Admissible nilpotent orbits of real and 𝑝-adic split exceptional groups

Abstract

We determine the admissible nilpotent coadjoint orbits of real and p p -adic split exceptional groups of types G 2 G_2 , F 4 F_4 , E 6 E_6 and E 7 E_7 . We find that all Lusztig-Spaltenstein special orbits are admissible. Moreover, there exist non-special admissible orbits, corresponding to “completely odd” orbits in Lusztig’s special pieces. In addition, we determine the number of, and representatives for, the non-even nilpotent p p -adic rational orbits of G 2 G_2 , F 4 F_4 and E 6 E_6 .