Decrease of Fourier coefficients of stationary measures

Let $μ$ be a Borel probability measure on $\mathrm{SL}_2(\mathbb R)$ with a finite exponential moment, and assume that the subgroup $Γ_μ$ generated by the support of $μ$ is Zariski dense. Let $ν$ be the unique $μ-$stationary measure on $\mathbb P^1_{\mathbb R}$. We prove that the Fourier coefficients $\widehatν(k)$ of $ν$ converge to $0$ as $|k|$ tends to infinity. Our proof relies on a generalized renewal theorem for the Cartan projection.