Measure rigidity for solvable group actions in the space of lattices

TL;DR

This paper proves that under certain conditions, all invariant ergodic probability measures for specific solvable group actions on the space of lattices are homogeneous, extending measure rigidity results in homogeneous dynamics.

Contribution

It establishes measure rigidity for a class of solvable group actions on the space of lattices, generalizing previous results to new group configurations.

Findings

All invariant ergodic measures are homogeneous under the given conditions.

The result applies to groups generated by unipotent and diagonalizable elements with specific eigenvalue properties.

The proof extends measure classification techniques in homogeneous dynamics.

Abstract

We study invariant probability measures on the homogeneous space $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ for the action of subgroups of $\mathrm{SL}_n(\mathbb R)$ of the form $SF$ where $F$ is generated by one parameter unipotent groups and $S$ is a one parameter $\mathbb R$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains an element with only one eigenvalue less than one (counted with multiplicity) and others bigger than one we prove that all the $SF$ invariant and ergodic probability measures on $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ are homogeneous.

Equations

$$\{\mathrm{diag}(e^{r_{1}t},\dots ,e^{r_{n-1}t},e^{-t}):t\in \mathbb{R}\}$$

$$\displaystyle\mathbf{V}\cong\mathrm{Res}_{\mathbb{F}/\mathbb{Q}}\big{(}\mathrm{Res}_{\mathbb{D}/\mathbb{F}}\mathbf{W}^{\oplus m}\big{)}$$

$$\hat{\rho }({\mathbf{H}}_{\mathbb{F}})\le {\mathrm{GL}}_k(\mathbb{D}).$$

$$V=\prod_{i=1}^{r+s}V_{i}\quad\mbox{where}\quad V_{i}=\left\{\begin{array}[]{ll}\big{(}{\mathbf{W}}^{\sigma_{i}}_{\mathbb{R}}\big{)}^{\oplus m}&1\leq i\leq r\\ \big{(}{\mathbf{W}}^{\sigma_{i}}\big{)}^{\oplus m}&r+1\leq i\leq r+s\end{array}\right.$$

$$\displaystyle H=\prod_{i=1}^{r+s}H_{i}\quad\mbox{where}\quad H_{i}=\left\{\begin{array}[]{ll}\rho^{\oplus m}(\mathbf{H})^{\sigma_{i}}_{\mathbb{R}}&1\leq i\leq r\\ \rho^{\oplus m}(\mathbf{H})^{\sigma_{i}}&r+1\leq i\leq r+s.\end{array}\right.$$

$$Z_G(H)=G\cap \prod _{i=1}^{r+s}{Z}_i,$$

$$V_i=V_i^{\prime }{\otimes }_{\mathbb{C}}{V}_i^{\prime \prime }$$

$$H_i=\hat{\rho }(\mathbf{H}{)}^{{\sigma }_i}\otimes {\mathrm{I}}_{V_i^{\prime \prime }}$$

$$V_i=V_i^{\prime }{\otimes }_{\mathbb{R}}{V}_i^{\prime \prime }\text{mbox}and{H}_i=\hat{\rho }(\mathbf{H}{)}_{\mathbb{R}}^{{\sigma }_i}\otimes {\mathrm{I}}_{V_i^{\prime \prime }}$$

$$V_i^{\prime }$$

$$D_{i}=\left\{\begin{array}[]{ll}\mathbb{R}&1\leq i\leq l\\ \mathbb{H}&l+1\leq i\leq r\\ \mathbb{C}&r+1\leq i\leq r+s.\end{array}\right.$$

$$Z_i=(\mathbb{C}{\mathrm{I}}_{V_i^{\prime }}{\otimes }_{\mathbb{C}}\text{mbox}{Hom}_{\mathbb{C}}(V_i^{\prime \prime },V_i^{\prime \prime }){)}^{\ast }={\mathrm{I}}_{V_i^{\prime }}\otimes {\mathrm{GL}}_{\mathbb{C}}(V_i^{\prime \prime })$$

$$Z_{i}=\big{(}\mathbb{R}\mathrm{I}_{V_{i}^{\prime}}\otimes_{\mathbb{R}}\mbox{Hom}_{\mathbb{R}}(V_{i}^{\prime\prime},V_{i}^{\prime\prime})\big{)}^{*}=\mathrm{I}_{V_{i}^{\prime}}\otimes\operatorname{GL}_{\mathbb{R}}(V_{i}^{\prime\prime}),$$

