Some Open Problems in the Theory of Kazhdan-Lusztig polynomials and Coxeter groups

Abstract

The central role played in the representation theory of reductive algebraic groups, semisimple Lie algebras and quantum groups by the Hecke algebra of the corresponding Weyl group, as the lieu where the basic combinatorial phenomena take place, has progressively become clear during the past two decades, chiefly through the works of George Lusztig. There has also been great progress in the purely combinatorial theory of Coxeter groups and Hecke algebras; we would like to cite for instance the papers of Kazhdan and Lusztig [17], Deodhar [5, 6, 7], Björner and Wachs [1], Dyer [9, 14], Brink and Howlett [3], and Brenti [2]. In particular, in [17] the celebrated Kazhdan-Lusztig polynomials were introduced, which are the “shadows” in the Hecke algebra of the perverse sheaves on the flag manifold; these polynomials will be the main theme in the present paper. They form a fascinating computational challenge, because of the difficulty of performing extended computations about them, because they and their subsequent generalizations provide the key to the understanding of the various representation theories involved, and because there are still many open questions about them, which seem hard to attack in the absence of more example material. For a review of the computational achievements in this area, the reader may consult [4].