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Holonomic systems on a flag variety associated to Harish-Chandra modules and representations of a Weyl group. (English) Zbl 0582.22011
Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 139-154 (1985).
[For the entire collection see Zbl 0561.00006.]
Let $$G$$ be a connected semi-simple complex algebraic group, $$G_{{\mathbb{R}}}^ a$$ connected real form of $$G$$, $$K$$ the complexification of a maximal compact subgroup of $$G_{{\mathbb{R}}}$$, $$X$$ the variety of Borel subgroups of $$G$$, and $${\mathcal D}_ X$$ the sheaf of regular differential operators on $$X$$. On the Grothendieck group of the category of coherent $${\mathcal D}_ X$$-modules with K-action there is a defined a natural homomorphism $$Ch$$, namely the taking of the characteristic cycle. The main theorem says that $$Ch$$ is $$W$$-equivariant for the action of the Weyl group W; the $${\mathbb{Q}}[W]$$-module $${\mathbb{Q}} \otimes_{{\mathbb{Z}}}$$ im $$Ch$$ is described as a direct sum of induced modules. The author gives a neat description of the various requisites for his proof.
Reviewer: H.de Vries

##### MSC:
 22E46 Semisimple Lie groups and their representations 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 20G10 Cohomology theory for linear algebraic groups 32C99 Analytic spaces