M) ×l′w Kw → M
′
is a surjection. Since Kw /lw
is a separable extension part (1) follows.

The first claim in part (2) is a direct corollary of Lemma 7.3(c) and the above
definitions. The other claims follow from the uniqueness of presentation as the
restriction of scalars, see e.g. [CGP10, Prop. A.5.14].
To see part (3), first recall that F′w (lw ) is Zariski dense in M. The claim thus follows
follows from [Mar90b, Ch. I, Cor. 1.4.8] and Lemma 7.3(1).

We now use the K-structure of M in order to provide a global model.

36

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

Recall from (7.2) that {M∗i : 1 ≤ i ≤ r} denotes the K-almost simple factors of M
and
r
r
Y
Y
RKi /K (M̌i ).
M∗i =
M=
i=1

i=1

Since K ⊂ Kv for all v, we have M∗i is a semisimple, simply connected, Kv -group
for all 1 ≤ i ≤ r and all v ∈ T . Therefore, there exists a partition {1, . . . , nv } =
I1,v ∪ · · · ∪ Ir,v so that for all 1 ≤ j ≤ r we have
Y
Y
R(Kj )v′ /Kv (M̌j ×Kj (Kj )v′ )
RKi,v /Kv (M†i,v ) =
(7.4)
M∗j ×K Kv =
i∈Ij,v

v ′ |v

′

where (Kj )v′ is the completion of Kj at v . In particular, for all i ∈ Ij,v we have
Ki,v = Kj Kv = (Kj )v′ for some v ′ |v, and M†i,v is isomorphic to M̌j ×Kj (Kj )v′ .
Put
Λ := M(kT ) ∩ Γ ⊂ M(OT ).

(7.5)

Then, Λ is a finite index subgroup of M(OT ), in particular, it is a lattice in M(kT ).
¯ and ∆
¯ ′ be the images of ∆ and
Define ∆ := F(lT ) ∩ Λ and ∆′ := F′ (lT ) ∩ Λ. Let ∆
′
ad
′
ad
¯ is Zariski dense in M .
∆ in M , respectively. Then, ∆
Let ∆w be the projection of ∆ to Fw (lw ) and put
(7.6)

∆′w := the projection of ∆′ to F′w (lw ).

¯ v and ∆
¯ ′v
Similarly, define ∆v and ∆′v for all v ∈ T . Using Lemma 7.3(d), we let ∆

Q
ad
m
†
†
v
denote the images of ∆v and ∆′v in the adjoint group j=1
Lj,v (lj,v ).

Lemma 7.6. There exists a unital, semisimple, subring
′

⊕rα=1 Kα′ ⊂ ⊕ri=1 Ki ,
so that the following hold.
(1) There exists a fiberwise absolutely almost simple, connected, simply con`r′
′
nected, ⊕rα=1 Kα′ -group E = α=1 Eα , and an isomorphism
ad

ψ:E

×

′
′
⊕rα=1 Kα

⊕ri=1 Ki

→

r
a

M̌ad
i

i=1


¯ ′ ⊂ ψ Ead (⊕r′ Kα′ ) .
so that ∆
α=1
′
nv
v †
(2) For every v ∈ T , we have ⊕rα=1 Kα′ ⊂ ⊕m
j=1 lj,v ⊂ ⊕i=1 Ki,v .
(3) For every v ∈ T , there is an isomorphism
φv : Ead ×⊕r′

′
α=1 Kα

v †
⊕m
j=1 lj,v →

mv
a

L†j,v

j=1

ad



′
¯ ′v ⊂ φv Ead (⊕rα=1
Kα′ ) where ι is the isomorphism introduced
so that ι−1 ∆
in Lemma 7.3(d).
¯ ′ = {γ1 , γ2 , . . .} and for each s ≥ 1 define the subgroup
Proof. Write ∆
¯ ′ := hγ1 , . . . , γs i.
∆
s

¯ ′s,v for all v ∈ T .
Similar to the above discussion, define ∆

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

37

¯ ′ is Zariski dense in Mad , and that by Corollary 7.4, we have ∆
¯ ′ is
Recall that ∆
v
`mv † ad
(fiberwise) Zariski dense in j=1 Lj,v . Therefore, there exists some s0 so that

¯ ′ is Zariski dense in `mv L† ad , and also ∆
¯ ′ is Zariski
for all s ≥ s0 we have ∆
s,v

j=1

j,v

s

dense in Mad . Throughout the proof we always assume that s ≥ s0 .