$${\mathrm{I}}_{V_i^{\prime }}\otimes b_i$$

$$Z_{i}=\big{(}\mathbb{H}\mathrm{I}_{V_{i}^{\prime}}\otimes_{\mathbb{R}}\mbox{Hom}_{\mathbb{R}}(V_{i}^{\prime\prime})\big{)}^{*}\cong\operatorname{GL}_{md/2}(\mathbb{H}\mathrm{I}_{V_{i}^{\prime}}),$$

$$g_i{c}_i{g}_i^{-1}={\mathrm{I}}_{V_i^{\prime }}\otimes b_i$$

$$h=\prod _{i=1}^{r+s}{h}_i\otimes {\mathrm{I}}_{V_i^{\prime \prime }}\text{mbox}andb=\prod _{i=1}^{r+s}{\mathrm{I}}_{V_i^{\prime }}\otimes b_i$$

$$\mu ={\int }_X{\mu }_x\mathrm{d}\mu (x)$$

$$V\stackrel{\mathrm{def}}{=}\{v\in {\mathbb{R}}^n:hv=v\text{mbox}forallh\in {\mathcal{R}}_u(H)\}$$

Full text

1. introduction

Let $n\geq 2$ be a positive integer. Let $G=\operatorname{SL}_{n}(\mathbb{R}),\Gamma=\operatorname{SL}_{n}(\mathbb{Z})$ and $X=G/\Gamma$. Let $S$ be a one parameter $\mathbb{R}$-diagonalizable subgroup of $G$, i.e. $S$ is conjugate in $G$ to a subgroup with diagonal matrices. Let $F$ be a subgroup of $G$ normalized by $S$ and generated by one parameter unipotent subgroups. This paper is about the classification of $SF$ invariant and ergodic probability measures on $X$.

This type of questions has already been studied earlier by Margulis-Tomanov [4] and Mozes [5] using Ratner’s measure rigidity theorem [7] for $F$ actions. In particular, it is proved in [4] and [5] that if $SF$ is an epimorphic subgroup of $H\leq G$ and $H$ is generated by one parameter unipotent subgroups, then any $SF$ invariant and ergodic probability measure is $H$ invariant and hence homogeneous by Ratner’s theorem. Recently, in a joint paper with Barak Weiss the second named author [8] proved measure rigidity for certain non-epimorphic sovable subgroup actions on $X$ for $n=3$. More precisely, it is proved in [8, Theorem 5] that if $SF$ is nonabelian and $S$ has no nonzero fixed vectors in $\mathbb{R}^{3}$, then $SF$ acts uniquely ergodically on $X$. It is noticed already in [8] that even for $n=4$, there are homogeneous and even non-homogeneous $SF$ invariant and ergodic probability measures under similar assumptions. Therefore, to get rigidity results one needs to put further restrictions on the group $S$.

A one parameter $\mathbb{R}$-diagonalizable subgroup of $G$ is said to be simple if it is conjugate to

[TABLE]

for a probability vector $(r_{1},\ldots,r_{n-1})\in\mathbb{R}^{n-1}$ with positive entries. The following is the main result of this paper.

Theorem 1.1**.**

Let $F$ be a subgroup of $G=\operatorname{SL}_{n}(\mathbb{R})$ $(n\geq 2)$ generated by one parameter unipotent subgroups. Let $S=\{a_{t}:t\in\mathbb{R}\}$ be a simple one parameter $\mathbb{R}$-diagonalizable subgroup of $G$ normalizing $F$. Then any $SF$ invariant and ergodic probability measure on $X=G/\Gamma$ $(\Gamma=\operatorname{SL}_{n}(\mathbb{Z}))$ is homogeneous.

