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

Let G be a simple simply 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 center dot 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 centralizers G(v) of vectors v is an element of V-min and stabilizers Stab(G)< v > of 1-spaces < v > subset of V-min are smooth; that is, when dim G(v) = dim g(v) or dim Stab(G)< v > = dim Stab(g)< v >.