¯ ′ satisfies [P98, Assump. 2.1]. Therefore, by [P98,
For all s ≥ s0 , the group ∆
s
Thm. 3.6] we have the following. There exist

(a) a unital, semisimple, subring Υs ⊂ ⊕ri=1 Ki ,
(b) a fiberwise absolutely almost simple, connected, simply connected, Υs group Es ,
`r
ad
r
with nowhere vanishing
(c) an isogeny φs : Ead
s ×Υs ⊕i=1 Ki →
i=1 M̌i
derivative,
¯ ′ ⊂ φs (Ead (Υs )). Moreover, Υs is unique and (Es , φs ) is unique up to a
so that ∆
s
s
unique Υs -isomorphism.
We first note that in view of our assumption in small characteristics, see (M), it
follows from [P98, Thm. 1.7(b)] that the isogeny φs in (c) is an isomorphism.
Let Υ′s denote the total ring of quotients of


¯′ .
tr(ρ1 (γ)), . . . , tr(ρr (γ)) : γ ∈ ∆
s

where ρi is the unique non-trivial subrepresentation of the adjoint representation
rs′
′
of M̌ad
i . Write Υs = ⊕α=1 Bs,α .
r′

′
s
Ks,α
where
Then by [P98, Prop. 3.10], we have Υs = ⊕α=1
(

′
), Es,α 6= (2, SL2 )
Bs,α
if char(Ks,α
′

(7.7)
Ks,α
:=
.
′
), Es,α = (2, SL2 )
{c : c2 ∈ Bs,α } if char(Ks,α

We get from (7.7) and (a) that Υs ⊂ Υs+1 ⊂ · · · ⊂ ⊕ri=1 Ki . Therefore, there is
some s1 ≥ s0 so that for all s ≥ s1 we have Υ := Υs = Υs+1 . Let s ≥ s1 for the
rest of the argument.
′

Let us write Υ = ⊕rα=1 Kα′ . We claim that there exists a fiberwise absolutely almost
`r′
simple, connected, simply connected, Υ-group, E = α=1 Eα , and an isomorphism
(7.8)

ψ : Ead ×Υ ⊕ri=1 Ki →

r
a

M̌ad
i

i=1

so that (Es , φs ) is uniquely isomorphic to (E, ψ) for all s ≥ s1 . To see this, note
¯ ′ . Hence, (Es , φs ) is uniquely
that (Υ, Es+1 , φs+1 ) satisfies (a), (b), and (c) for ∆
s
isomorphic to (Es+1 , φs+1 ) which implies the assertion with (E, ψ) := (Es1 , φs1 ).
We now claim that
(7.9)

¯ ′ ⊂ ψ(Ead (Υ)) = ψ
∆

Qr′

α=1


Eα (Kα′ ) .

To see this, note that for all s ≥ s1 we have φs = ψ ◦ fs where fs is a unique
ad
ad
¯′
Υ-isomorphism between Ead
s and E . Also recall that ∆s ⊂ φs (Es (Υ)) for all s.
Since fs is an Υ-isomorphism, we have

ad
fs Ead
s (Υ) = E (Υ)

38

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY


¯ ′ ⊂ ψ Ead (Υ) for all s, and (7.9) follows. Thus, we have
for all s; we get that ∆
s
′
shown that part (1) holds for Υ = ⊕rα=1 Kα′ and (E, ψ).
′

We now show that part (2) in the lemma also holds for Υ = ⊕rα=1 Kα′ . By
Lemma 7.3, for v = w, and by the definition, with ι = id otherwise, we have
m
m


