Unitary Highest Weight-Modules of Inductive Limit Lie Algebras and Groups

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Representations of inductive limits of finite-dimensional Lie groups and algebras am our main objects of study. Such representations can be constructed by taking inductive limits of the representations of the finite-dimensional groups or algebras. For (finite dimensional) Lie groups and algebras of Hermitian symmetric type, Enright-Howe-Wallach classified those highest weight representations which are unitarizable. In this paper we consider the limits of the algebras u(p, n), so*(2n) and sp(n, R), the limits of the corresponding maximal compact subgroup and the limits of the corresponding generalized Verma modules. We give necessary and sufficient conditions for the inductive limit representation to be unitarizable. In all three cases we show that a necessary condition for unitarizability is that the highest weight is eventually constant. We also give sufficient conditions (depending on the constant) which will ensure that the limit is unitarizable.