Structures cellulaires sur les algèbres de Hecke : théorie de Kazhdan-Lusztig, algèbres de Schur, algèbres de Cherednik – CELLULHECKE

In the last decade, the theory of representations of finite reflection groups and Hecke algebras has been developed in many different directions: algebra (representations of finite of p-adic or finite reductive groups and of Lie algebras, invariant theory, canonical bases of quantum groups, rational Cherednik algebras...), geometry (singularities of flag varieties, cohomology of Deligne-Lusztig varieties), algebraic topology (braid groups, knot theory). The present project aims to enlighten the links between the theory of representations of finite reductive groups and the canonical bases of quantum groups. Roughly speaking, the starting point of these ideas is the LLT conjecture (Lascoux-Leclerc-Thibon [LLT], 1996), proved in 1997 by Ariki [A], which relates the decomposition matrices of the finite general linear groups and the canonical basis of the representation of the quantum affine special linear Lie algebra on the Fock space. For other finite reductive groups, some new conjectures (X. Yvonne [Y]) came to improve the theory. One of the aim of this project is to involve the latest theoretical development in the theory of Hecke algebras with unequal parameters and Cherednik algebras in the understanding of these conjectures. Our aim is not only esthetic (unification, structural explanations for numerical facts): we also aim to make concrete progress in the proofs of these conjectures and to apply these results to finite reductive groups. These questions have been studied in Besançon since 2002 (and the visit of L. Iancu for one semester) by C. Bonnafé and, since 2005, by N. Jacon.