Y
Y

†
L†j,v (lj,v
)
Rl† /li,v (L†i,v ) (li,v ) = ι
(7.10)
[Fv , Fv ](l) = F′v (l) = ι
j=1

Therefore,

Υ′s

i,v

j=1

is contained in the total ring of quotients of the ring generated by


tr(ρ1 (g)), . . . , tr(ρnv (g)) : g ∈ F′v (l)

v
Ki,v , where ρi is the unique, non-trivial, subquotient of the adjoint reprein ⊕ni=1
sentation of (M†i )ad , see [Mar90b, Ch. I, Cor. 1.4.8]. This and Lemma 7.5(3) imply
part (2) in view of (7.4) and (7.7).

We now turn to the proof of (3). Note that in view of (7.4), the isomorphism ψ
in (7.8) extends to an isomorphism
v
Ki,v →
ψv : Ead ×Υ ⊕ni=1

nv
a

M†i

i=1

for all v ∈ T .

ad

,

Let ι be as in Lemma 7.3(d), for v = w, and be the identity, otherwise. Then
Lemma 7.3(d), Lemma 7.5, and the definitions imply that ι induces an isomorphism
ῑv :

mv
a

(L†j,v )ad

× ⊕ mv l†

j=1 j,v

j=1

v
Ki,v
⊕ni=1

→

nv
a

i=1

M†i

ad

.

We claim that
φv := (ῑv )−1 ◦ ψv

(7.11)
satisfies part (3) in the lemma.

First note that the above definitions imply that
φv = (ῑv )−1 ◦ ψv : Ead ×⊕r′

′
α=1 Kα

v
Ki,v →
⊕ni=1

mv
a

v
Ki,v
(L†j,v )ad ×⊕mv l† ⊕ni=1

j=1

j=1 j,v

v
Ki,v -algebraic groups. Therefore, and in view of part (2),
is an isomorphism of ⊕ni=1
v †
part (3) will follow if we show that φv is defined over ⊕m
j=1 lj,v .

To see this, first note that
′

¯ ′v ) ⊂ Ead (⊕rα=1 Kα′ ),
ψv−1 (∆
¯ ′v ) is Zariski dense in Ead . The definitions, thus, imply that
and ψv−1 (∆

′
¯ ′v ) = φv ◦ ψv−1 (∆
¯ ′v ) ⊂ φv Ead (⊕rα=1
(7.12)
(ῑv )−1 (∆
Kα′ )

v †
⊂ φv Ead (⊕m
j=1 lj,v ) ,
where we used part (2) in the last inclusion.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

39

We now recall from Corollary 7.4 that
¯ ′v ) ⊂
(ῑv )−1 (∆

mv
Y

L†j,v

j=1

` mv

†
ad
j=1 (Lj,v ) .

is fiberwise Zariski dense in
v †
Zariski dense in Ead ×Υ ⊕m
j=1 lj,v .

ad

†
(lj,v
)

v †
Moreover, Ead (⊕m
j=1 lj,v ) is fiberwise

†
Therefore, (7.12) implies that φv is fiberwise defined over lj,v
, completing the proof
of part (3) and the lemma.


By Lemma 7.6, there is a partition {1, . . . , mv } = J1,v ∪ · · · ∪ Jr′ ,v so that Kα′ is
Q
†
. In particular, we have ∆′ = α ∆′α ,
a unital, semisimple, subring of ⊕j∈Jα,v lj,v
ψ = (ψα )α , φv = (φα,v )α , etc.
Abusing the notation, we let • = (•v )T for • = ι, ψ, φ, etc.
Corollary 7.7.

