The closure of a sheet is not always a union of sheets

TL;DR

This paper demonstrates that in the context of algebraic group actions, specifically for semisimple groups acting on their Lie algebras, the closure of a G-sheet may not always be a union of sheets, countering common assumptions.

Contribution

It provides explicit examples showing that the closure of a G-sheet can fail to be a union of sheets in the adjoint action setting.

Findings

Counterexamples of sheets with non-union closures

Clarification of the behavior of G-sheets under closure

Insight into the structure of algebraic group actions

Abstract

In this note we answer to a frequently asked question. If G is an algebraic group acting on a variety V, a G-sheet of V is an irreducible component of V^(m), the set of elements of V whose G-orbit has dimension m. We focus on the case of the adjoint action of a semisimple group on its Lie algebra. We give two families of examples of sheets whose closure is not a union of sheets in this setting.