Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$, and (H4): $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$.
Let $k \in \mathbb{N}$.
(a) Show that $M \circ T^k = (-1)^k T^k \circ M$.
(b) Deduce that $\operatorname{Im}(T^k)$ and $\operatorname{ker}(T^k)$ are stable under $M$.

Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.

Let $f \in \mathcal{H}(U)$. Show that if $f$ is $\mathcal{C}^\infty$ on $U$, then every partial derivative of any order of $f$ belongs to $\mathcal{H}(U)$.

For every integer $k\geq 1$, recall that $$P_k = \{v\in\mathcal{H}\cap V_\mathbb{Q} \text{ such that } kv\in V_\mathbb{Z}\}.$$ Show that the set $P_k$ is invariant under $\Gamma$.

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$ with $\ker(u) = \operatorname{Vect}(U)$, and the matrix $H_t$. We denote by $p : E \rightarrow E$ the orthogonal projection onto $\ker(u)$. Let $t \in \mathbf{R}_+$. Show that $p(H_t X) = p(X)$.

Show that, for all $\rho > 0$ and all $m , n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } ) , \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { m , n } ( \mathbb { R } ) \right)$ are closed under addition.

Show that, for all $\rho > 0$ and all $n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } )$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ are closed under multiplication.

Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.