†
is identified with
(1) The closure of Kα′ in ⊕v∈T ⊕j∈Jα,v lj,v
u
⊕u∈Tα ⊕sb=1
k̃α,b,u

where Tα is a finite set of places in Kα′ .
¯ ′α ), and put
(2) Let 1 ≤ α ≤ r′ . Set Ω′α := ψα−1 (∆


Ωα := π −1 Ω′α ∩ π Eα (Kα′ ) ,

where π : Eα → Ead
α is the covering map. Then, Ωα is an arithmetic lattice
Q Q
in u b Eα (k̃α,b,u ).
(3) ∆′ is a finite index subgroup of ∆.
Proof. Part (1) is a consequence of the definitions.
We now turn to the proof of part (2). First note that since φα,v is an isomorphism,
we have
Y

Y † ad †
†
(7.13)
φα,v
Ead
Lj,v (lj,v ).
α (lj,v ) =
j∈Jα

j∈Jα

In view of (7.6) and Lemma 7.3(1), thus, we have
Y Y
Y

Y
ad † 
† 
¯ ′α =
L†j,v (lj,v
∆
φα,v
) ∩Λ =
Ead
∩ Λ,
α (lj,v )
T j∈Jα

T

j∈Jα

The above, part (1), and (7.9) imply that
su
Y
Y

¯ ′α =
∆
∩ Λ,
φα,v
Ead
α (k̃α,b,u )
Tα

b=1

where we used the embedding in part (1).

Recall now that Λ has finite index in M(OT ). These observations, together with
the fact that ψα is induced from a ⊕ri=1 Ki -isomorphism, see (7.8), imply that Ω′α
is commensurable with Ead
α (OTα ). Part (2) thus follows, see e.g. [BP89, §1].
To see part (3) we argue as in the proof of Theorem
Qr3.3(2). Indeed, for every
γ ∈ ∆, conjugation by γ defines an automorphism of i=1 M(Ki ) which preserves

40

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY


Qr′
′
ψ α=1 Ead
α (Kα ) . The claim thus follows from part (2) and the fact that arithmetic
groups have finite index in their normalizer, see e.g. [BP89, §1].

Let the notation be as in Lemma 7.3 and Lemma 7.6; let φ̃v , etc. denote the
canonical lift of φv , etc. to the simply connected covering group. Define
Y
Y † †  YY
†
(7.14)
L†α :=
φ̃−1
Lj,v (lj,v ) =
Eα (lj,v
),
α,v
v∈T

†

and let L :=
We also put

Q

j∈Jα,v

T Jα,v

†
α Lα .

Eα† :=
and write E † :=

Q

α

su
Y Y

Eα (k̃α,b,u ),

u∈Tα b=1

Eα† .

Let λ be the cocharacter of Fw which appears in Theorem 6.9. We get from the
discussion in §3, see (3.1), that F + (λ) ⊂ F′w (lw ).
Recall from (7.1) that D′ = F + (λ)∆. Put D′′ := F + (λ)∆′ ⊂ F′ (lT ) and define


†
−1
′′
−1
′′
(7.15)
D := φ̃−1
v ◦ ιv (D ) v∈T = ψ̃v (D ) v∈T ⊂ L .
Q ′
Lemma 7.8. Let Ω := rα=1 Ωα . Then Ω ⊂ D is a lattice in D.

Proof. First note that Ω ⊂ D follows from the definitions. Now, by Theorem 6.9,
we have ∆ is a lattice in D′ ; therefore, ∆′ is a lattice in D′′ by Corollary 7.7(3).
This, together with the fact that the above maps are topological isomorphisms,
implies the claim.

In the rest of this section, we will use Corollary 7.7 and Lemma 7.8 to finish the
proof of Theorem 7.2.
Define C ⊂ {1, . . . , r′ } to be the following subset.
(7.16)

α ∈ C if and only if the projection of λ(Gm ) to L†α is noncentral.

Recall that Eα is a connected, simply connected, absolutely almost simple, Kα′ group for all 1 ≤ α ≤ r′ . Then, [Mar90b, Ch. 1, Prop. 1.5.4 and Thm. 2.3.1] and
Lemma 7.3(1) imply that for every α ∈ C there exists a (maximal) nonempty subset
Jα (λ) ⊂ Jα,w so that
Y Y
†
Eα (lj,w
) ⊂ D.
(7.17)
α∈C j∈Jα (λ)

The following argument applies to any α ∈ C and any j ∈ Jα (λ); for the sake of an
explicit exposition, however, we assume 1 ∈ Jα (λ) for some α ∈ C and work with
this α and j = 1.
†
Fix Q, a compact open subgroup of L† . Since Eα (l1,w
) ⊂ D is a normal subgroup of
†
†
†
L and Q is compact and open in L , we get that Eα (l1,w
)Q is an open and closed
†
subgroup of L .