The starting point of the proof of Theorem 1.1 is the same as that of [4] and [5] where $\mu$ is written as a convex comination of $F$ invariant and ergodic probability measures. Note that any $F$ invariant and ergodic probability measure $\nu$ on $X$ is homogeneous by Ratner’s theorem, i.e. there is a closed subgroup $H$ of $G$ and $x_{g}\in X$ (here $x_{g}$ is the coset $g\Gamma$ for $g\in G$) such that $\nu$ is supported on $Hx_{g}$ and is $H$ invariant. We show that $a_{t}\nu$ converges to zero in the space of finite measures on $X$ as $t$ tends to infinity or minus infinity unless $g^{-1}Hg$ is a semisimple algebraic group virtually defined over $\mathbb{Q}$ and its action on $\mathbb{R}^{n}$ has only one $\mathbb{Q}$-isotropic type. The main new ingredient here is the use of Tits [9] on the structure of irreducible representations to show that the centralizer of $H$ in $G$ (denoted by $Z_{G}(H)$) is an abelian group. The proof is then completed by a standard argument.

Acknowledgements

The authors would like to thank MSRI for its hospitality during Spring 2015.

2. Irreducible representations over $\mathbb{Q}$

This section is about the structure of $\mathbb{Q}$-rational representations of semisimple algebraic groups defined over $\mathbb{Q}$. We adopt the convention of [1] to identify algebraic varieties or groups with their rational points over complex numbers and denote them by boldfaced capital letters.

Let $\mathbf{V}$ be an $n$-dimensional vector space over $\mathbb{C}$ endowed with a $\mathbb{Q}$-structure. Let $\mathbf{G}=\operatorname{SL}(\mathbf{V})$ be the special linear group with the natural $\mathbb{Q}$-structure such that linear action of $\mathbf{G}$ on $\mathbf{V}$ is a $\mathbb{Q}$-rational representation. Let $\mathbf{H}$ be a nontrivial connected semisimple subgroup of $\mathbf{G}$ defined over $\mathbb{Q}$. Recall that the linear action of $\mathbf{H}$ on $\mathbf{V}$ is completely determined by the induced Lie algebra representations which is completely reducible over $\mathbb{Q}$ (see [11, Theorem 7.8.11]). So $\mathbf{V}$ is a direct sum of $\mathbf{H}$ invariant $\mathbb{Q}$-irreducible subspaces. We say the linear action of $\mathbf{H}$ on $\mathbf{V}$ has only one $\mathbb{Q}$-isotropic type if all these irreducible subspaces are isomorphic to each other over $\mathbb{Q}$.

For a field $\mathbb{F}$ containing $\mathbb{Q}$ and a variety $\mathbf{X}$ defined over $\mathbb{Q}$, we let $\mathbf{X}_{\mathbb{F}}$ be the set of $\mathbb{F}$-rational points of $\mathbf{X}$. Let $V=\mathbf{V}_{\mathbb{R}}$, $G=\mathbf{G}_{\mathbb{R}}$ and $H=\mathbf{H}_{\mathbb{R}}$. The main result of this section is the following proposition.

Proposition 2.1**.**

Let $S$ be a simple one parameter $\mathbb{R}$-diagonalizable subgroup of $G$. Suppose $S$ is contained in $HZ_{G}(H)$ and the linear action of $\mathbf{H}$ on $\mathbf{V}$ has only one $\mathbb{Q}$-isotropic type. Then $Z_{G}(H)$ is an abelian group.

Proof.

Since the linear action of $\mathbf{H}$ on $\mathbf{V}$ has only one $\mathbb{Q}$-isotropic type, there exists a positive integer $m$ such that $\mathbf{V}$ is isomorphic to the direct sum of $m$ copies of a certain $\mathbb{Q}$-irreducible representation of $\mathbf{H}$. According to [9, Theorem 7.2], there exists a central simple division algebra $\mathbb{D}$ over a number field $\mathbb{F}$, a complex vector space $\mathbf{W}$ defined over $\mathbb{F}$ with $\mathbf{W}_{\mathbb{F}}$ having a right $\mathbb{D}$-module structure and an absolutely irreducible $(\mathbb{D},\mathbb{F})$-representation $\rho:\mathbf{H}\to\operatorname{GL}(\mathbf{W})$ such that

[TABLE]

in the category of $\mathbb{Q}$-rational representations of $\mathbf{H}$. According to Tits’ notation, the morphism $\rho$ is defined over $\mathbb{F}$ and $\rho(\mathbf{H}_{\mathbb{F}})$ acts $\mathbb{D}$-linear on $\mathbf{W}_{\mathbb{F}}$. The notion of absolute irreducibility will be explained later.

