Kazhdan-Lusztig Conjecture and Holonomic Systems

In [7], D. Kazhdan and G. Lusztig gave a conjecture on the multiplicity of simple modules which appear in a Jordan-H61der series of the Verma modules. This multiplicity is described in the terms of Coxeter groups and also by the geometry of Schubert cells in the flag manifold (see [8]). The purpose of this paper is to give the proof of their conjecture. The method employed here is to associate holonomic systems of linear differential equations with R.S. on the flag manifold with Verma modules and to use the correspondance of holonomic systems and constructible sheaves. Let G be a semi-simple Lie group defined over • and g its Lie algebra. We take a pair (B,B-) of opposed Borel subgroups of G and let T=B~Bbe a maximal torus and W the Weyl group. Let b, b and f the corresponding Lie algebras and 9l the nilpotent radical of b. Let us denote by Jg the category of holonomic systems with R.S. on X=G/B whose characteristic varieties are contained in the union of the conormal bundles of Xw=BWB/B (we W). On the other hand, let (9 denote the category of finitely-generated U(g)-modules which are Tl-finite. By (gtrlv we denote the category of the modules in (9 with the trivial central character. We shall prove that J / / a n d (~trlv are equivalent by the correspondances S0l ~--*F(X;gJI) and M~--~,~| Here ~ is the sheaf of differential operators on X. Let us denote by M w the Verma module with highest weight -w(p)-p and let ~Jl w be the dual g -module of ~codimXwt/~ ~ Then, ~ w and Mw ~ [X~] ~,~X]" correspond by the above correspondence. For any 9 J l e ~ , we can calculate the character of F(X; 93l) by the formula