ON THE MAXIMAL SPECTRAL TYPE OF NILSYSTEMS

Let (G/Γ, Ra) be an ergodic k-step nilsystem for k ≥ 2. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that L2 (G/Γ) 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–Maass [J. Anal. Math. 124 (2014), pp. 261–295] for 2-step nilsystems and a result by Stepin [Uspehi Mat. Nauk 24 (1969), pp. 241–242] for nilsystems G/Γ with connected, simply connected G.