Vinberg's \theta-groups in positive characteristic and Kostant-Weierstrass slices

TL;DR

This paper extends Vinberg's heta-group theory to positive characteristic fields, establishing invariants as polynomial rings and confirming the existence of Kostant-Weierstrass slices in classical cases.

Contribution

It generalizes Vinberg's heta-group results to positive characteristic and proves the existence of KW-sections for classical graded Lie algebras.

Findings

Ring of invariants is polynomial

Existence of KW-sections confirmed in classical cases

Relationship between little Weyl group and Weyl group clarified

Abstract

We generalize the basic results of Vinberg's \theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or good characteristic), confirming a conjecture of Popov in this case.