On universal subspaces for Lie groups

TL;DR

This paper investigates the concept of universal subspaces in Lie group representations, revealing that while certain topological obstructions are sufficient for universality, they are not always necessary, especially in specific group and representation contexts.

Contribution

The paper demonstrates that the topological obstruction condition is not necessary for universality in general, and provides results for solvable groups and Levi subgroups.

Findings

Non-vanishing obstruction class is not necessary for universality in general.

For solvable complex Lie groups, the only universal subspace is the entire space.

Universality for Levi subgroups depends on specific group and representation properties.

Abstract

Let $U$ be a finite dimentional vector space over $\mathbb R$ or $\mathbb C$, and let $\rho:G\to GL(U)$ be a representation of a connected Lie group $G$. A linear subspace $V\subset U$ is called universal if every orbit of $G$ meets $V$. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Dokovi\'{c} approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of $V$, and asked whether it is also necessary under certain conditions. In this article, we show that the answer to the question is negative in general, but we discuss some important situations where the answer is positive. We show that if $G$ is solvable and $\rho:G\to GL(U)$ is a complex representation, then the only universal complex subspace is $U$ itself. We also investigate the question of universality for a Levi subgroup.