Let G be a classical group with natural module V and Lie algebra $\mathfrak{g}$ over an algebraically closed field K of good characteristic. For rational irreducible representations $f:G\to \mathrm{GL}(W)$ occurring as composition factors of $V\otimes V^{⁎}$, ${\wedge }^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d}f(e)$ for all nilpotent elements $e\in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of e on $V\otimes V^{⁎}$, ${\wedge }^2(V)$, 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 $u\in G$.

令G为具有自然模V和李代数的古典群$\mathfrak{G}$在具有良好特征的代数封闭场K上。对于理性不可约表示$F：G\to \mathrm{GL}（\mathrm{w\; ^}）$ 作为...的构成因素出现 $V\otimes V^{⁎}$， ${\wedge }^2（V）$和 ${\mathrm{小号}}^2（V）$，我们描述的约旦范式 $\mathrm{d}F（Ë）$ 对于所有幂等元素 $Ë\in \mathfrak{G}$。根据e的作用的Jordan块大小给出描述。$V\otimes V^{⁎}$， ${\wedge }^2（V）$和 ${\mathrm{小号}}^2（V）$，其递归公式是已知的。我们的结果与早期工作（Proc。Amer。Math。Soc。，2019）类似，在该研究中，我们考虑了这些相同的表示形式，并描述了约旦的正态形式。$F（ü）$ 对于每个单能元素 $ü\in G$。