Let $d^{2}=[\mathbb{D}:\mathbb{F}]$ and $k=[\mathbf{W}_{\mathbb{F}}:\mathbb{D}]$. We will show that $m=d=1$, which will imply the proposition immediately. We would like to understand the group $H$ and its centralizer by choosing a good basis of $V$ using (2.1). To make the presentation simpler we assume the left and right hand sides of (2.1) are equal. We will give explicit descriptions of $\mathbb{D},\mathbf{W}$ and of restriction of scalars using several isomorphisms.

By choosing an isomorphism $\mathbb{D}\otimes_{\mathbb{F}}\mathbb{C}\cong\mathrm{M}_{d}(\mathbb{C})$ and identifying $\mathbb{D}$ with $\mathbb{D}\otimes 1$ in the natural way we assume that $\mathbb{D}$ is a subring of $\mathrm{M}_{d}(\mathbb{C})$. We assume $\mathbf{W}_{\mathbb{F}}=\mathbb{D}^{k}$ and $\mathbf{W}=\mathrm{M}_{d}(\mathbb{C})^{k}$ where $\mathrm{M}_{d}(\mathbb{C})^{k}$ is naturally identified with the set of $kd$-by-$d$ matrices by viewing its elements blockwise. Therefore, the group of right $\mathrm{M}_{d}(\mathbb{C})$-module automorphisms of $\mathbf{W}$ can be naturally identified with $\operatorname{GL}_{kd}(\mathbb{C})$ which acts on $\mathbf{W}$ from the left. Let $\widehat{\rho}$ be the composite of $\rho$ and this identification, that is, $\widehat{\rho}(\mathbf{H})$ is a subgroup of $\operatorname{GL}_{kd}(\mathbb{C})$. We consider $\widehat{\rho}$ as a morphism of algebraic groups defined over $\mathbb{F}$ with

[TABLE]

We note that the absolute irreducibility of $\rho$ refers to the fact that $\widehat{\rho}(\mathbf{H})$ acts irreducibly on $\mathbb{C}^{kd}$.

Let $\sigma_{1},\ldots,\sigma_{r}$ be the real embeddings of $\mathbb{F}$ and $\sigma_{r+1},\sigma_{r+1}^{\prime}~{},\ldots,\sigma_{r+s},\sigma_{r+s}^{\prime}$ be the pairs of complex embeddings of $\mathbb{F}$. Each $\sigma_{i}$ defines a functor from the category of varieties defined over $\mathbb{F}$ to the category of varieties defined over $\mathbb{F}^{\sigma_{i}}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\{\sigma_{i}(x):x\in\mathbb{F}\}$ (see [1, §0.14.1] for definitions). For a variety $\mathbf{X}$ defined over $\mathbb{F}$ we let $\mathbf{X}^{\sigma_{i}}$ be the image of this functor. Also let $\mathbb{D}^{\sigma_{i}}=\mathbb{D}\otimes_{\mathbb{F}}\mathbb{F}^{\sigma_{i}}$ where the left $\mathbb{F}$-module structure of $\mathbb{F}^{\sigma_{i}}$ comes from the left multiplication of $\sigma_{i}(\cdot)$.

According to (2.1) and [12, Theorem 1.3.2] we have

[TABLE]

and

[TABLE]

Moreover,

[TABLE]

where $Z_{i}$ is the centralizer of $H_{i}$ in $\operatorname{GL}_{\mathbb{R}}(V_{i})$. It suffices to prove that all the $Z_{i}$ are abelian groups.

For $r+1\leq i\leq r+s$, the complex part of the above decomposition is easy to understand. Since $\mathbb{D}^{\sigma_{i}}\otimes_{\mathbb{F}^{\sigma_{i}}}\mathbb{C}=\mathrm{M}_{d}(\mathbb{C})$, we have a decomposition

[TABLE]

where $V_{i}^{\prime}=\mathbb{C}^{kd}$ and $V_{i}^{\prime\prime}=\mathbb{C}^{md}$ such that the group

[TABLE]

where $\mathrm{I}_{V_{i}^{\prime\prime}}$ denotes the identity map of $V_{i}^{\prime\prime}$.

