On distinguished orbits of reductive representations

TL;DR

This paper introduces the concept of 'nice spaces' in real reductive representations to identify distinguished orbits containing critical points, providing new criteria and applications in geometric and algebraic contexts.

Contribution

It defines 'nice spaces' for real reductive representations, offers a new criterion for distinguished orbits, and generalizes Nikolayevsky's basis criterion with applications to forms and nilmanifolds.

Findings

Elementary proof of Atiyah-Guillemin-Sternberg convexity theorem in this context

Sufficient condition for orbit being distinguished in nice spaces

Characterizations of nice spaces via weights of the representation

Abstract

Let $G$ be a real reductive Lie group and ${\tau}:G \longrightarrow GL(V)$ be a real reductive representation of $G$ with (restricted) moment map $m_{\ggo}: V-{0} \longrightarrow \ggo$. In this work, we introduce the notion of "nice space" of a real reductive representation to study the problem of how to determine if a $G$-orbit is "distinguished" (i.e. it contains a critical point of the norm squared of $m_{\ggo}$). We give an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in our particular case and we use it to give an easy-to-check sufficient condition for a $G$-orbit of a element in a nice space to be distinguished. In the case where $G$ is algebraic and $\tau$ is a rational representation, the above condition is also necessary (making heavy use of recent results of M. Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterium. We also provide useful characterizations of nice spaces in terms of the weights of $\tau$. Finally, some applications to ternary forms and minimal metrics on nilmanifolds are presented.