$\DeclareMathOperator\SL{SL}$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.