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Journal of Lie Theory 11 (2001), No. 2, 339--353
Copyright Heldermann Verlag 2001

Direct Limits of Zuckerman Derived Functor Modules

Amber Habib
Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, 8th Mile Mysore Road, Bangalore 560059, India


We construct representations of certain direct limit Lie groups $G=\lim G^n$ via direct limits of Zuckerman derived functor modules of the groups $G^n$. We show such direct limits exist when the degree of cohomology can be held constant, and discuss some examples for the groups $Sp(p,\infty)$ and $SO(2p,\infty)$, relating to the discrete series and ladder representations. We show that our examples belong to the ``admissible'' class of Ol'shanski{\u\i}, 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.

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