On the maximal spectral type of nilsystems

Let $(G/\Gamma,R_a)$ be an ergodic $k$-step nilsystem for $k\geq 2$. We adapt an argument of Parry to show that $L^2(G/\Gamma)$ decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host, Kra and Maass for $2$-step nilsystems and a result by Stepin for nilsystems $G/\Gamma$ with connected, simply connected $G$.