Analytic localization of group representations
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 0621.22014)], \textit{W. Schmid} introduced a functor of ``minimal globalization'' from the category of Harish-Chandra modules of (\({\mathfrak g},K)\) into the category of representations of \(G_ 0\). By the construction, minimal globalizations are DNF spaces. The second main result of this paper is as follows: Let D(\({\mathcal U}_{\theta},G_ 0)\) be the full triangulated subcategory of D(\({\mathcal U}_{\theta})\) generated by minimal globalizations of Harish- Chandra modules. Let D(\({\mathcal D}_{\lambda},G_ 0)\) be the full triangulated subcategory of D(\({\mathcal D}_{\lambda})\) generated by \(G_ 0\)-equivariant \({\mathcal D}_{\lambda}\)-modules whose restrictions to any \(G_ 0\)-orbit are locally free of finite rank over the structure sheaf. Then, for regular \(\lambda\), the functors \(L\Delta_{\lambda}\) and \(R\Gamma\) induce equivalences of these categories.
scientific article; zbMATH DE number 4148438
scientific article