A peculiar property of exceptional Weyl groups (Q1819189)

Let \(W\) be an irreducible Weyl group and \(P\) be a parabolic subgroup of \(W\). Consider the irreducible components of \(\text{Ind}^W_P\mathbf{1}_P\), the induced representation of the trivial representation \(\mathbf{1}_P\) of \(P\) to \(W\). It is known that the trivial representation \(\mathbf{1}_W\) and the reflection representation \(\Lambda\) always occur as irreducible components of \(\text{Ind}^W_P\mathbf{1}_P\) for any proper parabolic subgroup \(P\) of \(W\), and that these are the only two that appear for \(W\) of classical types [see \textit{R. Donagi}, Astérisque 218, 145-175 (1993; Zbl 0820.14031)]. In the paper under review, the author shows that when \(W\) is of exceptional type, there is at least one irreducible representation \(\rho\) of \(W\), besides the trivial and reflection representation, such that \(\rho\) is a component of \(\text{Ind}^W_P\mathbf{1}_P\) for all proper parabolic subgroups \(P\) of \(W\). More precisely, when \(W\) is of exceptional type other than \(E_8\), such a \(\rho\) is unique, which is the nontrivial irreducible representation in the symmetric square of \(\Lambda\); when \(W\) is of type \(E_8\), \(\rho\) can be one of the three, which are in the symmetric square, symmetric cube and the symmetric quartic of \(\Lambda\), respectively.