†
†
Lemma 7.9. Eα (l1,w
)Q ∩ Ω is a lattice in Eα (l1,w
)Q.

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

41

†
Proof. Put R := Eα (l1,w
) and let Q′ := Q∩D. Then Q′ is a compact open subgroup
of D. Recall that RQ is an open and closed subgroup of L† . Put R′ := RQ ∩ D.
Then R ⊂ R′ and R′ is an open and closed subgroup of D. We first show that
Σ := R′ ∩ Ω is a lattice in R′ .

Recall from Lemma 7.8 that Ω is a lattice in D. Since R′ is an open subgroup of
D we have R′ Ω is an open and closed subset of D. Therefore, the orbit map is a
homeomorphism from R′ Ω/Ω onto its image in D/Ω equipped with the subspace
topology. Now the restriction of the D-invariant measure on D/Ω to R′ Ω/Ω pulls
back to the R′ -invariant measure on R′ /Σ. Hence, Σ is a lattice in R′ as we claimed.
Since RQ and R′ are unimodular groups, RQ/R′ has an RQ-invariant measure.
Moreover, RQ/R′ is compact, hence, RQ/R′ has a finite RQ-invariant measure.
We showed in the previous paragraph that Σ is a lattice in R′ . Therefore, Σ is a
lattice in RQ, see e.g. [Rag72, Lemma 1.6]. The claim now follows since (RQ) ∩ Ω
is discrete and contains Σ.

†
Let p : L† → Eα (l1,w
) be the natural projection.

†
†
Corollary 7.10. p (Eα (l1,w
)Q) ∩ Ω is a lattice in Eα (l1,w
).

†
†
Proof. The map p : Eα (l1,w
)Q → Eα (l1,w
) is surjective and has compact kernel. Therefore, p is a surjective, proper map. This and Lemma 7.9 imply that
†
†
p (Eα (l1,w
)Q) ∩ Ω is a lattice in Eα (l1,w
).


Recall from Corollary 7.7(1) that

†
u
.
⊕Tα ⊕sb=1
k̃α,b,u ⊂ ⊕T ⊕Jα,v lj,v

Therefore, for every v ∈ T and any j ∈ Jα,v , there exist some u(v) = uα (v) ∈ Tα ,
some 1 ≤ b(j) = bα,v (j) ≤ su(v) , and a (continuous) embedding of fields
†
τα,j,v : k̃α,b(j),u(v) → lj,v
.

Lemma 7.11. Recall that we fixed one α ∈ C and assumed that 1 ∈ Jα (λ). Let us
write uα (w) = u0 ; without loss of generality we
 may assume bα,w (1) = 1. Suppose
v ∈ T and j ∈ Jα,v are so that uα (v), bα,v (j) = (u0 , 1). Then v = w and j = 1.

Proof. Let S ⊂ T be the set of all v so that the equation

uα (v), bα,v (j) = (u0 , 1)

′
has a solution if and only if v ∈ S. For all v ∈ S, let Jα,v
⊂ Jα,v be the set of all
′
j so that bα,v (j) = 1 if and only if j ∈ Jα,v . We need to show that S = {w} and
′
that Jα,w
= {1}.

†
†
By Lemma 7.9, we have Eα (l1,w
)Q ∩ Ω is a lattice in Eα (l1,w
)Q.

†
Recall that Ω ⊂ E † . Moreover, Eα (l1,w
) is not compact since λ is non-central.

†
†
Hence, Eα (l1,w )Q ∩ E is unbounded.

Therefore, there exists a sequence {cn } ⊂ k̃α,1,u0 so that {τα,1,w (cn )} is unbounded,
′
but, {τα,j,v (cn )} is bounded for all w 6= v ∈ S and all j ∈ Jα,v
, and for all 1 6=

42

AMIR MOHAMMADI AND ALIREZA SALEHI GOLSEFIDY

′
j ∈ Jα,w
when v = w. This contradicts the fact that {τα,j,v }’s are continuous, and
finishes the proof.

