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

Let be a simple simply connected exceptional algebraic group of type G₂, F₄, E₆ or E₇ 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 on the minimal induced module V_min of g. We also establish when the centralizers G_ν of vectors ν ∈ V_min and stabilizers Stab_G〈ν〉 of 1-spaces 〈ν〉 are ⊂ V_min smooth; that is, when dim G_ν = dim g_ν or dim Stab_G〈ν〉 = dim Stab_g〈ν〉.