  We study the structure of minimal parabolic subgroups of the classical
infinite dimensional real simple Lie groups, corresponding to the classical
simple direct limit Lie algebras. This depends on the recently developed
structure of parabolic subgroups and subalgebras that are not necessarily
direct limits of finite dimensional parabolics. We then discuss the use of that
structure theory for the infinite dimensional analog of the classical principal
series representations. We look at the unitary representation theory of the
classical lim--compact groups $U(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in
order to construct the inducing representations, and we indicate some of the
analytic considerations in the actual construction of the induced
representations.

Representation Theory