Let
H
o
m
0
(
Γ
,
G
)
\mathsf {Hom}^{0}(\Gamma ,G)
be the connected component of the identity of the variety of representations of a finitely generated nilpotent group
Γ
\Gamma
into a connected compact Lie group
G
G
, and let
X
0
(
Γ
,
G
)
\mathsf {X}^0(\Gamma ,G)
be the corresponding moduli space. We show that there exists a natural
O
u
t
(
Γ
)
\mathsf {Out}(\Gamma )
-invariant measure on
X
0
(
Γ
,
G
)
\mathsf {X}^0(\Gamma ,G)
and that whenever
O
u
t
(
Γ
)
\mathsf {Out}(\Gamma )
has at least one hyperbolic element, the action of
O
u
t
(
Γ
)
\mathsf {Out}(\Gamma )
on
X
0
(
Γ
,
G
)
\mathsf {X}^0(\Gamma , G)
is mixing with respect to this measure.