Linear maps preserving orbits

TL;DR

This paper classifies certain orbits in complex vector spaces under group actions, focusing on those that are not characteristic or semi-characteristic, extending previous work on orbit structure in reductive group representations.

Contribution

It provides a classification of non-(semi)-characteristic orbits for connected complex reductive groups acting on vector spaces, generalizing Ra"is's concept.

Findings

Classification of non-characteristic orbits in many cases

Extension of Ra"is's orbit classification framework

Insights into orbit structure for reductive group actions

Abstract

Let H\subset\GL(V) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let v\in V and let G=\{g\in\GL(V)\mid gHv = Hv\}. Following Ra\"is we say that the orbit Hv is \emph{characteristic for H} if the identity component of G is H. If H is semisimple, we say that Hv is \emph{semi-characteristic} for H if the identity component of G is an extension of H by a torus. We classify the H-orbits which are not (semi)-characteristic in many cases.