Unipotent elements forcing irreducibility in linear algebraic groups

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p>0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if $u$ is a regular unipotent element, then $X$ cannot be contained in a proper parabolic subgroup of $G$. We generalize their result and show that if $u$ has order $p$, then except for two known examples which occur in the case $(G,p)=(C_2,2)$, the subgroup $X$ cannot be contained in a proper parabolic subgroup of $G$. In the case where $u$ has order $>p$, we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

33 pages

References in corpus (2)

• Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
• Invariant forms on irreducible modules of simple algebraic groups

Cited by in corpus (3)

• Hesselink normal forms of unipotent elements in some representations of classical groups in characteristic two
• Adjoint Jordan blocks for simple algebraic groups of type $C_{\ell }$ in characteristic two
• Representatives for unipotent classes and nilpotent orbits