[Paper Review] Topological and homological properties of the orbit space of a compact linear Lie group with commutative connected component

O. G. Styrt|arXiv (Cornell University)|Jul 23, 2016

Advanced Algebra and Geometry1 references3 citations

TL;DR

This paper investigates whether the orbit space $V/G$ of a compact linear Lie group $G$ acting on a Euclidean space $V$ is a topological or homological manifold, focusing on the case where $G^0$ (the connected component of identity) is commutative (i.e., a torus). Using representation theory, weight systems, and the structure of the adjoint action, the authors establish that $V/G$ is a homological manifold if and only if $G$ is generated by elements in $\Omega$ and satisfies specific conditions on the adjoint representation and root system. The key result is a complete characterization of such groups for which the orbit space is a homological manifold.

ABSTRACT

The problem in question is whether the quotient space of a compact linear group is a topological manifold and whether it is a homological manifold. In the paper, the case of an infinite group with commutative connected component is considered.

Motivation & Objective

• To determine whether the orbit space $V/G$ of a compact linear Lie group $G$ acting on a Euclidean space $V$ is a topological or homological manifold.
• To analyze the case where the connected component $G^0$ is commutative (i.e., a torus), which simplifies the representation theory and allows for a classification of such actions.
• To characterize the group $G$ such that $V/G$ is a homological manifold, using algebraic and topological invariants like the adjoint representation and weight systems.
• To establish necessary and sufficient conditions on $G$ and its action for $V/G$ to be a homological manifold, particularly when $G$ is infinite and $G^0$ is a torus.
• To resolve the structure of stabilizers and the role of elements in $\Omega$ and $\Omega'$, especially in relation to the adjoint action and the existence of complex reflections.

Proposed method

• The authors use the fact that $G^0$ is a torus to decompose the representation $G \curvearrowright V$ into irreducible components, each of dimension 1 or 2, and associate weights in $\mathfrak{g}^*$ to these components.
• They define the set $\Omega = \{g \in G : \omega(g) \in \{0,2\}\}$, where $\omega(g) = \operatorname{rk}(E - g) - \operatorname{rk}(E - \operatorname{Ad}(g))$, to identify elements with specific fixed-point behavior.
• The set $\Omega'$ is introduced to capture elements with $\omega(g) = 4$ and $\omega(g^5) = 0$, which helps in analyzing the structure of the group and its action.
• The adjoint representation $\operatorname{Ad}(G) \subset \mathbf{O}(\mathfrak{g})$ is used to analyze the group's structure, particularly showing that $\operatorname{Ad}(G) = \{\pm E\}$ under certain conditions.
• The paper applies results from representation theory and the theory of reflection groups, especially the classification of finite groups generated by pseudoreflections and the role of the Poincaré group.
• The proof relies on analyzing stabilizers $G_v$, their commutator subgroups, and the condition $G_v = \langle G_v \cap \Omega \rangle$, which is crucial for the homological manifold property.

Experimental results

Research questions

• RQ1 Under what conditions is the orbit space $V/G$ of a compact linear Lie group $G$ acting on a Euclidean space $V$ a homological manifold?
• RQ2 When $G^0$ is a torus, what structural properties must $G$ satisfy for $V/G$ to be a homological manifold?
• RQ3 How does the adjoint representation $\operatorname{Ad}(G)$ constrain the possible group structures of $G$ in this setting?
• RQ4 What is the role of the sets $\Omega$ and $\Omega'$ in determining whether $V/G$ is a homological manifold?
• RQ5 Can the group $G$ be generated by elements in $\Omega$ when $V/G$ is a homological manifold, and what does this imply about the representation?

Key findings

• If $G$ is an infinite compact linear Lie group with commutative connected component $G^0$, then $V/G$ is a homological manifold if and only if $G = \langle \Omega \rangle$, $\dim_{\mathbb{C}} V = \|P\| = 3$, and $\operatorname{Ad}(G) = \{\pm E\}$.
• The orbit space $V/G$ is a homological manifold precisely when the representation $G \curvearrowright V$ is reducible and $G$ is generated by elements in $\Omega$.
• The condition $\operatorname{Ad}(G) = \{\pm E\}$ is necessary and sufficient for the orbit space to be a homological manifold when $m = 1$ and no complex reflections are present.
• If $G$ is irreducible and contains no complex reflections, then $V/G$ cannot be a homological manifold, leading to a contradiction unless the representation is reducible.
• The stabilizer $G_v$ of any point $v \in V$ with finite isotropy satisfies $G_v = \langle G_v \cap \Omega \rangle$ if $V/G$ is a homological manifold.
• The group $G$ is generated by $G^0$ and $\Omega$, and since $\operatorname{Ad}(g) = -E$ for some $g \in \Omega$, it follows that $G = \langle \Omega \rangle$.