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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

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