\(D\)-modules and representation theory of Lie groups
type: Item
instance of: scholarly article
title: \(D\)-modules and representation theory of Lie groups
zbMATH Open document ID: 0823.22013
author: Masaki Kashiwara
publication date: 1994-11-10T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75049
review text: 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.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 14L35
zbMATH DE Number: 683181
zbMATH Keywords: representations
zbMATH Keywords: Harish-Chandra modules
zbMATH Keywords: \(G_ \mathbb{R}\)-equivariant sheaves
zbMATH Keywords: \(K\)-equivariant sheaves
zbMATH Keywords: flag manifolds
zbMATH Keywords: representations of real semisimple Lie groups
reviewed by: V. Lakshmibai
MaRDI profile type: Publication
OpenAlex ID: W2328453693
Wikidata QID: Q115159064
cites work: Q3688545
cites work: Q3933105
cites work: Q3320507
cites work: Differential operators on homogeneous spaces. I: Irreducibility of the associated variety for annihilators of induced modules
cites work: A property of the characters of irreducible representations of real semisimple Lie groups
cites work: Representations of a Semisimple Lie Group on a Banach Space. I
cites work: Q4108530
cites work: The invariant holonomic system on a semisimple Lie algebra
cites work: Q3999005
cites work: Q4869245
cites work: Q4198960
cites work: Orbits on affine symmetric spaces under the action of parabolic subgroups
cites work: Matsuki correspondence for sheaves
cites work: Q3820841
cites work: Q3758136
cites work: Q3139083
cites work: Globalizations of Harish-Chandra modules
cites work: Remarks on real nilpotent orbits of a symmetric pair
DOI: 10.5802/AIF.1385
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 683181