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

TL;DR

This paper determines the Jordan normal form of nilpotent elements in certain irreducible representations of classical groups over fields of good characteristic, extending previous work on unipotent elements.

Contribution

It provides explicit descriptions of Jordan blocks for nilpotent elements in specific irreducible representations, using recursive formulas related to the action on tensor, wedge, and symmetric powers.

Findings

Explicit Jordan block sizes for nilpotent elements in the representations.

Extension of previous unipotent element results to nilpotent elements.

Recursive formulas for Jordan block sizes are utilized.

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 $f: G \rightarrow \operatorname{GL}(W)$ 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., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u \in G$.