  In this note we characterize the distinguished boundary of the symmetrized
polydisc and thereby develop a model theory for $\Gamma_n$-isometries along the
lines of \cite{AY}. We further prove that for invariant subspaces of
$\Gamma_n$-isometries, similar to the case $n=2$ \cite{S}, Beurling-Lax-Halmos
type representation holds.
