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Localization and standard modules for real semisimple Lie groups. I: The duality theorem. (English) Zbl 0699.22022
Let G be a connected semisimple real matrix group with maximal compact subgroup K. In this paper, the authors are concerned with relating two fundamental constructions of standard Harish-Chandra modules, each of which is closely related to the flag variety $${\mathcal B}$$ of G.
On the one hand, we have the Beilinson-Bernstein modules $$H^ q({\mathcal B},j_+({\mathcal O}_ Q({\mathcal F})))$$ which arise via the $${\mathcal D}$$-module push-forward of a $$K_ C$$-homogeneous line bundle $${\mathcal F}$$ along the $$K_ C$$-orbit Q. Here $${\mathcal D}$$ is a q-equivariant twisted sheaf of differential operators on $${\mathcal B}$$ which, when restricted to Q, acts on the sections of $${\mathcal F}$$. These push-forwards may have higher cohomology groups, each of which is a Harish-Chandra module. What we have just described are the Beilinson-Bernstein modules corresponding to the data (Q,$${\mathcal F},{\mathcal D},q).$$
On the other hand, consider triples (S,$${\mathcal E},q)$$, where S is a G- orbit in $${\mathcal B}$$, $${\mathcal E}$$ is a G-homogeneous line bundle on S and q is an integer. This datum corresponds to a standard Zuckerman module in degree q.
By work of Matsuki, there is an order reversing bijective pairing between G-orbits and $$K_ C$$-orbits. Suppose S is a G-orbit dual to the $$K_ C$$-orbit $$Q_ S$$. This pairing can be extended to make the correspondence (S,$${\mathcal E})\to (Q_ S,{\mathcal F}_{{\mathcal E}},{\mathcal D}_{{\mathcal E}})$$ bijective, where $${\mathcal D}_{{\mathcal E}}$$ is determined $${\mathcal E}$$. The main theorem of this paper is the duality theorem: The Beilinson-Bernstein module attached to $$(Q_ S,{\mathcal F}_{{\mathcal E}},{\mathcal D}_{{\mathcal E}},q)$$ is canonically dual, in the category of Harish-Chandra modules, to the standard Zuckerman module attached to (S,$${\mathcal E}^*\otimes \Omega_{{\mathcal B}},s-q)$$, where $$s=\dim_{{\mathbb{R}}}(Q_ S\cap S)-\dim_{{\mathbb{C}}}(Q_ S)$$. In a future Part II of this paper, the authors plan to use the duality theorem to relate the Langlands classification, the Vogan-Zuckerman classification and the Beilinson-Bernstein classification directly and explicitly.

##### MSC:
 2.2e+47 Semisimple Lie groups and their representations
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##### References:
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