Journal of Lie Theory, Vol. 11, No. 2, pp. 339-353 (2001)

Direct Limits of Zuckerman Derived Functor Modules

Amber Habib

Stat Math Unit
Indian Statistical Institute
Bangalore Centre
8th Mile, Mysore Road
Bangalore 560059, India

Abstract: 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{\ui}, 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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