Properties of $h$ In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and the complex number $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$. a) Prove that $h$ is diagonalizable. b) Let $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \operatorname{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$. c) Calculate, for $k$ relative integer, $h^k \circ u \circ h^{-k}$ and, for $l$ natural integer, $h \circ u^l \circ h^{-1}$.
\textbf{Properties of $h$}\\
In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and the complex number $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$.
a) Prove that $h$ is diagonalizable.
b) Let $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \operatorname{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$.
c) Calculate, for $k$ relative integer, $h^k \circ u \circ h^{-k}$ and, for $l$ natural integer, $h \circ u^l \circ h^{-1}$.