Jordan blocks of nilpotent elements in some irreducible representations of classical groups in good characteristic

Abstract

Let G be a classical group with natural module V and Lie algebra g over an algebraically closed field K of good characteristic. For rational irreducible representations f : G → GL(W ) occurring as composition factors of V ⊗ V * , ∧ 2 (V ), and S 2 (V ), we describe the Jordan normal form of df (e) for all nilpotent elements e ∈ g. The description is given in terms of the Jordan block sizes of the action of e on V ⊗ V * , ∧ 2 (V ), and S 2 (V ), for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 2019), where we considered these same representations and described the Jordan normal form of f (u) for every unipotent element u ∈ G.