Sheets, slice induction and G2(2) case

TL;DR

This paper investigates the structure of sheets in symmetric Lie algebras using Slodowy slices, introduces a new notion of slice induction, and provides detailed analysis and desingularization of G2-type sheets.

Contribution

It introduces the concept of slice induction for nilpotent orbits and applies it to analyze sheets in G2 symmetric Lie algebras, including their singularities and desingularization.

Findings

Slice induction coincides with parabolic induction in Lie algebras.

Characterization of singular loci in G2 symmetric Lie algebra sheets.

A desingularization of G2 sheets in so7.

Abstract

In this paper, we study sheets of symmetric Lie algebras through their Slodowy slices. In particular, we introduce a notion of slice induction of nilpotent orbits which coincides with the parabolic induction in the Lie algebra case. We also study in more details the sheets of the non-trivial symmetric Lie algebra of type G2. We characterize their singular loci and provide a nice desingularisation lying in so7.