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Complex geometry and the asymptotics of Harish-Chandra modules for real reductive Lie groups. I


Authors: Luis G. Casian and David H. Collingwood
Journal: Trans. Amer. Math. Soc. 300 (1987), 73-107
MSC: Primary 22E46; Secondary 22E47
DOI: https://doi.org/10.1090/S0002-9947-1987-0871666-5
MathSciNet review: 871666
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Abstract: Let $ G$ be a connected semisimple real matrix group. It is now apparent that the representation theory of $ G$ is intimately connected with the complex geometry of the flag variety $ \mathcal{B}$. By studying appropriate orbit structures on $ \mathcal{B}$, we are naturally led to representation theory in the category of Harish-Chandra modules $ \mathcal{H}\mathcal{C}$, or the representation theory of category $ \mathcal{O}'$. The Jacquet functor $ J:\mathcal{H}\mathcal{C} \to \mathcal{O}'$ has proved a useful tool in converting " $ \mathcal{H}\mathcal{C}$ problems" into " $ \mathcal{O}'$ problems," which are often more tractable. In this paper, we advance the philosophy that the complex geometry of $ \mathcal{B}$, associated to $ \mathcal{H}\mathcal{C}$ and $ \mathcal{O}'$, interacts in a natural way with the functor $ J$, leading to deep new information on the structure of Jacquet modules. This, in turn, gives new insight into the structure of certain nilpotent cohomology groups associated to Harish-Chandra modules. Our techniques are based upon many of the ideas present in the proof of the Kazhdan-Lusztig conjectures and Bernstein's proof of the Jantzen conjecture.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0871666-5
Article copyright: © Copyright 1987 American Mathematical Society

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