We consider the partition of a finite Coxeter group W into left cells with respect to a weight function L. In the equal parameter case, Lusztig has shown that the representations carried by the left cells are precisely the so-called constructible ones. We show that this holds for general L, if the conjectural properties (P1)–(P15) in Lusztig’s book on Hecke algebras with unequal parameters hold...

The lowest two-sided cell of the extended affine Weyl group We is the set {w ∈ We : w = x · w0 · z, for some x, z ∈ We}, denoted W(ν). We prove that for any w ∈ W(ν), the canonical basis element C w can be expressed as 1 [n]!χλ(Y )C ′ v1w0 C w0v2 , where χλ(Y ) is the character of the irreducible representation of highest weight λ in the Bernstein generators, and v1 and v −1 2 are what we call ...

We construct an endoscopic decomposition for local L-packets associated to irreducible cuspidal Deligne–Lusztig representations. Moreover, the obtained decomposition is compatible with inner twistings.

on the set of F-rational points of X given by the alternating sum of traces of Fr, the Frobenius action on stalks of the cohomology sheaves HiF . He then went on to initiate an ambitious program of giving geometric (= sheaf theoretic) meaning to various classical algebraic formulas via the above “functions-faisceaux” correspondence F 7→ χ F . This program got a new impetus with the discovery of...

Let (W,S) be a Coxeter system and H the associated Hecke algebra with unequal parameters. The Laurent polynomials Ms y,w and py,w for y, w ∈ W and s ∈ S play an important role in the representations of H. We study the properties of Ms y,w and py,w, the relations among them, as well as with the left, right and two-sided cells of W . In his book [5], Lusztig gave a systematic introduction to the ...

We refine an idea of Deodhar, whose goal is a counting formula for Kazhdan–Lusztig polynomials. This consequence simple observation that one can use the solution Soergel's conjecture to make ambiguities involved in defining certain morphisms between Soergel bimodules characteristic zero (double leaves) disappear.

Let Wa be an irreducible affine Weyl group with W0 the associated Weyl group. The present paper is to study the second lowest two-sided cell Ωqr of Wa. Let nqr be the number of left cells of Wa in Ωqr. We conjecture that the equality nqr = 1 2 |W0| should always hold. When Wa is either e An−1, n > 2, or of rank 6 4, this equality can be verified by the existing data (see 0.3). Then the main res...

This is an overview article on finite-dimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima.

We show that the Littlewood-Richardson coeecients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for aane symmetric groups. These q-analogues of Littlewood-Richardson multiplicities coincide with those previously introduced in 21] in terms of ribbon tableaux.