[Paper Review] Separating Vector Bundle Sections by Invariant Means

Gestur Ólafsson, Joseph A. Wolf|arXiv (Cornell University)|Oct 19, 2012

Advanced Algebra and Geometry1 references3 citations

TL;DR

This paper establishes that bounded right uniformly continuous sections of homogeneous Hilbert bundle over $G/H$ can be separated by invariant means when $H$ is amenable, proving that principal series representation spaces are completions of these sections rather than quotient spaces. The key contribution is a topological separation result using means on $G/H$, ensuring non-degenerate norms via invariant means, which simplifies the construction of induced representations in infinite-dimensional Lie groups.

ABSTRACT

We sharpen the construction of representation space in the paper "Principal Series Representations of Infinite Dimensional Lie Groups II: Construction of Induced Representations". We show that the principal series representation spaces constructed there, are completions of spaces of sections of Hilbert bundles rather than completions of quotient spaces of sections.

Motivation & Objective

• To refine the construction of representation spaces in infinite-dimensional Lie group representations, particularly for principal series representations.
• To show that representation spaces are completions of bounded right uniformly continuous sections of Hilbert bundles, not quotient spaces.
• To eliminate the need for quotient space constructions in the representation theory of infinite-dimensional Lie groups.
• To establish that invariant means on $G/H$ can separate non-zero sections, ensuring faithfulness of the norm structure.
• To simplify the proof of unitary equivalence in flag-closed parabolic induction by avoiding quotient comparisons.

Proposed method

• Use of bounded right uniformly continuous sections $f \in RUC_b(G/H; \mathbb{E}_\tau)$ valued in a Hilbert space bundle associated to a unitary representation $\tau$ of $H$.
• Definition of seminorms $\nu_\mu(f) = \mu(\|f\|)$ using means $\mu$ on $G/H$, leveraging amenability of $H$.
• Application of the fixed point property of amenable groups to construct a mean $\mu_f$ on $G/H$ such that $\nu_{\mu_f}(f) \neq 0$ for non-zero $f$
• Use of evaluation functionals $\delta_x$ and their weak-* closure to define a compact convex set $S$ of means with $\sigma(\|f\|) = 1$
• Construction of a Fréchet space completion of $RUC_b(G/P; \mathbb{E}_\tau)$ using a countable family of rational point evaluations $\delta_{xP}$ with $x \in G_\mathbb{Q}$
• Definition of a Hilbert space structure via a weighted sum $\langle f,h \rangle = \sum_{m \geq 1} 2^{-m} \langle f(x_m), h(x_m) \rangle$ using an enumeration of $G_\mathbb{Q}$

Experimental results

Research questions

• RQ1 Can bounded right uniformly continuous sections of a homogeneous Hilbert bundle over $G/H$ be separated by means on $G/H$ when $H$ is amenable?
• RQ2 Is the representation space of a principal series representation of an infinite-dimensional Lie group isomorphic to the completion of $RUC_b(G/P; \mathbb{E}_\tau)$ rather than a quotient space?
• RQ3 Does the use of invariant means on $G/H$ yield a non-degenerate seminorm on $RUC_b(G/H; \mathbb{E}_\tau)$ for non-zero sections?
• RQ4 Can a Fréchet or Hilbert space completion of $RUC_b(G/P; \mathbb{E}_\tau)$ be constructed using only rational points in $G_\mathbb{Q}$?
• RQ5 Does the group action on $RUC_b(G/P; \mathbb{E}_\tau)$ extend continuously to the Hilbert or Fréchet completions defined via rational points?

Key findings

• For any non-zero $f \in RUC_b(G/H; \mathbb{E}_\tau)$, there exists a mean $\mu \in \mathcal{M}(G/H)$ such that $\nu_\mu(f) \neq 0$, proving separation by invariant means.
• The representation space $\Gamma_{\mathcal{M}}(G/H; \mathbb{E}_\tau)$ is the locally convex topological vector space completion of $RUC_b(G/H; \mathbb{E}_\tau)$, not a quotient space.
• The principal series representations $\mathrm{Ind}_P^G(\tau)$ on $\Gamma_{\mathcal{M}}(G/P; \mathbb{E}_\tau)$ are constructed directly from sections, avoiding quotient constructions.
• A Fréchet space completion of $RUC_b(G/P; \mathbb{E}_\tau)$ is achieved using the countable family of evaluation means $\delta_{xP}$ for $x \in G_\mathbb{Q}$, with the topology defined by these seminorms.
• A pre-Hilbert space structure is defined on $RUC_b(G/P; \mathbb{E}_\tau)$ via $\langle f,h \rangle = \sum_{m \geq 1} 2^{-m} \langle f(x_m), h(x_m) \rangle$, yielding a Hilbert space completion.
• The action of $G$ on $RUC_b(G/P; \mathbb{E}_\tau)$ does not extend continuously to the Hilbert space completion defined via the rational point means.