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 $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f:G\to \mathrm{GL}(W)$ occurring as composition factors of $V\otimes V^{\ast }$, ${\wedge }^2(V)$, 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 $V\otimes V^{\ast }$, ${\wedge }^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u\in G$.

to appear in J. Pure Appl. Algebra

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Cited by in corpus (1)

• Adjoint Jordan blocks for simple algebraic groups of type $C_{\ell }$ in characteristic two