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\(D\)-modules and representation theory of Lie groups. (English) Zbl 0823.22013

Let \(G_{\mathbb{R}}\) be a real semisimple Lie group with finite center. Let \(K_ \mathbb{R}\) be a maximal compact subgroup of \(G_ \mathbb{R}\). Let \(G\) (respectively \(K\)) be the complexification of \(G_ \mathbb{R}\) (respectively \(K_{\mathbb{R}}\)). In this paper, the representation theory of \(G_ \mathbb{R}\) is studied using the geometry of the flag manifold of \(G\). Using Beilinson-Bernstein’s results one has a correspondence between {representations of \(G_ \mathbb{R}\)} and {Harish-Chandra modules}. On the other hand, using Mirkovic-Uzawa-Vilonen’s results, one has a correspondence between {\(G_ \mathbb{R}\)-equivariant sheaves} and {\(K\)- equivariant sheaves} (on the flag manifold of \(G\)). The author establishes a correspondence between {representations of \(G_ \mathbb{R}\)} and {\(G_ \mathbb{R}\)-equivariant sheaves} on the one hand, and {Harish- Chandra modules} and {\(K\)-equivariant sheaves} on the other, thus making the epicture complete.
This paper makes an important contribution to the representation theory of real semi-simple Lie groups.

MSC:

22E46 Semisimple Lie groups and their representations
14M15 Grassmannians, Schubert varieties, flag manifolds
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
14L35 Classical groups (algebro-geometric aspects)
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