Suppose $\psi$ a symmetric symplectic matrix(identify to symplectic map in $\mathbb{R}^{2n}$), and $V$ a Lagrangian subspace in $\mathbb{R}^{2n}$. Do we have $\psi(V)=V$?
I'm trying to prove the product property of maslov index for Lagrangian space(base on the definition in the Introduction to Symplectic Topology by Mcduff and Salamon). In explicitly:
$\mu(\psi(t) V(t))=2\mu(\psi(t))+\mu(V(t))$
Here $\psi(t)$ a loop of symplectic map, $V(t)$ a loop in Lagrangian subspace, $\mu(X)$ means maslov index of $X$.
Now I reduce this problem into the linear algebra problem.But I have no idea to prove it.
Any reference about this property is also welcome.
Thanks in advance.