Reciprocal spectrum of a symplectic matrix (${\chi }_M(x)=x^{2m}{\chi }_M(x^{-1})$)

For a symplectic matrix $M$ of size $2m$ with determinant one, the defining identity $M^{T}JM=J$ gives $M^{-1}=J^{-1}{M}^{T}J$. Since similarity and transposition preserve the characteristic polynomial, ${\chi }_{M^{-1}}={\chi }_M$. Factoring $xI-M=M(x{M}^{-1}-I)$ then proves the displayed reciprocal identity for $x\ne 0$. If ${\chi }_M(z)=(z-\lambda {)}^{d}R(z)$ with $R(\lambda )\ne 0$, this identity gives ${\chi }_M(x)=x^{2m-d}(1-\lambda x{)}^{d}R(1/x)$ near $x={\lambda }^{-1}$. Thus reciprocal eigenvalues have the same algebraic multiplicity. The fixed points $\pm 1$ require the stronger parity argument in even multiplicities at fixed points of reciprocal root pairing.