Weyl groups for non-classical restricted Lie algebras and the Chevalley restriction theorem

TL;DR

This paper extends the concept of Weyl groups to non-classical restricted Lie algebras of Cartan type, computing the associated groups and exploring their implications for structure and invariants.

Contribution

It provides explicit computations of the Weyl group S(g) for Cartan type Lie algebras and explores applications to weight decompositions and polynomial invariants.

Findings

Computed S(g) for Cartan type Lie algebras.

Established links between S(g) and weight space decompositions.

Identified conditions for the existence of generic tori.

Abstract

Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S(g) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.