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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.

22E46 Semisimple Lie groups and their representations
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