The symplectic characteristic polynomial

We introduce the notion of the symplectic characteristic polynomial of an endomorphism of a symplectic vector space. This is a polynomial in two variables and can be considered as a generalization of the characteristic polynomial of the endomorphism in the context of symplectic linear algebra. One of the goal of this paper is to prove that the symplectic characteristic polynomial is a complete symplectic invariant of symplectically diagonalizable endomorphisms.