Search for a stable complement In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$. Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$. a) Verify that $u$ and $p$ commute. We denote $$\bar{p} = \frac{1}{N}\sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$ b) Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$. c) Deduce that $\bar{p}$ is a projector and that its image is $W$. d) Prove carefully that $\bar{p}$ commutes with $u$ and $h$. e) Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.
\textbf{Search for a stable complement}\\
In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$. Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$.
a) Verify that $u$ and $p$ commute.
We denote
$$\bar{p} = \frac{1}{N}\sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$
b) Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.
c) Deduce that $\bar{p}$ is a projector and that its image is $W$.
d) Prove carefully that $\bar{p}$ commutes with $u$ and $h$.
e) Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.