Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

Abstract

Abstract Let G be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field F of characteristic p ≥ 0 {p\geq 0} , and let u ∈ G {u\in G} be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation of G. Then the Jordan normal form of ϕ ⁢ ( u ) {\phi(u)} contains at most one non-trivial block if and only if G is of type G 2 {G_{2}} , u is a regular unipotent element and dim ⁡ ϕ ≤ 7 {\dim\phi\leq 7} . Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].