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Let $G$ be a connected semisimple Lie group without compact factor, $\mu$ be a Borel probability measure on $G$ such that the group generated by $\mathrm{supp}(\mu)$ is Zariski dense in $G$. For simplicity, we assume that $\mathrm{supp}(\mu)$ is compact. Let $G=KAN$ be an Iwasawa decomposition, $P=MAN$ be the minimal parabolic subgroup. Then there exists a unique $\mu$-stationary probability measure $\nu$ on the Furstenberg boundary $G/P$. There also exists a Borel map $\xi:G^\mathbb{N}\to G/P$ such that for $\mu^{\otimes\mathbb{N}}$-a.e. $\omega=(g_1,g_2,\ldots)\in G^\mathbb{N}$, the sequence of measures $(g_1\cdots g_n)_*\nu$ converges weakly to the Dirac measure $\delta_{\xi(\omega)}$.

Now consider an irreducible representation $\rho:G\to GL(V)$, where $V$ is a finite-dimensional real vector space. Let $\chi:A\to(0,\infty)$ be the highest weight for this representation, and let $V_\chi\subset V$ be the weight subspace for $\chi$. Note that $PV_\chi=V_\chi$. I am interested in the following result: For $\mu^{\otimes\mathbb{N}}$-a.e. $\omega=(g_1,g_2,\ldots)\in G^\mathbb{N}$, the limit of any convergent subsequence of $$\frac{\rho(g_1\ldots g_n)}{\|\rho(g_1\ldots g_n)\|}$$ has image $\xi(\omega)V_\chi$, where $\|T\|=\sup_{\|v\|=1}\|Tv\|$.

This should be a classical result on random walks on semisimple groups. I saw it from the paper of Benoist and Quint But I cannot find a reference for it. Could someone be so kind to indicate a proof or point out a reference? Thanks in advance.

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Dear Ionekite,

I believe the answer to your question is already in Benoist-Quint paper: in the proof of Proposition 5.2 of their article, you see that this result essentially goes back to Furstenberg, Furstenberg and Kesten, and Goldscheid and Margulis.



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