The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V ). This paper classifies all the algebras obtained by applying Drinfeld’s construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we...

In this paper we are looking at the problem of determining the composition factors for the affine graded Hecke algebra via the computation of Kazhdan-Lusztig type polynomials. We review the algorithms of [9, 10], and use them in particular to compute, at every real central character which admits tempered modules, the geometric parameterization, the Kazhdan-Lusztig polynomials, the composition s...

In this paper, we study the Kazhdan–Lusztig cells of a Coxeter group W in a “relative” setting, with respect to a parabolic subgroup WI ⊆ W . This relies on a factorization of the Kazhdan–Lusztig basis {Cw} of the corresponding (multi-parameter) Iwahori–Hecke algebra with respect to WI . We obtain two applications to the “asymptotic case” in type Bn, as introduced by Bonnafé and Iancu: we show ...

We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M , in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We al...

We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne– Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes Xh. Boyarchenko’s two conjectures ...

Four statistics, ls, rb, rs, and lb, previously studied on all partitions of { 1, 2, ..., n }, are applied to non-crossing partitions. We consider single and joint distributions of these statistics and prove equidistribution results. We obtain qand p, q-analogues of Catalan and Narayana numbers which refine the rank symmetry and unimodality of the lattice of non-crossing partitions. Two unimoda...

In this paper, we investigate the structure and representations of the quantum group U(∞) = Uυ(gl∞). We will present a realization for U(∞), following Beilinson–Lusztig–MacPherson (BLM) [1], and show that the natural algebra homomorphism ζr from U(∞) to the infinite q-Schur algebra S(∞, r) is not surjective for any r > 1. We will give a BLM type realization for the image U(∞, r) := ζr(U(∞)) and...

Let W be a Coxeter group of type ̃ An−1. We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial Px,w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).