The Structure of W-graphs Arising in Kazhdan-Lusztig Theory.

Abstract

This thesis is primarily about the combinatorial aspects of Kazhdan-Lusztig theory. Central to this area is the notion of a W-graph, a certain weighted directed graph which encodes a representation of the Iwahori-Hecke algebra of a Coxeter group. The most important examples were given in the original work of Kazhdan and Lusztig in 1979; from these graphs the Kazhdan-Lusztig polynomials are obtained via a weighted path count. In the first part, we consider ``parallel transport'' relations among edge weights. Some of these relations, namely those coming from simply-laced Weyl groups, appeared in the same paper of Kazhdan and Lusztig. We introduce additional ones corresponding to doubly-laced Weyl groups, and, as an application, prove Green's 0-1 conjecture in type B. In the second part we clarify the structure of W-graphs corresponding to minuscule and quasi-minuscule quotients of finite Weyl groups. The W-graphs for minuscule quotients can be deduced, on a case-by-case basis, from previous work on the associated Kazhdan-Lusztig polynomials; we give a type-independent proof of a weaker result that these graphs can be characterized by simple combinatorial rules. For quasi-minuscule quotients, we compute the graphs for all finite Weyl groups except for Lie type D (where we give a conjectural answer). We also compute the parabolic Kazhdan-Lusztig polynomials for the type A quasi-minuscule quotient. The last part concerns the conjecture that in Lie type A, the only strongly connected W-graphs which satisfy a weak set of conditions known as ``admissibility'' are the Kazhdan-Lusztig examples. We prove a partial result that the symmetrically weighted edges of such a graph are the same as the symmetrically weighted edges of some Kazhdan-Lusztig examples.