Reading Dynamical Systems IV: Symplectic Geometry and its Applications by Arnol'd and Novikov I found the following assertion:
The $n$-dimensional monodromy operator of a Hamiltonian system[..]lies in some torus $T^n \subset Sp(2n, \mathbb R)$. We may consider $T^n$ as the diagonal subgroup of the group of unitary transformations of the space $\mathbb C^n.$
I'm fairly new to the concept of "diagonal subgroup", and looking on the web has not be very enlightening, at least in this case...does this "diagonal subgroup" have anything to do with the diagonalization of symplectic matrices? Any clue would be very appreciated!
A diagonal subgroup refers to a maximal torus, for example $S^1 \times \dots \times S^1 \subseteq U(n)$ as "literal" diagonal matrices of the form $diag(t_1,\dots, t_n)$. In the case of $Sp(n, \mathbb{R})$ we hace that it consits of matrices of the form $$ \left(\begin{array}{cc} a & b\\ c & d \end{array}\right) $$ where $a^Tc =c^Ta, b^Td = d^Tb$ and $a^Td - c^Tb = Id$. We have a natural embedding of $U(n) = Sp(n,\mathbb{R}) \cap O(2n, \mathbb{R}) \subseteq Sp(n,\mathbb{R})$ this gives a natural maximal torus in $Sp(n, \mathbb{R})$ given by matrices of the form $$ \left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) $$ where $x,y$ are diagonal matrices such that the matrix $x + iy= diag(z_1, \dots, z_n)$ satisfies $\lvert z_i\rvert =1$.