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Representation theory and D-modules on flag varieties. (English) Zbl 0705.22010

Orbites unipotentes et représentations. III. Orbites et faisceaux pervers, Astérisque 173-174, 55-109 (1989).
[For the entire collection see Zbl 0682.00007.]
This paper is a lecture on the study of representations of Lie groups through the geometry of flag varieties. Let \({\mathbb{G}}\) be a complex reductive group and let X be its flag variety. Let \({\mathfrak g}\) be the Lie algebra of \({\mathbb{G}}\), \({\mathfrak t}\) the Cartan subalgebra and \(\Delta\) the root system. For \(\lambda\in {\mathfrak t}^*\), let \(\chi_{\lambda}\) be the corresponding character of the center Z(\({\mathfrak g})\) of the universal enveloping algebra U(\({\mathfrak g})\). We normalize this so that \(\chi_{\lambda}=\chi_{w\lambda}\) for w in the Weyl group W. For \(\lambda\in {\mathfrak t}^*\), set \(U_{\lambda}({\mathfrak g})=U({\mathfrak g})/U({\mathfrak g})Ker \chi_{\lambda}\). Then we can construct a twisted ring of differential operators \({\mathcal D}_{\lambda}\) on X such that \(\Gamma\) (X;\({\mathcal D}_{\lambda})=U_{\lambda}({\mathfrak g})\). Let \({\mathbb{G}}_{{\mathbb{R}}}\) be a real semisimple group, \({\mathbb{K}}_{{\mathbb{R}}}\) a maximal compact subgroup of \({\mathbb{G}}_{{\mathbb{R}}}\) and let \({\mathbb{G}}\) and \({\mathbb{K}}\) be their complexifications. Let \({\mathfrak g}\) and \({\mathfrak k}\) be their Lie algebras. Then, by Harish-Chandra [Am. J. Math. 86, 534- 564 (1964; Zbl 0161.338)], any admissible representation of \({\mathbb{G}}_{{\mathbb{R}}}\) is described by (\({\mathfrak g},{\mathbb{K}})\)-modules, so called Harish-Chandra modules. Moreover, a Harish-Chandra module with infinitesimal character \(\chi_{\lambda}\) is described by a \({\mathbb{K}}\)- equivariant \({\mathcal D}_{\lambda}\)-module. The structure of irreducible \({\mathbb{K}}\)-equivariant \({\mathcal D}_{\lambda}\)-mdules \({\mathcal M}\) can be described by using the geometry of \({\mathbb{K}}\)-orbits. Especially, when \(\lambda\) is not regular, the author obtains new results.
Reviewer: M.Muro

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
14M15 Grassmannians, Schubert varieties, flag manifolds
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
17B35 Universal enveloping (super)algebras
22E60 Lie algebras of Lie groups