Local Lyapunov spectrum rigidity of nilmanifold automorphisms

We study the regularity of a conjugacy between an Anosov automorphism \begin{document}$ L $\end{document} nilmanifold id="M2">\begin{document}$ N/\Gamma and volume-preserving, id="M3">\begin{document}$ C^1 $\end{document}-small perturbation id="M4">\begin{document}$ f $\end{document}. We say that id="M5">\begin{document}$ is locally Lyapunov spectrum rigid if this id="M6">\begin{document}$ C^{1+} whenever id="M7">\begin{document}$ id="M8">\begin{document}$ has same volume as id="M9">\begin{document}$ For id="M10">\begin{document}$ with simple spectrum, we show local rigidity equivalent to id="M11">\begin{document}$ satisfying both irreducibility condition ordering on its exponents.