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