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$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel probability measure on the homogeneous space $G/\Gamma$ that is left-invariant w.r.t $G$. I wonder where I can find the proof that the action of $G$ is ergodic?

If this is true in more general settings say when $G$ is a simple Lie group as in the $G=\SL(n,\mathbb R)$ case, please let me know.

Many comments below mentioned the Howe-Moore theorem. But I am aware of the fact (as a corollary of Howe-Moore) that every unbounded subgroup of $G$ also acts ergodically/mixingly after we proved that $G$ itself acts ergodically.

$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel probability measure on the homogeneous space $G/\Gamma$ that is left-invariant w.r.t $G$. I wonder where I can find the proof that the action of $G$ is ergodic?

If this is true in more general settings say when $G$ is a simple Lie group as in the $G=\SL(n,\mathbb R)$ case, please let me know.

$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel probability measure on the homogeneous space $G/\Gamma$ that is left-invariant w.r.t $G$. I wonder where I can find the proof that the action of $G$ is ergodic?

If this is true in more general settings say when $G$ is a simple Lie group as in the $G=\SL(n,\mathbb R)$ case, please let me know.

Many comments below mentioned the Howe-Moore theorem. But I am aware of the fact (as a corollary of Howe-Moore) that every unbounded subgroup of $G$ also acts ergodically/mixingly after we proved that $G$ itself acts ergodically.

\DeclareMathOperator
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Ergodicity of the action of $SL$\operatorname{SL}(n,\mathbb R)$ on $SL$\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$

Let$\DeclareMathOperator\SL{SL}$Let $G:=SL(n,\mathbb R)$$G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma:=SL(n,\mathbb Z)$$\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel probability measure on the homogeneous space $G/\Gamma$ that is left-invariant w.r.t $G$. I wonder where I can find the proof that the action of $SL(n,\mathbb R)$$G$ is ergodic?

If this is true in more general settings say when $G$ is a simple Lie group as in the $G=SL(n,\mathbb R)$$G=\SL(n,\mathbb R)$ case, please let me know.

Ergodicity of the action of $SL(n,\mathbb R)$ on $SL(n,\mathbb R)/SL(n,\mathbb Z)$

Let $G:=SL(n,\mathbb R)$ and $\Gamma:=SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel probability measure on the homogeneous space $G/\Gamma$ that is left-invariant w.r.t $G$. I wonder where I can find the proof that the action of $SL(n,\mathbb R)$ is ergodic?

If this is true in more general settings say when $G$ is a simple Lie group as in the $G=SL(n,\mathbb R)$ case, please let me know.

Ergodicity of the action of $\operatorname{SL}(n,\mathbb R)$ on $\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$

$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel probability measure on the homogeneous space $G/\Gamma$ that is left-invariant w.r.t $G$. I wonder where I can find the proof that the action of $G$ is ergodic?

If this is true in more general settings say when $G$ is a simple Lie group as in the $G=\SL(n,\mathbb R)$ case, please let me know.

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