Levi decomposition of nilpotent centralisers in classical groups

TL;DR

This paper verifies that connected centralisers of nilpotent elements in orthogonal and symplectic groups possess Levi decompositions in even characteristic, supporting classification of reductive quotients.

Contribution

It confirms the Levi decomposition property for centralisers in classical groups in even characteristic, aligning with existing classification frameworks.

Findings

Connected centralisers have Levi decompositions in even characteristic.

Supports classification of reductive quotients in classical groups.

Provides theoretical validation for prior isomorphism class identifications.

Abstract

We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].