\(D\)-modules and representation theory of Lie groups
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.
representations of real semisimple Lie groups
scientific article; zbMATH DE number 683181
scientific article