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

Abstract

Let G G be a connected semisimple real matrix group. It is now apparent that the representation theory of G G is intimately connected with the complex geometry of the flag variety B \mathcal {B} . By studying appropriate orbit structures on B \mathcal {B} , we are naturally led to representation theory in the category of Harish-Chandra modules H C \mathcal {H}\mathcal {C} , or the representation theory of category O ′ \mathcal {O}’ . The Jacquet functor J : H C → O ′ J:\mathcal {H}\mathcal {C} \to \mathcal {O}’ has proved a useful tool in converting " H C \mathcal {H}\mathcal {C} problems" into " O ′ \mathcal {O}’ problems," which are often more tractable. In this paper, we advance the philosophy that the complex geometry of B \mathcal {B} , associated to H C \mathcal {H}\mathcal {C} and O ′ \mathcal {O}’ , interacts in a natural way with the functor J 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.