†
.
Lemma 7.12. With the notation as in Lemma 7.11, we have k̃α,1,u0 = l1,w

Proof. Recall from Corollary 7.7(1) that
†
u
,
⊕Tα ⊕sb=1
k̃α,b,u ⊂ ⊕T ⊕Jα,v lj,v
†
u
is a finitely generated ⊕Tα ⊕sb=1
k̃α,b,u -module.
and that ⊕T ⊕Jα,v lj,v
†
The projection p is fiberwise defined over lj,v
. Put

†
P := p E .

†
Then, by [BZ76, A.2] we get P is a constructible subgroup of Eα (l1,w
). Hence, P
†
is a closed subgroup of Eα (l1,w ).

Recall that Ω ⊂ E † ; therefore,


†
p (Eα (l1,w
)Q) ∩ Ω ⊂ P.


†
†
Moreover, by Corollary 7.10, we have p (Eα (l1,w
)Q) ∩ Ω is a lattice in Eα (l1,w
).

†
All together, we get P is a non-discrete, closed subgroup of Eα (l1,w
) with finite covolume, see [Rag72, Lemma 1.6]. In view of this and the fact that Eα is a connected,
absolutely almost simple, simply connected, Kα′ -group, we get from [Mar90b, Ch. 1,
Thm. 2.3.1 and Ch. 2, Thm. 5.1] that
†
P = Eα (l1,w
).

(7.18)

Using the definition of p, we thus get the following.
ker(p)Eα (k̃α,1,u0 ) = ker(p)E †
=L
=

†

Lemma 7.11
(7.18)

†
ker(p)Eα (l1,w
).

†
†
, see e.g. [Mar90b,
) which implies k̃α,1,u0 = l1,w
Therefore, Eα (k̃α,1,u0 ) = Eα (l1,w
Ch. 1, Prop. 2.5.5].


A similar argument as in Lemma 7.11 and Lemma 7.12 implies the following.
Lemma 7.13. Let α ∈ C.


(1) Let j0 ∈ Jα (λ). Then uα (v), bα,v (j) = uα (w), bα,w (j0 ) if and only if
v = w and j = j0 .
(2) There is a subset Iα (λ) ⊂ {1, . . . , suα (w) } and a bijection
bα : Jα (λ) → Iα (λ)
†
so that lj,w
= k̃α,bα (j),uα (w) for all j ∈ Jα (λ).

` ′
′
Proof of Theorem 7.2.
• Let ⊕rα=1 Kα′ and E = rα=1 Eα be as in Lemma 7.6.
• Let C be `
as in (7.16), and define J as in Theorem 7.2(2) for this C.
• Let E′ = α∈C Eα .

MEASURE RIGIDITY FOR SOLVABLE GROUP ACTION

43

u
• Let A = ⊕α∈C Aα where Aα = ⊕Tα ⊕sb=1
k̃α,b,u is as in Corollary 7.7(1).

Then, ψ in Lemma 7.6(2) induces a ⊕J Ki -isomorphism, f , between E′ ×⊕C Kα′ ⊕J Ki

`
and J M̌i . Moreover, A is the closure of ⊕C Kα′ in ⊕i∈J Ki ⊗K (⊕T Kv ) .

By Lemma 7.13, for all α ∈ C, we have

†
Eα (lj,w
) = Eα (k̃α,bα (j),uα (w) ).

Therefore, in view of the strong approximation theorem [Mar90b, Ch. 2, Thm. 6.7]
and Corollary 7.7 we get that
Y
Y
D=
Eα†
Ωα ,
α∈C

α6∈C

see (7.17). This and (7.15) finish the proof.



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Mathematics Dept, University of California, San Diego, CA 92093-0112
E-mail address: ammohammadi@ucsd.edu
Mathematics Dept, University of California, San Diego, CA 92093-0112
E-mail address: golsefidy@ucsd.edu

