A Peculiar Property of Exceptional Weyl Groups

Let W be a Weyl group and P W, a parabolic subgroup. In this paper, we give the decomposition of the permutation representation Ind_P^W 1 into irreducibles for each exceptional W and maximal parabolic P. We find that there is an 'extra' common irreducible component which appears for exceptional groups and not for classical groups. This work is motivated by the study of Prym varieties and integrable systems.