For the real part we first note that the central simple algebra $\mathbb{D}^{\sigma_{i}}\otimes_{{\mathbb{F}}^{\sigma_{i}}}\mathbb{R}$ is equal to $\mathrm{M}_{d}(\mathbb{R})$ or $\mathrm{M}_{d/2}(\mathbb{H})$ where in the latter case $\mathbb{H}$ is the Hamilton quaternions and $d$ has to be an odd number. By choosing an embedding $\mathbb{H}\to\mathrm{M}_{2}(\mathbb{C})$ we assume that $\mathrm{M}_{d/2}(\mathbb{H})$ is a subring of $\mathrm{M}_{d}(\mathbb{C})$. Suppose for $1\leq i\leq l$ we are in the case of $\mathrm{M}_{d}(\mathbb{R})$ and for $l+1\leq i\leq r$ we are in the case of $\mathrm{M}_{d/2}(\mathbb{H})$. Then

[TABLE]

where

[TABLE]

For $1\leq i\leq r$ (resp. $r+1\leq i\leq r+s$) let $R_{i}$ be the ring of $\mathbb{R}$-linear spans of $\widehat{\rho}(\mathbf{H})^{\sigma_{i}}_{\mathbb{R}}$ (resp. $\widehat{\rho}(\mathbf{H})^{\sigma_{i}}$) in the ring of $\mathbb{R}$-linear transformations of $V_{i}^{\prime}$. Since $\widehat{\rho}(\mathbf{H})$ acts irreducibly on $\mathbb{C}^{kd}$, the space $V_{i}^{\prime}$ is an irreducible $R_{i}$-module. For every $i$ there is a division algebra $D_{i}$ such that the opposite division algebra $D_{i}^{\circ}$ is isomorphic to set of $R_{i}$-module morphisms of $V_{i}^{\prime}$. By Wedderburn-Artin theorem (see e.g. [3, §1]), the ring $R_{i}$ is isomorphic to a matrix ring over $D_{i}$ and $V_{i}$ is isomorphic to a free $D_{i}$ module of finite rank. Since $\widehat{\rho}(\mathbf{H})$ acts irreducibly on $\mathbb{C}^{kd}$, we have

[TABLE]

We can understand $Z_{i}$ using (2.7), (2.5) and (2.6). For $r+1\leq i\leq r+s$,

[TABLE]

where $(\cdot)^{*}$ denotes the invertible elements of a ring. Moreover, if $c_{i}\in Z_{i}$ is diagonalizable over $\mathbb{R}$, then we have a decomposition $c_{i}=\mathrm{I}_{V_{i}^{\prime}}\otimes b_{i}$ where $b_{i}$ is an $\mathbb{R}$-diagonal element of $\operatorname{GL}_{\mathbb{C}}(V_{i}^{\prime\prime})$ (here we view $\operatorname{GL}_{\mathbb{C}}(V_{i}^{\prime\prime})$ as the real points of an algebraic group coming from the restriction of scalars). For $1\leq i\leq l$

[TABLE]

and every $\mathbb{R}$-diagonalizable $c_{i}\in Z_{i}$ can be decomposed as

[TABLE]

where $b_{i}\in\operatorname{GL}_{\mathbb{R}}(V_{i}^{\prime\prime})$ is $\mathbb{R}$-diagonalizable. For $l+1\leq i\leq r$

[TABLE]

where $\mathbb{H}\mathrm{I}_{V_{i}^{\prime}}$ acts on $V_{i}^{\prime}$ via right multiplications of matrices. Suppose $c_{i}\in Z_{i}$ is $\mathbb{R}$-diagonalizable, then there exists $g_{i}\in Z_{i}$ such that

[TABLE]

where $b_{i}\in\operatorname{GL}_{\mathbb{R}}(V_{i}^{\prime\prime})$ is $\mathbb{R}$-diagonalizable.

Now we turn to the proof of the conclusion. Since $S$ is contained in $HZ_{G}(H)$, the group $S$ is the direct product $S_{1}S_{2}$ of one parameter $\mathbb{R}$-diagonalizable subgroups $S_{1}\leq H$ and $S_{2}\leq Z_{G}(H)$. Since $S$ is simple, there exists $a\in S$ such that the linear action of $a$ on $\mathbb{R}^{n}$ has only one eigenvalue $\lambda<1$ counted with multiplicity and all the other eigenvalues are bigger than $1$. We write $a=hb$ where $h\in S_{1}$ and $b\in S_{2}$. According to the discussions previously, by possibly replacing $S$ by $gSg^{-1}$ for some $g\in\prod_{i=1}^{r+s}Z_{i}$ we have

[TABLE]

where all the $h_{i}$ and $b_{i}$ are $\mathbb{R}$-diagonalizable elements of $\operatorname{GL}_{\mathbb{R}}(V_{i}^{\prime})$ and $\operatorname{GL}_{\mathbb{R}}(V_{i}^{\prime\prime})$ respectively.

