  We study the regularity of a conjugacy between an Anosov automorphism $L$ of
a nilmanifold $N/\Gamma$ and a volume-preserving, $C^1$-small perturbation $f$.
We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is
$C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as
$L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum
rigidity is equivalent to $L$ satisfying both an irreducibility condition and
an ordering condition on its Lyapunov exponents.
