## On the left cell representations of

Iwahori-Hecke algebras
of finite Coxeter groups

### Andrew Mathas

#### Keywords

Iwahori-Hecke algebras, Coxeter groups,
Kazhdan-Lusztig polynomials.

#### Status

*J. London Math. Soc.*, ** 54** (1996), 475-488.

MR97h:20008.
#### Abstract

In this paper we investigate the left cell representations
of the Iwahori-Hecke algebras associated to a finite Coxeter group
*W*. Let *w*_{0} be the element of longest length in
*W*. Our main result shows that *T*_{w0}
acts (essentially) as an involution upon the Kazhdan-Lusztig
basis of a cell representation. We describe some properties of this involution,
use it to further describe the left cells, and finally
show how to realize each cell representation as a submodule of *H*.
Our results rely upon certain positivity properties of the structure
constants of the Kazhdan-Lusztig bases of the Hecke algebra and so have
not yet been shown to apply to all finite Coxeter groups.

*Andrew Mathas *

15th May 1996.