There exists $i_{0}$ with $1\leq i_{0}\leq r+s$ such that $\lambda$ is an eigenvalue of $a$ restricted to $V_{i_{0}}$. Since each eigenvalue of $a$ restricted to $V_{i}$ has multiplicity at least $4$ (resp. $2$) for $l<i\leq r$ (resp. $r+1\leq i\leq r+s$), we have $i_{0}\leq l$. We assume without loss of generality that $i_{0}=1$. Since $\det h_{i}=1$ for all $i$, $\lambda$ has multiplicity $1$, all the other eigenvalues of $a$ are bigger than $1$ and (2.8), we have $\det b_{i}>1$ for $i>1$. Since $\det a=1$ we have $0<\det b_{1}\leq 1$ (with equality in case $\mathbb{F}=\mathbb{Q}$). This and the simplicity of $\lambda$ implies that $\min\{kd,md\}=1$. Therefore $d=1$. Since $\mathbf{H}$ is a semisimple algebraic group, we have $k>1$ and hence $m=1$. Using previously discussed structures of $Z_{G}(H)$ and $m=d=1$, we have $l=r$ and $Z_{G}(H)$ is an abelian group.

∎

3. measure rigidity

The aim of this section is to prove Theorem 1.1. Let $G,\Gamma,X,S,F$ be as in Theorem 1.1. For every $g\in G$ we use $x_{g}$ to denote the coset $g\Gamma\in X$. Also, we use $x_{\Gamma}$ to denote the coset $\Gamma\in X$. We endow $G$ and $\mathbb{R}^{n}$ with $\mathbb{Q}$-structures such that their $\mathbb{Q}$-rational points are $\operatorname{SL}_{n}(\mathbb{Q})$ and $\mathbb{Q}^{n}$ respectively. A subgroup $H$ of $G$ is said to be virtually defined over $\mathbb{Q}$ if it has finite index in the group of real points of a $\mathbb{Q}$-rational algebraic group. The following is a version of Ratner’s theorem [7], see also the work [2, Prop. 1.1] of Borel-Prasad and [10, Theorem 2] of Tomanov.

Theorem 3.1**.**

Let $\nu$ be an $F$ invariant and ergodic probability measure on $X$. Then there exists a connected subgroup $H$ of $G$ virtually defined over $\mathbb{Q}$ with semisimple Levi factor and a point $x_{g}\in X$ such that the measure $\nu$ is invariant under the action of $gHg^{-1}\geq F$ and $\nu$ is concentrated on the closed subset $gHx_{\Gamma}$.

Lemma 3.2**.**

Suppose there is a proper $\mathbb{Q}$-rational subspace $V$ of $\mathbb{R}^{n}$ invariant under $SH$ where $H$ is a subgroup of $G$ normalized by $S$ . Then either $\{a_{t}hx_{\Gamma}:t\geq 0\}$ is divergent for every $h\in H$ or $\{a_{t}hx_{\Gamma}:t\leq 0\}$ is divergent for every $h\in H$.

Proof.

By assumption, there exists an integer $k$ with $1\leq k\leq n-1$ and a basis $v_{1},\ldots,v_{k}$ of $V$ consisting of rational vectors. Since the space $V$ is $SH$ invariant, the vector $w=v_{1}\wedge\cdots\wedge v_{k}\in\wedge^{k}\mathbb{Q}^{n}$ is an eigenvector of $S$. By assumption the one parameter group $S$ is simple, so there exists a nonzero real number $\lambda$ such that $a_{t}w=e^{\lambda t}w$. Therefore $a_{t}w\to 0$ as $t\to\infty$ or $t\to-\infty$. We assume without loss of generality that $a_{t}w\to 0$ as $t\to\infty$. Note that for any $h\in H$ one has $hw=sw$ for some nonzero real number $s$, which implies that $a_{t}hw\to 0$ as $t\to\infty$. It follows from the Mahler’s compactness criterion (see e.g. [13, Propositon 3.1]) that the trajectory $\{a_{t}hx_{\Gamma}:t\geq 0\}$ is divergent for every $h\in H$. ∎

Proof of Theorem 1.1.

