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In this paper we extend the computations in parts I and II of this series ofpapers and complete the proof of a conjecture of Lehrer and Solomon expressingthe character of a finite Coxeter group W acting on the pth graded component ofits Orlik-Solomon algebra as a sum of characters induced from linear charactersof centralizers of elements of W for groups of rank seven and eight. Forclassical Coxeter groups, these characters are given using a formula that isexpected to hold in all ranks.

Computations for Coxeter arrangements and Solomon's descent algebra III: Groups of rank seven and eight