Finite orbit modules for parabolic subgroups of exceptional groups.
Let \(G\) be a reductive linear algebraic group defined over an algebraically closed field, \(P\) a parabolic subgroup of \(G\) and \(P_u\) be its unipotent radical. Considering the adjoint action of \(P\) on the Lie algebra \(L_u\) of \(P_u\), each term \(L^{(n)}_u\) of the descending central series of \(L_u\) is stable under this action. All instances when \(P\) acts on \(L^{(n)}_u\) for \(n\geq 0\) with a finite number of orbits were determined for classical \(G\) and for \(G\) of exceptional type. The authors extend these results to groups of type \(F_4\) and \(E_6\); for each parabolic subgroup \(P\) of \(G\) of these types, \(n\geq 1\), they determine whether \(P\) acts on \(L^{(n)}_u\) with a finite number of orbits or, otherwise, they determine whether \(P\) still acts with a dense orbit. For groups \(G\) of type \(E_7\) or \(E_8\) they investigate the case of a Borel subgroup.
scientific article; zbMATH DE number 2124638