Direct Limits of Zuckerman Derived Functor Modules
We construct representations of certain direct limit Lie groups G=lim Gn via direct limits of Zuckerman derived functor modules of the groups Gn. We show such direct limits exist when the degree of cohomology can be held constant, and discuss some examples for the groups Sp(p,) and SO(2p,), relating to the discrete series and ladder representations. We show that our examples belong to the "admissible" class of Ol'shanskii, and also discuss the globalizations of the Harish-Chandra modules obtained by the derived functor construction. The representations constructed here are the first ones in cohomology of non-zero degree for direct limits of non-compact Lie groups.