  We study representations of the classical infinite dimensional real simple
Lie groups $G$ induced from factor representations of minimal parabolic
subgroups $P$. This makes strong use of the recently developed structure theory
for those parabolic subgroups and subalgebras. In general parabolics in the
infinite dimensional classical Lie groups are are somewhat more complicated
than in the finite dimensional case, and are not direct limits of finite
dimensional parabolics. We extend their structure theory and use it for the
infinite dimensional analog of the classical principal series representations.
In order to do this we examine two types of conditions on $P$: the flag-closed
condition and minimality. We use some riemannian symmetric space theory to
prove that if $P$ is flag-closed then any maximal lim-compact subgroup $K$ of
$G$ is transitive on $G /P$\,. When $P$ is minimal we prove that it is
amenable, and we use properties of amenable groups to induce unitary
representations $\tau$ of $P$ up to continuous representations $\Ind_P^G(\tau)$
of $G$ on complete locally convex topological vector spaces. When $P$ is both
minimal and flag-closed we have a decomposition $P = MAN$ similar to that of
the finite dimensional case, and we show how this gives $K$--spectrum
information $\Ind_P^G(\tau)|_K = \Ind_M^K(\tau|_M)$.
