Enhanced orbit embedding

TL;DR

This paper investigates conditions under which orbit space maps are injective in the context of algebraic group actions, focusing on symmetric subgroups and enhanced Lie algebra representations, providing new examples and analyzing obstructions.

Contribution

It establishes sufficient conditions for injectivity of orbit space maps for symmetric subgroups and constructs new examples of enhanced Lie algebras and representations.

Findings

Provided criteria for orbit space map injectivity.

Constructed new examples of enhanced Lie algebras.

Analyzed obstructions to orbit space inclusion.

Abstract

Let $ \tilde{G} $ be an algebraic group acting on a variety $ \tilde{L} $, and $ G \subset \tilde{G} $ a subgroup which leaves a subvariety $ L \subset \tilde{L} $ stable. For a $ G $-orbit $ O_G = G u (u \in L) $ in $ L $, we can associate an orbit $ O_{\tilde{G}} = \tilde{G} u $ of $ \tilde{G} $ so that we get a map $ L/G \to \tilde{L}/\tilde{G} $ between orbit spaces, though this map is usually not injective. In this note, when $ G $ is a symmetric subgroup arising from an involutive anti-automorphism, we give certain sufficient conditions for the map $ L/G \to \tilde{L}/\tilde{G} $ to be injective after the method of Ohta (2008). Our main concern here is to produce examples of enhanced Lie algebras (or enhanced $ \theta $-representations). We also analyze an obstruction which prevents the orbit space inclusion.