# zbMATH — the first resource for mathematics

The characteristic cycles of holonomic systems on a flag manifold related to the Weyl group algebra. (English) Zbl 0611.22008
The authors give a relation between the characteristic cycles of the holonomic systems on G/B entering into the proof of the Kazhdan-Lusztig conjecture and the Springer representations of the Weyl group: W operates on the K-group of the holonomic systems: $$w\cdot [{\mathfrak M}_ y]=[{\mathfrak M}_{wy}]$$ for $$w\in W$$ ($${\mathfrak M}_ w$$ is the holonomic system corresponding to the Verma module with highest weight -w($$\rho)$$-$$\rho)$$; W operates on top-degree cycles on the conormal variety of the Schubert cells by a representation constructed by Kazhdan-Lusztig in connection with the Springer representation. The main result of the paper is equivalent to the assertion that the map which associates to each holonomic system its characteristic cycle is a W-isomorphism for these W- actions.
Reviewer: W.Rossmann

##### MSC:
 22E46 Semisimple Lie groups and their representations 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
Full Text:
##### References:
 [1] Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex classical groups. Math. Ann.259, 153-199 (1982) · Zbl 0489.22010 [2] Barbasch, D., Vogan, D.: Primitive ideals and orbital integrals in complex exceptional groups. J. Algebra80, 350-382 (1983) · Zbl 0513.22009 [3] Beilinson, A., Bernstein, J.: Localisation deg-modules. Comptes Rendus292A, 15-18 (1981) · Zbl 0476.14019 [4] Borho, W., Brylinski, J.-L.: Differential operators on homogeneous spaces. I. Invent. Math.69, 437-476 (1982) · Zbl 0504.22015 [5] Borho, W., MacPherson, R.: Représentations des groupes de Weyl et homologie d’intersection pour les variétés de nilpotents. Comptes Rendus292, 707-710 (1981) · Zbl 0467.20036 [6] Brylinski, J.-L., Kashiwara, M.: Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math.64, 387-410 (1981) · Zbl 0473.22009 [7] Hotta, R.: On Joseph’s construction of Weyl group representations; Tohoku Math. J.36, 49-74 (1984) · Zbl 0545.20029 [8] Joseph, A.:W-module structure in the primitive spectrum of the enveloping algebra of a complex semisimple Lie algebra. In: Noncommutative harmonic analysis, Lecture Notes in Math.728, 116-135. Berlin Heidelberg New York: Springer 1979 [9] Joseph, A.: Goldie rank in the enveloping algebra of a semisimple Lie algebra I, II. J. Algebra65, 269-316 (1980) · Zbl 0441.17004 [10] Joseph, A.: On the variety of a highest weight module; to appear in J. Algebra · Zbl 0539.17006 [11] Joseph, A.: On the associated variety of a primitive ideal; preprint (1983) [12] Kashiwara, M.: Index theorem for a maximally overdetermined system of linear differential equations. Proc. Japan Acad.49, 803-804 (1973) · Zbl 0305.35073 [13] Kashiwara, M.: Systems of microdifferential equations. Progress in Mathematics34. Basel: Birkhäuser 1983 · Zbl 0521.58057 [14] Kashiwara, M.: The Riemann-Hilbert problem for holonomic systems; preprint (1983), submitted to Publ. RIMS. Kyoto University [15] Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math.53, 165-184 (1979) · Zbl 0499.20035 [16] Kazhdan, D., Lusztig, G.: Schubert varieties and Poincaré duality. Proc. Symp. in Pure Math.36, 185-203 (1980) · Zbl 0461.14015 [17] Kazhdan, D., Lusztig, G.: A topological approach to Springer’s representations. Advances in Math.38, 222-228 (1980) · Zbl 0458.20035 [18] Lusztig, G.: A class of irreducible representations of a Weyl group. Proc. Kon. Nederl. Akad.A 82, 323-335 (1979) · Zbl 0435.20021 [19] Lusztig, G., Vogan, D.: Singularities of closures ofK-orbits on flag manifolds. Invent. Math.71, 365-379 (1983) · Zbl 0544.14035 [20] Sabbah, C.: Quelques remarques sur la theorie des classes de Chern des espaces analytiques singuliers; preprint (1983) [21] Springer, T.A.: Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Invent. Math.36, 173-207 (1976) · Zbl 0374.20054 [22] Springer, T.A.: Quelques applications de la cohomologie d’intersection. Séminaire Bourbaki, expos 589. Astérisque 92-93, 249-273 (1982) [23] Steinberg, R.: Conjugacy classes in algebraic groups. Lecture Notes in Math.366. Berlin Heidelberg New York: Springer 1974 · Zbl 0281.20037 [24] Tanisaki, T.: Representation theory of complex semisimple Lie algebras andD. In: Reports of the 5-th seminar on algebra II. 67-163 (1983) (in Japanese) [25] Vogan, D.: Ordering of the primitive spectrum of a semisimple Lie algebra. Math. Ann.248, 195-203 (1980) · Zbl 0422.17005 [26] Hotta, R.: A local formula for Springer’s representation; to appear in Advanced Studies in Pure Math.6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.