  The Green function of the spectral ball is constant over the isospectral
varieties, is never less than the pullback of its counterpart on the
symmetrized polydisk, and is equal to it in the generic case where the pole is
a cyclic (non-derogatory) matrix. When the pole is derogatory, the inequality
is always strict, and the difference between the two functions depends on the
order of nilpotence of the strictly upper triangular blocks that appear in the
Jordan decomposition of the pole. In particular, the Green function of the
spectral ball is not symmetric in its arguments. Additionally, some estimates
are given for invariant functions in the symmetrized polydisc, e.g.
(infinitesimal versions of) the Carath\'eodory distance and the Green function,
that show that they are distinct in dimension greater or equal to $3$.
