We use the affine Robinson–Schensted correspondence to describe structure of bidirected edges in Kazhdan–Lusztig cells type A. Equivalently, we give a comprehensive description Knuth equivalence classes permutations.

The introduction of l-adic cohomology by M. Artin and Grothenieck was motivated by the Weil conjecture concerning the number of rational points of algebraic varieties over finite fields. Since its introduction in the early 60s, l-adic cohomology theory has been used in the representation theory of finite Chevalley groups in a “misterious way”. The first application of l-adic cohomology and inte...

Let G be a reductive group over a finite field Fq and let G(Fq) be its group of Fq-rational points. In 1976 P. Deligne and G. Lusztig (following a suggestion of V. Drinfeld for G = SL(2)) associated to any maximal torus T in G defined over Fq certain remarkable series (virtual) representations of the group G(Fq). These representations were realized in é tale l-adic cohomology of certain algebra...

In this paper, the author studies when some monomials are in the canonical basis of the quantized enveloping algebra corresponding to a simply laced semisimple finite dimensional complex Lie algebra. 0. Introduction To any graph Γ, Lusztig has associated in [L1] and [L2] an algebra U− over Z[v, v−1] provided with a canonical basis B. In the case that Γ is the Dynkin graph of a simply laced semi...

In this and the next lecture we will describe an entirely different application of Hecke algebras, now to the category O. In the first section we will define the Kazhdan-Lusztig basis in the Hecke algebra of W and explain how to read the multiplicities in the category O from this basis (the Kazhdan-Lusztig conjecture proved independently by Beilinson-Bernstein and Brylinski-Kashiwara). In the r...

We prove that certain parabolic Kazhdan-Lusztig polynomials calculate the graded decomposition matrices of v-Schur algebras given by the Jantzen filtration of Weyl modules, confirming a conjecture of Leclerc and Thibon.

We give explicit formulas for the Kazhdan-Lusztig P and Rpolynomials for permutations coming from the variety F1,n−1 of incomplete flags consisting of a line and a hyperplane.

Inspired by a recent paper of Garcia, Stichtenoth and Xing [2000, Compositio Math. 120, 137–170], we investigate the quotient curves of the Deligne-Lusztig curve associated to the Suzuki group Sz(q).

If f f denotes the truncated Lusztig Fourier transform, we show that image by of normalized characteristic function a Coxeter element is alte...