On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilisers

TL;DR

This paper investigates the structure of minimal modules for exceptional Lie algebras over fields of positive characteristic, focusing on Jordan blocks and stabilizers to understand their algebraic and geometric properties.

Contribution

It provides a detailed classification of Jordan blocks for nilpotent elements and determines conditions for smoothness of centralizers and stabilizers in minimal modules.

Findings

Classification of Jordan blocks for nilpotent orbits

Criteria for smoothness of centralizers and stabilizers

Insights into the representation theory of exceptional Lie algebras

Abstract

Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \g. We also establish when the centralisers G_v of vectors v\in V_min and stabilisers \Stab_G<v> of 1-spaces <v>\subset V_min are smooth; that is, when \dim G_v=\dim\g_v or \dim \Stab_G<v>=\dim\Stab_\g<v>.