Adjoint Jordan Blocks

TL;DR

This paper explores the relationship between nilpotent elements and unipotent classes in algebraic groups over fields of positive characteristic, establishing partition equivalences under certain conditions and providing new proofs for related algebraic structures.

Contribution

It proves that the partitions of adjoint actions of nilpotent and unipotent elements are the same when the unipotent element has order p, extending known results to classical and G_2 types without order restrictions.

Findings

Partitions of ad(X) and Ad(u) are identical when u has order p.

New elementary proof for the independence of structure constants in formal group law representation rings.

Established bijection between nilpotent orbits and unipotent classes in good characteristic.

Abstract

Let G be a quasisimple algebraic group over an algebraically closed field of characteristic p>0. We suppose that p is very good for G; since p is good, there is a bijection between the nilpotent orbits in the Lie algebra and the unipotent classes in G. If the nilpotent X in Lie(G) and the unipotent u in G correspond under this bijection, and if u has order p, we show that the partitions of ad(X) and Ad(u) are the same. When G is classical or of type G_2, we prove this result with no assumption on the order of u. In the cases where u has order p, the result is achieved through an application of results of Seitz concerning good A_1 subgroups of G. For classical groups, the techniques are more elementary, and they lead also to a new proof of the following result of Fossum: the structure constants of the representation ring of a 1-dimensional formal group law F are independent of F.