We give explicit combinatorial product formulas for the maximal parabolic Kazhdan-Lusztig and R-polynomials of the symmetric group. These formulas imply that these polynomials are combinatorial invariants, and that the KazhdanLusztig ones are nonnegative. The combinatorial formulas are most naturally stated in terms of Young’s lattice, and the one for the Kazhdan-Lusztig polynomials depends on ...

We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the pos-itivity of certain structure constants for the associated Kazhdan–Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading ter...

According to [2], the finite-dimensional restricted simple Lie algebras over an algebraically closed field of characteristic p > 7 are either classical or of Cartan type. Lusztig has a conjecture (see [8, p. 294]) which provides a formula for the dimensions of the simple modules for the classical algebras. (In this paper, modules are assumed to be left, finitedimensional and restricted.) Anders...

We describe all the left cells of a-values 5, 6 in the affine Weyl group e E8 in the paper. More precisely, we show that each of those left cells is left-connected, verifying a conjecture of Lusztig in our case. We find all the distinguished involutions in those left cells which occur as the vertices of the corresponding distinguished involution graphs. We also get all the corresponding left ce...

2 Highest weight modules 4 2.1 Kac-Moody Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Integral Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Category of highest weight modules . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Enright functor for non-integral weights . . . . . . . . . . . . . . . . . . . . 11 2.5 Embeddings of Verma m...

We construct families of hyperelliptic curves over Q of arbitrary genus g with (at least) g integral elements in K2. We also verify the Beilinson conjectures about K2 numerically for several curves with g = 2, 3, 4 and 5. The first few sections of the paper also provide an elementary introduction to the Beilinson conjectures for K2 of curves.

To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category AFF(C)κ of smooth modules (in the sense of Kazhdan and Lusztig [13]) of finite length over the corresponding affine Kac–Moody algebra in the case of central charge less than the critical level. Equivalent characterizations ...

In 1986, George Lusztig stated some interesting properties of character sheaves with non-zero restriction to the unipotent variety and proved them for the group SO2n+1(Fq) under the assumption that the characteristic p of Fq is good. In February 1992, the author has given lectures on that proof of Lusztig in the Seminar “Faisceaux Caractères” at the École Normale Supérieure and it was no doubt ...