Subspace of a quotient space is Lagrangian

Question

Let $(V, \omega)$ be a symplectic vector space.
Let $W \subset V$ be coisotropic and let $U \subset V$ be Lagrangian. Show that the quotient space $ ((U \cap W)+W^{\perp})/W^{\perp} \subset W/W^{\perp}$ is a Lagrangian subspace.

My ideas:
I want to show that the subspace is isotropic and coisotropic. I have already managed to show that it is isotropic. But no idea how to tackle coisotropy.

Thanks in advance for any help !

Answer 1

You need to use the facts that for any subspaces $S, T$ of $V$, $(S\cap T)^\perp = S^\perp+T^\perp$ and $(S+T)^\perp = S^\perp\cap T^\perp$. It follows that $$ ((U\cap W)+W^\perp)^\perp = (U^\perp+W^\perp)\cap W = (U^\perp\cap W)+(W^\perp\cap W) = (U\cap W)+W^\perp $$ where we have used the facts that $U^\perp = U$ and $W^\perp\subset W$ in the last equality. Hence $L := (U\cap W)+W^\perp$ is a Lagrangian subspace of $V$, which also satisfies $W^\perp\subset L\subset W$.

Now given any subspace $F$ satisfying $W^\perp\subset F \subset W$, taking the symplectic complement of these inclusions implies that $W^\perp\subset F^\perp \subset W$. By definition of the induced symplectic structure on $W/W^\perp$, we then get that the symplectic complement of $F/W^\perp$ is just $F^\perp/W^\perp$. It follows that $L/W^\perp$ is Lagrangian in $W/W^\perp$.