Analytic localization of group representations
type: Item
instance of: scholarly article
title: Analytic localization of group representations
zbMATH Open document ID: 0701.22005
DOI: 10.1016/0001-8708(90)90062-R
author: Henryk Hecht
author: Joseph L. Taylor
published in: Advances in Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The authors develop an analytic version of the localization theory of \textit{A. Beilinson} and \textit{J. Bernstein} [C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)]. Let \(G_ 0\) be a connected semisimple Lie group with finite center, K its maximal compact subgroup and \({\mathfrak g}\) its complexified Lie algebra. Let X be the flag variety of \({\mathfrak g}\) considered as an analytic manifold. To each \(\lambda\) in the dual of a Cartan subalgebra \({\mathfrak h}\) one attaches a sheaf \({\mathcal D}_{\lambda}\) of twisted differential operators on X, with the property that its global sections \(\Gamma\) (X,\({\mathcal D}_{\lambda})\) are given by the quotient \({\mathcal U}_{\theta}\) of the enveloping algebra \({\mathcal U}({\mathfrak g})\) by the ideal generated by the kernel of the infinitesimal character attached to the Weyl group orbit of \(\lambda\). The algebra \({\mathcal U}_{\theta}\) has the natural structure of a dual of a nuclear Fréchet (or DNF) space as the direct limit of finite-dimensional subspaces \(F_ p{\mathcal U}_{\theta}\) of elements of order \(\leq p\). Let \({\mathcal M}({\mathcal U}_{\theta})\) be the category of DNF \({\mathcal U}_{\theta}\)-modules. Analogously, one defines the category \({\mathcal M}({\mathcal D}_{\lambda})\) of DNF \({\mathcal D}_{\lambda}\)-sheaves. The localization function is defined in this situation as \(\Delta_{\lambda}(V)={\mathcal D}_{\lambda}{\hat \otimes}_{{\mathcal U}_{\theta}}V\), where \({\hat \otimes}\) is the topological tensor product. One then defines the corresponding derived categories D(\({\mathcal U}_{\theta})\) and D(\({\mathcal D}_{\lambda})\) and the derived functors of localization \(L\Delta_{\lambda}\) and cohomology \(R\Gamma\) between them. The first main result is that, if \(\lambda\) is regular, these functors are mutually inverse equivalences of categories. In his paper [in ``Elie Cartan et les mathématiques d'aujourd'hui'', Astérisque 1985, 311-321 (1985; Zbl 062…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4148438
zbMATH Keywords: connected semisimple Lie group
zbMATH Keywords: flag variety
zbMATH Keywords: Cartan subalgebra
zbMATH Keywords: twisted differential operators
zbMATH Keywords: enveloping algebra
zbMATH Keywords: infinitesimal character
zbMATH Keywords: localization function
zbMATH Keywords: topological tensor product
zbMATH Keywords: derived categories
zbMATH Keywords: derived functors of localization
zbMATH Keywords: equivalences of categories
zbMATH Keywords: Harish- Chandra modules
zbMATH Keywords: category of representations
reviewed by: Dragan Milicic
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4148438