Linear maps preserving orbits
Gerald W. Schwarz

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.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models
