We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Let $\mathcal { S }$ be the set defined in question 13: $$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$ Show that the set of elements of $\mathcal { S }$ whose characteristic polynomial $P$ has roots of multiplicity at most 2 in $\mathbb { C }$ is dense in $\mathcal { S }$. Hint: You may use $r \left( P ^ { \prime } \right)$ where the map $r$ is defined in question 11.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Let $\mathcal { S }$ be the set defined in question 13:
$$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$
Show that the set of elements of $\mathcal { S }$ whose characteristic polynomial $P$ has roots of multiplicity at most 2 in $\mathbb { C }$ is dense in $\mathcal { S }$.
\textit{Hint: You may use $r \left( P ^ { \prime } \right)$ where the map $r$ is defined in question 11.}