Résumé
In his article Brenti (J. Am. Math. Soc. 11(2): 480–497, 1973), Brenti gave a non-recursive formula for the Kazhdan–Lusztig polynomials of a Coxeter group and proved it by combinatorial methods [cf. Theorem 4.1 of Brenti (J. Am. Math. Soc. 11(2): 480–497, 1973)]. The goal of this note is to give a geometric interpretation of this formula for the Coxeter groups that are isomorphic to the Weyl group of a split group over a finite field. This interpretation rests on a result of Morel (J. Am. Math. Soc. 21(1): 23–61, 2008) (Theorem 3.3.5) that expresses the intermediate extension of a pure perverse sheaf as a “weight truncation” of the usual direct image.
This is a preview of subscription content, log in via an institution to check access.
Références
Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981). Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)
Białynicki-Birula A.: Some theorems on actions of algebraic groups. Ann. Math. (2) 98, 480–497 (1973)
Brenti F.: Lattice paths and Kazhdan-Lusztig polynomials. J. Am. Math. Soc 11(2), 229–259 (1998)
Cohomologie l-adique et fonctions L. Lecture Notes in Mathematics, vol. 589. Springer, Berlin (1977). Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5), Edité par Luc Illusie
Deligne, P.: Cohomologie étale. Lecture Notes in Mathematics, vol. 569. Springer, Berlin (1977). Séminaire de Géométrie Algébrique du Bois-Marie SGA 41øer 2, Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier
Demazure, M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4) 7, 53–58 (1974). Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I
Deodhar V.V.: On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells. Invent. Math. 79(3), 499–511 (1985)
Haerterich, M.: The t-equivariant cohomology of bott-samelson varieties (2004)
Kazhdan D., Lusztig G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979)
Kazhdan, D., Lusztig, G.: Schubert varieties and Poincaré duality. In: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979). Proc. Sympos. Pure Math., XXXVI. Am. Math. Soc., Providence (1980), pp. 185–203
Morel, S.: Complexes pondérés sur les compactifications de Baily-Borel: le cas des variétés de Siegel. J. Am. Math. Soc. 21(1), 23–61 (2008, electronic)
Springer T.A.: A purity result for fixed point varieties in flag manifolds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31(2), 271–282 (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Morel, S. Note sur les polynômes de Kazhdan–Lusztig. Math. Z. 268, 593–600 (2011). https://doi.org/10.1007/s00209-010-0685-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-010-0685-6