Complex Geometry and the Asymptotics of Harish-Chandra Modules for Real Reductive Lie Groups. I

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.