On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilisers

Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \g. We also establish when the centralisers G_v of vectors v\in V_min and stabilisers \Stab_G<v> of 1-spaces <v>\subset V_min are smooth; that is, when \dim G_v=\dim\g_v or \dim \Stab_G<v>=\dim\Stab_\g<v>.