Decrease of Fourier coefficients of stationary measures

Let mu be a Borel probability measure on SL2(R) with a finite exponential moment, and assume that the subgroup Gamma(mu) generated by the support of mu is Zariski dense. Let. be the unique mu-stationary measure on P-1. We prove that the Fourier coefficients nu(k) of nu converge to 0 as vertical bar k vertical bar tends to infinity. Our proof relies on a generalized renewal theorem for the Cartan projection.