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

Let Gbe a classical group with natural module Vand Lie algebra gover an algebraically closed field K of good characteristic. For rational irreducible representations f: G -> GL(W) occurring as composition factors of V circle times V*, Lambda(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 eon V circle times V*, Lambda(2)(V), and S-2(V), for which recursive formulae are known. Our results are in analogue to earlier work [Proc. Amer. Math. Soc. 147 (2019) 4205-4219], where we considered these same representations and described the Jordan normal form of f(u) for every unipotent element u. G. (C) 2021 Elsevier B.V. All rights reserved.