On the Left Cell Representations of Iwahori-Hecke Algebras of Finite Coxeter Groups

In this paper we investigate the left cell representations ofthe Iwahori-Hecke algebras associated to a finite Coxeter groupW. Our main result shows that , where w₀ is the element of longest length in W, acts essentiallyas an involution upon the canonical bases of a cell representation.We describe some properties of this involution, use it to furtherdescribe the left cells, and finally show how to realize eachcell representation as a submodule of . Our results rely uponcertain positivity properties of the structure constants ofthe Kazhdan-Lusztig bases of the Hecke algebra and so have notyet been shown to apply to all finite Coxeter groups.