Let $\mu$ be an $SF$ invariant and ergodic probability measure on $X$. It follows from Poincare recurrence that for almost every $x\in X$

1. (1)

Neither of the trajectories $\{a_{t}x:t\geq 0\}$ or $\{a_{t}x:t\leq 0\}$ is divergent.

Let

[TABLE]

be the ergodic decomposition of $\mu$ into $F$ invariant and ergodic components. According to [4] and [5] we can find a connected subgroup $H$ of $G$ and a measurable subset $X^{\prime}$ of $X$ with full $\mu$ measure such that for every $x\in X^{\prime}$ the following properties hold:

1. (2)

$\mu_{x}$ is $H$ invariant and supported on the closed subset $Hx$. 2. (3)

$S\leq N_{G}^{1}(H)^{\circ}$ where $N_{G}^{1}(H)$ consists of $g\in N_{G}(H)$ with the property that the conjugation by $g$ preserves the Haar measure of $H$. Moreover, $\mu(N_{G}^{1}(H)^{\circ}x)=1$.

By possibly removing a measure zero subset from $X^{\prime}$, we assume that (1) holds for all $x\in X^{\prime}$.

According to (2) and Theorem 3.1 there is a connected subgroup $H^{\prime}$ of $G$ virtually defined over $\mathbb{Q}$ with semisimple Levi factor and $g\in G$ with $x_{g}\in X^{\prime}$ such that $H=gH^{\prime}g^{-1}$. By replacing $S$ by $g^{-1}Sg$, $F$ by $g^{-1}Fg$ and $\mu$ by $g^{-1}\mu$ we can without loss of generality assume that $H=H^{\prime}$ and $x_{\Gamma}\in X^{\prime}$.

Let $\mathscr{R}_{u}(H)$ be the maximal unipotent normal subgroup of $H$. Then $H$ has a connected semisimple subgroup $H_{0}$ virtually defined over $\mathbb{Q}$ such that $H=H_{0}\ltimes\mathscr{R}_{u}(H)$. Note that $\Gamma\cap H_{0}$ and $\Gamma\cap\mathscr{R}_{u}(H)$ are lattices in $H_{0}$ and $\mathscr{R}_{u}(H)$ respectively. As a connected unipotent group, the space of $\mathscr{R}_{u}(H)$ invariant vectors

[TABLE]

is nonzero and defined over $\mathbb{Q}$. Since $S$ normalizes $H$, the space $V$ is also $S$-invariant. If $\mathscr{R}_{u}(H)$ is nontrivial, then $V$ is a proper subspace of $\mathbb{R}^{n}$. Hence Lemma 3.2 implies that either $\{a_{t}x_{\Gamma}:t\geq 0\}$ or $\{a_{t}x_{\Gamma}:t\leq 0\}$ is divergent. This contradicts (1). Therefore $\mathscr{R}_{u}(H)$ is trivial and $H=H_{0}$ is semisimple.

Let $Z=Z_{G}(H)$. Note that $N_{G}(H)$ and $HZ$ have the same connected component, see [6]. So by (3) we have $S\leq HZ$. The natural representation of $H$ on $\mathbb{R}^{n}$ is defined over $\mathbb{Q}$ and is completely reducible over $\mathbb{Q}$. Since $S$ is contained in $HZ$, every maximal $\mathbb{Q}$-rational H-invariant subspace which has only one $\mathbb{Q}$-isotropic type is $S$-invariant. By Lemma 3.2 and (1) the representation of $H$ on $\mathbb{R}^{n}$ has only one $\mathbb{Q}$-isotropic type. Therefore Proposition 2.1 implies that $Z$ is an abelian group. In view of (3) we get $\mu(ZHx)=1$. By pulling $\mu$ back to $ZH/\Gamma_{1}$ where $\Gamma_{1}=ZH\cap\Gamma$ we get an $SF$-invariant and ergodic probability measure $\mu_{1}$ on $ZH/\Gamma_{1}$. Note that $SF$ is generated by Ad-unipotent one parameter subgroups of $ZH$. It follows from Ratner’s theorem that $\mu_{1}$ is a homogeneous measure. Therefore $\mu$ is homogeneous.

∎