ON THE KAZHDAN-LUSZTIG CELLS IN TYPE E8

In 1979, Kazhdan and Lusztig introduced the notion of "cells" (left, right and two-sided) for a Coxeter group W, a concept with numerous applications in Lie theory and around. Here, we address algorithmic aspects of this theory for finite W which are important in applications, e.g., run explicitly through all left cells, determine the values of Lusztig's a-function, identify the characters of left cell representations. The aim is to show how type E-8 (the largest group of exceptional type) can be handled systematically and efficiently, too. This allows us, for the first time, to solve some open questions in this case, including Kottwitz' conjecture on left cells and involutions.